text:
\section{Introduction}

Generation of vortex dipoles in dilute Bose-Einstein condensates (BECs) when a
superfluid moves past an obstacle with a velocity greater than a critical
velocity (speed of sound locally) has been established by various numerical
studies \cite{Frisch}-\cite{Raman}. It has been established, both
theoretically and numerically, that even the flow of the BEC below bulk sound
speed can result in the flow velocity greater than the local sound speed at
the equator of the obstacle. It implies that the critical velocity for the
creation of vortex-antivortex pairs is lower than the bulk sound speed. Above
this critical speed vortex-antivortex pairs are created periodically,
resulting in an oscillatory drag force. This is due to the phase difference
between the main flow and stationary wake behind the obstacle. Whenever this
phase difference grows to $2\pi$ a vortex-antivortex pair is created.
Recently, Neely et al. \cite{Neely} observed the generation of vortex dipoles
in a pancake-shaped $^{87}$Rb condensate, when the condensate was moved past
a Gaussian obstacle, created by blue-detuned laser beam, above critical
velocity. Real time dynamics of single vortex line and vortex dipoles has
also been studied experimentally \cite{Freilich}. The flow of the miscible
binary and spinor condensates across a Gaussian obstacle potential has also
been investigated theoretically in Refs.\cite{Susanto,Gladush,Rodrigues}.
Vortex dipoles or vortex rings are also produced as one of the the decay
products of dark solitons (a notch or density depression in the condensate
with a phase slip of $\pi$ across it), due to the onset of a transverse
instability in 2D and 3D geometries, namely snake instability. The decay of
dark soliton into vortex rings has already been observed experimentally
\cite{Anderson, Dutton}. In this context, it should be noted that motion of
the obstacle across a 1D BEC generates grey solitons
\cite{Hakim,Radouani,Carretero}. The snake instability is also responsible for
the decay of ring soliton into a necklace of vortex dipoles \cite{Theocharis}.

In the present work, we study the flow of phase-separated binary condensates
across a Gaussian obstacle numerically. We consider two experimentally
realizable binary condensates: the first of these has $^{85}$Rb and $^{87}$Rb
and the second has the two hyperfine states of $^{87}$Rb as the two
constituent species. In both these condensates, one of the scattering length
can be tuned by using magnetic Feshbach resonances \cite{Papp, Tojo}.
This allows us to choose the tunable scattering lengths values suitable for
generation of coreless vortex dipoles. We observe the generation of vortex
dipoles for the flow velocity greater than the critical velocity. In our
simulations while moving the obstacle, its strength is continuously decreased.
We consider two cases: (a) the obstacle laser potential, initially in the
outer component, is moved towards the center, and (b) the obstacle laser
potential, initially in the inner component, moves towards the interface of
the binary condensate. With the successful realization of vortex dipoles in
single component condensates using obstacle potential \cite{Neely}, the
present studies give more than enough reasons to extend these experimental
studies to binary condensates.

\section{Binary condensates and vortex dipoles}

\subsection{Meanfield description of binary condensates}

The dynamics of binary condensate at $T=0$K can be very well described by
a set of coupled GP equations
\begin{equation}
 \left[ \frac{-\hbar^2}{2m}\nabla^2 + V_i({\mathbf r},t) +
 \sum_{j=1}^2U_{ij}|\Psi_j({\mathbf r},t)|^2 -
 i\hbar\frac{\partial}{\partial t}\right]\Psi_i ({\mathbf r},t) = 0
 \label{eq.gp1}
\end{equation}
in mean field approximation, where $i = 1, 2$ is the species index. Here
$U_{ii} = 4\pi\hbar^2a_{ii}/m_i$, where $m_i$ is the mass and $a_{ii}$ is
the $s$-wave scattering length, is the intra-species interaction,
$U_{ij}=2\pi\hbar^2a_{ij}/m_{ij}$, where $m_{ij}=m_i m_j/(m_i+m_j)$ is the
reduced mass and $a_{ij}$ is the inter-species scattering length, is the
inter-species interaction, and $V_i({\mathbf r})$ is the trapping potential
experienced by $i$th species. In the present work, we consider binary
condensate consisting either of $^{85}$Rb and $^{87}$Rb for which
$m_1\approx m_2$. Furthermore, we also consider identical trapping potential
for both the species, the total potential is then $i$th species
\begin{equation}
 V({\mathbf r},t) = \frac{m_1\omega^2}{2}(x^2 + \alpha ^2 y^2 +
                      \beta ^2 z^2) + V_{\rm obs}(x,y,t),
\label{eq.pots}
\end{equation}
where $V_{\rm obs}(x,y,t) =
V_0 (t)\exp\lbrace -2([x-x_0(t)]^2+y^2)/w_0^2\rbrace$ is the potential
created by blue detuned laser beam, and $\alpha$ and $\beta$ are anisotropy
parameters. To rewrite the coupled GP equations in scaled units, define the
oscillator length of the trapping potential
\begin{equation}
   a_{\rm osc} = \sqrt{\frac{\hbar}{m\omega}},
\end{equation}
and consider $\hbar\omega$ as the unit of energy. We then divide the
Eq.(\ref{eq.gp1}) by $\hbar\omega$ and apply the transformations
\begin{equation}
\tilde{{\mathbf r}}   =  \frac{{\mathbf r}}{a_{\rm osc}},
~\tilde{t} =  t\omega, \text{and} ~
\phi_{i}(\tilde{{\mathbf r}},\tilde{t})=   \sqrt{\frac{a_{\rm osc}^3}{N_i}}
\Psi_i({\mathbf r},t).
\end{equation}
The transformed coupled GP equation in scaled units is
\begin{equation}
  \left[ -\frac{\tilde{\nabla}^2}{2} + V_i(\tilde{{\mathbf r}},\tilde{t}) +
  \sum_{j=1}^2\tilde{U}_{ij}|\phi_j(\tilde{{\mathbf r}},\tilde{t})|^2 -
   i\frac{\partial}{\partial \tilde{t}}\right]
  \phi_i (\tilde{{\mathbf r}},\tilde{t}) = 0,
\label{eq.gp2}
\end{equation}
where $\tilde{U}_{ii} = 4\pi a_{ii}N_i/a_{\rm osc}$,
$\tilde{U}_{ij} = 4\pi a_{ij}N_j/a_{\rm osc}$, and
$\tilde{\nabla}^2 = a_{\rm osc}^2\nabla^2$. From here on, we will represent the
scaled quantities without tilde unless stated otherwise.

   In pancake-shaped (highly oblate) traps, trapping frequency along axial
direction is much greater than the radial directions,
i.e., $\beta\gg\alpha\sim1$; in which case Eq.(\ref{eq.gp2}) can be reduced
to two dimensional form by substituting $\phi(\mathbf{r},t)=
\psi(x,y,t)\xi(z)\exp({-i\beta t/2})$ \cite{Muruganandam}. Here
$\xi=(\beta/(2\pi))^{1/4}\exp(-\beta z^2/4)$, the ground state wave function
in axial direction. The reduced two dimensional form of coupled GP equation in
pan-cake shaped traps is
\begin{eqnarray}
  \left[ -\frac{1}{2}\left(\frac{\partial^2}{\partial x^2} +
  \frac{\partial^2}{\partial y^2}\right) + \frac{x^2+\alpha_i^2y^2}{2} +
  U(x,y,t) + \right.\\
 \sum_{j=1}^2 u_{ij}|\psi_j({\mathbf r},t)|^2 \left. -
  i\frac{\partial}{\partial t}\right]
  \psi_i ({\mathbf r},t) = 0,\nonumber
\label{eq.gp3}
\end{eqnarray}
where $ u_{ii} = 2 a_{ii}N_i\sqrt{2\pi\beta_i}/a_{\rm osc}$ and
$ u_{ij} = 2 a_{ij}N_j\sqrt{2\pi\beta_i}/a_{osc}$. In the present work, we
consider $u_{12}>\sqrt{u_{11}u_{22}}$ so that the ground state of the binary
condensate is phase-separated. The ground states of the binary condensates
in the phase separated domain have been studied in Refs.\cite{Ho}-\cite{Gautam2}
In pancake-shaped traps the binary condensates has cylindrical interface
separating the two components \cite{Gautam1}.

\subsection{Vortex dipole trajectory in a single component condensate}
   The total velocity field experienced by each vortex in vortex dipole is the
vector sum of two component fields. One of these fields is the velocity field
generated on each vortex due to inhomogeneous nature of the condensate in
the trapping potential. This inhomogeneity generated velocity field is
responsible for the rotation of an off center vortex around the trap center
\cite{Jackson-2,Svidzinsky}. In addition to this, each vortex creates a
velocity field varying inversely with the distance from its center, which
is experienced by the other vortex of vortex dipole. If $(x,y)$ and $(x,-y)$
are the locations of the positively and negatively charged vortices of the
vortex dipole, respectively, then the total velocity field experienced by the
positively charged vortex of the vortex dipole
\begin{equation}
 \mathbf v(x,y) = \omega_{\rm pr}\hat k \times \mathbf r + \frac{1}{2y}\hat i,
\end{equation}
where $\omega_{\rm pr}$ is the rotational frequency of a vortex with charge
$+1$ in the condensate. In terms of component velocities, the previous equation
can be written as
\begin{eqnarray}
 \frac{dx}{dt} & = & -\omega_{\rm pr}y + \frac{1}{2y}, \nonumber\\
 \frac{dy}{dt} & = & \omega_{\rm pr}x.
 \label{vdipole_trejectory1}
\end{eqnarray}
This coupled set of differential equations describes the trajectory of the
positively charged vortex of the vortex dipole. In the present work, we move
the obstacle along $x$-axis, leading to the generation of vortex dipole located
symmetrically about $x$-axis. Since $y$ is small (and positive) for the
positively charged vortex of the vortex dipole at the instant of generation,
one can neglect $-\omega_{\rm pr}y$ in first equation of coupled
Eqs.\ref{vdipole_trejectory1} to obtain
\begin{eqnarray}
 \frac{dx}{dt} & \approx & \frac{1}{2y}, \nonumber\\
 \frac{dy}{dt} & = & \omega_{\rm pr}x.
 \label{vdipole_trejectory2}
\end{eqnarray}
Eqs.\ref{vdipole_trejectory2} imply that vortex initially to the left
of the origin moves towards origin with increasing speed along
positive $x$-axis but decreasing speed along negative $y$-axis.
At $x=0$, $dy/dt=0$ and beyond this vortex moves away from origin
with decreasing speed along $x$-axis but increasing speed along positive
$y$-axis. Near the origin $\omega_{\rm pr}x\ll1/2y$, then the velocity of
the vortex can be approximated as
\begin{equation}
 \mathbf v(x,y) \approx \frac{1}{2y}\hat i.
\end{equation}
In this case the velocity of the vortex is mainly induced one, and it moves
with induced velocity towards positive $x$ direction. On the other hand if
$y$ is too large, one can neglect $1/2y$ term in Eq.\ref{vdipole_trejectory1}
to obtain
\begin{equation}
 \mathbf v(x,y) \approx \omega_{\rm pr}\hat k \times \mathbf r.
\end{equation}
In this case, the velocity of the vortex is mainly due to the inhomogeneous
nature of the condensate, resulting in the rotation with frequency
$ \omega_{\rm pr}$. The transition between vortex induced velocity to
inhomogeneity induced velocity occurs when $dx/dt = 0$, i.e., velocity
along x-axis changes sign. From Eqs.\ref{vdipole_trejectory1}, it occurs
at $y = \sqrt{1/2\omega_{\rm pr}}$.

\section{Obstacle modified condensate density}

 In the phase separated domain of binary condensates, the species with lower
repulsion energy forms a core  and the other species forms a shell around it.
For convenience, we identify the former and later as the first and second
species, respectively.  With this labelling, interaction energies
$u_{11}<u_{22} $ and for equal populations this implies $a_{11}<a_{22}$. There
is large difference of the local coherence length $\xi_i=1/\sqrt{2n_iu_{ii}}$
when $u_{11}\ll u_{22}$. This results in different density modifications
for the two species when an obstacle beam is applied. The dynamics is most
sensitive to the coherence length difference while the obstacle is crossing
the interface layer of the binary condensate.

  To analyze the density perturbations in presence of the obstacle beam,
let $R_{\rm in}$ be the radius of the inner species or the interface boundary.
And, let $R_{\rm out}$ be the radial extent of the outer species. In the
absence of the obstacle beam, the chemical potential of first species is
$\mu_1=R_{\rm in}^2/4 + u_{11}/(\pi  R_{\rm in}^2)$. Here, the second term is
correction arising from the presence of the second species. The chemical
potential of the second species is $R_{\rm out}^2/2$. Both of these are in
scaled units. We consider the obstacle beam initially ($t=0$) located
at $(-R_{\rm out},0)$ and traversing towards the center with velocity
$v_{\rm obs}$. As it moves, its intensity is ramped down at the rate $\eta$.
The location of the beam at a later time is
\begin{equation}
   x_0(t) = -R_{\rm out} + v_{\rm ob}t,
\end{equation}
and  intensity of the beam is
\begin{equation}
  V_0(t) = V_0(0) -\eta t,
\end{equation}
where  $V_0(0)$ is the initial intensity of the obstacle beam. At the starting
point, the total potential $V(R_{\rm out},0,0) > R_{\rm out}^2/2$ and the
density of the outer species $|\psi_2|^2$ is zero around the center of the
obstacle beam as is shown in Fig. \ref{plot2}(a).
\begin{figure}[ht]
\includegraphics[width=8.5cm] {plot2.png}
\caption{The stationary state wave functions of $^{85}$Rb-$^{87}$Rb binary
condensate obtained from imaginary time propagation. For (a)
the obstacle is located near $R_{\rm out}$, for (b) the obstacle is located at
the interface, and for (c) it is well inside the bulk of the inner species.}
\label{plot2}
\end{figure}
However, as it traverses the
condensates with decreasing intensity, at some later time
$t'$, $V(x_0(t'),0,0) < R_{\rm out}^2/2$. Density $|\psi_2|^2$ is then finite
within the obstacle. For compact notations, hereafter we drop the explicit
notation of time dependence while writing $x_0(t)$ and  $V_0(t)$.

\subsection{Density modification at interface}

  As the obstacle beam approaches the interface, the intensity is weak to
expel out the outer species. However, when the beam enters the inner species
with any finite intensity $V(R_{\rm in}, 0,t) > \mu_1 $, it is sufficient to
make $|\psi_1|^2$ zero around the beam [see Fig.~\ref{plot2}(a)]. The interface
is modified and equipotential curve of the interface is
\begin{equation}
  \frac{x^2+y^2}{2} + V_0 \exp\left[ -2\frac{(x-x_0)^2+y^2}{w^2}\right]
      = \frac{R_{\rm in}^2}{2} .
\label{eq_pot_surface}
\end{equation}
Around the center of the obstacle beam, $\sqrt{(x-x_0)^2+y^2}<w/\sqrt{2}$,
and the obstacle potential is
\begin{equation}
  V_{\rm opt}\approx V_0-\frac{2V_0}{w^2}\left [ (x-x_0)^2+y^2\right].
 \label{app_obs_pot}
\end{equation}
Hence from Eq.(\ref{eq_pot_surface}) the equipotential curve in the
neighborhood of the obstacle beam is
\begin{equation}
 \left(x+\frac{C}{2A}\right)^2 + y^2 = \tilde{R}_{\rm in}^2,
\end{equation}
where
\begin{eqnarray}
    A  & = &\frac{1}{2}-\frac{2V_0}{w^2}, \nonumber \\
    C  & = & \frac{4V_0x_0}{w^2}, \nonumber  \\
    \tilde{R}_{\rm in}^2 & = & \frac{1}{A}\left (\frac{R_{\rm in}^2}{2}
       +\frac{C^2}{4A} + C\frac{x_0}{2} - V_0\right ).    \nonumber
\end{eqnarray}
Equipotential curve in the vicinity of the obstacle is thus modified into a
circle with center at ($-C/2A,0$) and radius $\tilde{R}_{\rm in}$. In our
numerical calculations, the Gaussian obstacle potential is quite narrow and
$w^2/(4V_0)\ll1$. Retaining only the leading order terms in $w^2/4V_0$,
the center is approximately located at $(x_0+w^2x_0/(4V_0), 0)$ and radius is
\begin{equation}
   \tilde{R}_{\rm in} \approx\sqrt{\frac{w^2}{2V_0}\left(1
   +\frac{w^2}{4V_0}\right)\left(V_0
              +\frac{x_0^2}{2}-\frac{R_{\rm in}^2}{2}\right)}.
\end{equation}
Sufficiently away from the obstacle $\sqrt{(x-x_0)^2+y^2}> w/\sqrt{2}$ and the
obstacle potential is negligible. The equipotential curve then remains
unchanged, $R_{\rm in}^2=x^2+y^2$. Thus an approximate piecewise expression of
the equipotential curve is
\begin{eqnarray}
  \left(x+\frac{C}{2A}\right)^2+y^2 &=& \tilde{R}_{\rm in}^2,
           \text{ if }  |R_{\rm in}-x_0|<\frac{w}{\sqrt{2}}, \nonumber \\
   x^2+y^2 &=& R_{\rm in}^2,  \text{ if } \sqrt{(x-x_0)^2 + y^2}
               >\frac{w}{\sqrt{2}}.  \nonumber
\end{eqnarray}
This curve defines the interface geometry of the inner species in the TF
approximation. Surface effects, however, smoothen the density profile around
the meeting point of the two regions as is shown in Fig. \ref{plot2}(b). The
obstacle beam is repulsive and expels the condensate around it such that bulk
density is regained over a distance of one healing length.

\subsection{Fully immersed obstacle beam}

 A critical requirement to form coreless vortices is complete immersion
of the obstacle beam within $n_1$. Based on the previous discussions, as the
beam approaches the origin, the last point of contact between the beam and
interface at $R_{\rm in}$ lies along $x$-axis. To determine the condition when
complete immersion occurs, consider the total potential along $x$-axis
\begin{eqnarray}
 V(x,0,t) &\approx & \frac{x^2}{2} + V_0(t) \left[1 -2\frac{(x-x_0(t))^2}{w^2}
                     \right.  \nonumber \\
           && \left .  + 4\frac{(x-x_0(t))^4} {w^4}\right],
\end{eqnarray}
where, the Gaussian beam potential is considered upto the second order term.
The expression is appropriate in the neighborhood of the beam and has one
local minima and maxima each. There is also a global minima, however, it is
not the correct solution as it lies in the domain where
$x>\frac{w}{\sqrt{2}}$. Correct global minima is the one
located at $x=0$ and associated with the harmonic potential. The obstacle is
considered well immersed when the local minima is located at the interfacial
radius $R_{\rm in}$. The local minima is a root of the polynomial
\begin{equation}
x^3+a_2 x^2 +a_1 x+ a_0 = 0,
\end{equation}
obtained from the condition $dV/dx=0$. Here,
\begin{eqnarray}
a_2 &=& -3 x_0, \nonumber\\
a_1 &=& -\frac{w^2}{2}+3 x_0^2+\frac{w^4}{8 V_0}, \nonumber\\
a_0 &=& \frac{1}{2} x_0 \left(w^2-2 x_0^2\right).    \nonumber
\end{eqnarray}
For the trapping potential parameters considered in the present work, all the
three roots are real. The root corresponding to local minima is
\begin{equation}
x_{\rm min}(t)  = -\frac{\left(s_++s_-\right)}{2}-\frac{a_2}{3}
               + i \frac{\sqrt{3}}{2} \left(s_+-s_-\right),
\end{equation}
where
\begin{eqnarray}
   s_{\pm}&=&\Bigg[-\frac{a_2^3}{27}+\frac{1}{6} \left(-3 a_0+
        a_1 a_2\right)\pm   \nonumber\\
   & &\left.\sqrt{\left(\frac{a_1}{3}-\frac{a_2^2}{9}\right)^3+
       \left(-\frac{a_2^3}{27}+\frac{1}{6} \left(-3 a_0+a_1 a_2\right)\right)
       ^2}\right]^{1/3}. \nonumber
\end{eqnarray}
and $x_{\rm min}$ is a function of time as the obstacle potential is time
dependent. However, for compact notations, like in $V_0$ and $x_0 $, hereafter
we drop the explicit notation of time dependence while writing
$x_{\rm min}(t)$. Value of $x_{\rm min}$ is real when it satisfies certain
conditions \cite{mathworld} and these are met for the experimentally realizable
parameters. The obstacle is then completely immersed when
$x_{\rm min}=R_{\rm in}$ and let $t_{\rm im}$ denote the time at which this
condition is satisfied. Once the obstacle beam is well inside the inner
species, within the beam $n_1$ is zero but $n_2$ is nonzero. It then forms a
second interface layer, assisted by the obstacle beam, which embeds a bubble
of the $n_2$ within $n_1$. Recollect, the first interface layer is located at
$R_{\rm in}$ and it is where $n_2$ encloses $n_1$. The second interface,
unlike the one at $R_{\rm in}$, is a deformed-ellipse and label it as
$\Gamma$. It touches $x$-axis at two points and $x_{\rm min}$ is the one
farther from the origin. From TF-approximation, the density of $n_2$ within
$\Gamma$ is
\begin{equation}
  n_{2\Gamma} (x, y, t_{\rm im}) = \frac{\mu_2 - V(x, y, t_{\rm im})}{u_{22}},
\end{equation}
where the subscript $\Gamma$ is to indicate the density is within $\Gamma$. At
$t_{\rm im}$, $n_{2\Gamma}$ is in equilibrium with the bulk of $n_2$, and let
the number of atoms in the be $N_{2\Gamma}$, which can be calculated from
$n_{2\Gamma}$.

  Except for deviations arising from mean field energy, the second interface
$\Gamma$ closely follows an equipotential surface.  From the conditions of
pressure balance at two interfaces
\begin{eqnarray}
 \mu_1-\sqrt{\frac{u_{11}}{u_{22}}}\mu_2 & = &
 \frac{R_{\rm in}^2}{2}\left(1-\sqrt{\frac{u_{11}}{u_{22}}}\right),\nonumber\\
 \mu_1-\sqrt{\frac{u_{11}}{u_{22}}}\mu_2 & = &
 V(x_{\rm min}, 0,t)\left(1-\sqrt{\frac{u_{11}}{u_{22}}}
 \right).
\label{pressure_balance}
\end{eqnarray}
The first equation is valid all along the interface at $R_{\rm in}$. However,
the second is derived at $(r_{\rm min},0)$, but to a good approximation
the condition is valid along $\Gamma$ as it is close to an equipotential
surface. The density of the first species at ($x_{\rm min}$,0) is
\begin{equation}
n_1(x_{\rm min})  = \frac{1}{\sqrt{u_{11}u_{22}}}\left[
         \frac{R_{\rm out}^2 - 2V(x_{\rm min}, 0,t)}{2} \right].
\end{equation}
Similarly density of second species at at ($x_{\rm min}$,0) is
\begin{equation}
 n_2(x_{\rm min})  =  \frac{1}{u_{22}}\left[\frac{R_{\rm out}^2
              - 2V(x_{\rm min}, 0,t)}{2}\right].
\end{equation}
Inside $\Gamma$, where the obstacle potential is dominant, $n_1$ decays
exponentially. Along the $x$-axis
\begin{equation}
  n_1(\delta x_-,0) =  n_1(x_{\rm min},0)\exp\left (-\frac{2}{\Lambda_1}
                       \delta x_-\right ),
\end{equation}
where, $\delta x_- $ is the distance of the point measured from
$\Gamma$. Similarly the density of the second species just outside
$\Gamma$ is
\begin{eqnarray}
n_2(\delta x_+,0) = n_2(x_{\rm min},0)\exp\left(-\frac{2}{\Lambda_2}
                    \delta x_+\right ),
\end{eqnarray}
where $\delta x_+$ is distance from $\Gamma$ but away from the obstacle
beam, and $\Lambda_2 $ is the penetration depth of the second species. In
these expressions the  penetration depth is
\begin{equation}
    \Lambda_i = \xi_i\left [ \frac{\sqrt{a_{11}a_{22}}}
          {a_{12}-\sqrt{a_{11}a_{22}}}\right ] ^{1/2}.
\end{equation}
Based on the densities at the interface region, we can incorporate the
interspecies interaction and calculate the effective potential of each species.

\subsection{Obstacle assisted bubble}

 At a time $\Delta t$ after the obstacle is immersed in $n_1$, the location
and amplitude of the obstacle potential are
\begin{eqnarray}
  x_0(t_{\rm im}+\Delta t) & = & -R_{\rm out} + v_{\rm ob}(t_{\rm im}
                               + \Delta t),  \\
  V_0(t_{\rm im}+\Delta t) & = & V_0(0) + v_{\rm ob}(t_{\rm im} + \Delta t).
\end{eqnarray}
Equilibrium TF $n_2$ within the obstacle potential at this instant of time is
\begin{equation}
  n_{2\Gamma} (x, y, t_{\rm im} + \Delta t ) = \frac{\mu_2
          - V(x, y, t_{\rm im} + \Delta t)}{u_{22}}.
 \label{n2_den}
\end{equation}
This, however, is higher than the density distribution at $t_{\rm im}$, that is
$n_{2\Gamma} (x, y, t_{\rm im} + \Delta t )> n_{2\Gamma} (x, y, t_{\rm im})$ as
the potential $ V $ is lower. This is on account of two factors: first,
the amplitude of the obstacle potential decreases with time; and second, the
harmonic oscillator potential is lower at $x_0(t_{\rm im}+\Delta t)$. The
number of atoms, however, does not change from the value at $t_{\rm im}$
unless there is a strong Josephson current. Density $n_{2\Gamma}$ is thus
below the equilibrium value once the obstacle beam is well within $n_1$.
This creates a stable bubble of $n_2$ assisted or trapped within the beam
and is transported through the $n_1$.

 Departure of $n_2$ from the equilibrium is not the only density evolution
within the beam. There is a progressive change of $n_{1\Gamma} $ as the beam
moves deeper into $n_1$. At time $t_{\rm im}$, when the obstacle is
completely immersed in $n_1$ the effective potential, experienced by $n_1$,
$V(x, y, t_{\rm im}) + n_{2\Gamma}u_{12}$ is larger than
$\mu_1$. So, $n_1$ is zero within the beam. However, if the rate of ramping
$\eta$ is such that at a later time
$V(x, y, t_{\rm im} + \Delta t) + n_{2\Gamma}u_{12}< \mu_1$, while the beam
is still within $n_1$, there is a finite $n_1$ within the beam.
\begin{equation}
 n_{1\Gamma} = \frac{\mu_1 - V(x, y, t_{\rm im} + \Delta t)-u_{12}
               n_{2\Gamma}}{u_{11}}
\end{equation}
As the growth of $n_{1\Gamma}$ is in its nascent stage, Eq.~(\ref{n2_den})is
still applicable. Since $a_{12}>\sqrt{a_{11}a_{22}}$ for the condensate, in
TF approximation both $n_{1\Gamma}$ and $n_{2\Gamma}$ can not be
simultaneously non-zero. At the same time, $n_{2\Gamma}$ is forbidden to
migrate to the bulk $n_2$ due to the $n_1$ generated potential barrier in the
region between interfaces $\Gamma$ and $R_{\rm in}$. To accommodate both
$n_1$ and $n_2$ within the beam, the shape of interface $\Gamma$ is
transformed to increase $n_{2\Gamma}$. So that $n_2$ is zero in certain regions
within the beam where the condition
$V(x, y, t_{\rm im} + \Delta t) + n_{2\Gamma}u_{12}<\mu_1$ is satisfied. This
mechanism is responsible for obstacle assisted transport of $n_2$ across $n_1$.

\subsection{Obstacle induced density instability }

 The harmonic potential increases once the obstacle crosses origin, but the
obstacle potential continues to decrease. A fine balance is achieved if
\begin{equation}
  \frac{dV_{\rm obs}}{dx} = \frac{\partial V_{\rm obs}}{\partial x} +
     \left ( \frac{dx}{dt}\right )^{-1}\frac{\partial V_{\rm obs}}{\partial t},
\end{equation}
the dynamical rate of change of obstacle potential, is same in magnitude and
opposite to the rate of change of the harmonic potential $dV_{\rm ho}/dx$.
It is, however, difficult to satisfy this condition across the entire obstacle
potential as the Gaussian beam has has different profile than the oscillator
trapping potential. But, the parameters may be tuned so that the condition is
satisfied at the center of the beam, then
\begin{equation}
  \frac{dV_{\rm ho}}{dx}  =  \frac{\eta}{v_{\rm obs}}.
\label{temp}
\end{equation}
In the present case $\eta$ is a constant and the
condition in Eq. (\ref{temp}) is satisfied only when the beam is at a
specific point $x_{\rm e}$, then
\begin{equation}
\left . \frac{dV_{\rm ho}}{dx}\right|_{x_{\rm e}}  =  \frac{\eta}{v_{\rm obs}}
\label{x_e}
\end{equation}
Before reaching $x_{\rm e}$, the potential within the beam decreases, but
increases after crossing it. At some point beyond $x_{\rm e}$, the total
potential experienced by $n_2$ bubble can become greater than $\mu_2$,
resulting in an unstable bubble.

\section{Stability of coreless vortex dipole}

  To analyze the energetic stability of coreless vortex dipoles in
phase-separated binary condensates, we adopt the following {\em ansatz} for
the wave function of inner and outer species
\begin{eqnarray}
  \psi_1(x,y) & = & a\left[i \left(x - v_1\right) + \left(y^2
                  - v_2^2\right)\right]\exp \left( - \frac{x^2
                  + y^2}{2}\right),\nonumber\\
\psi _2(x,y) & = & b\left(x^2 + y^2\right)\exp\left( - \frac{x^2
                  + y^2}{2}\right),
\label{ansatz}
\end{eqnarray}
where $(v_1,\pm v_2)$ is the location of the vortex dipole. With this choice
of {\em ansatz}, the vortex dipole in the inner species is a normal vortex
dipole. For simplicity of analysis, consider the trapping potential is
purely harmonic, i.e.
\begin{equation}
  V(x,y)  = \frac{x^2+y^2}{2}.
\end{equation}
Normalization yields the following constraints on $a$ and $b$
\begin{eqnarray}
  a & = & \left [ \frac{4}{5\pi+4\pi (v_1^2- v_2^2+ v_2^4)} \right ] ^{1/2},
                  \nonumber\\
  b & = &\sqrt{ \frac{1}{2\pi} },
\label{b}
\end{eqnarray}
and reduces the number of unknowns to two $v_1$ and $v_2$. The energy of the
binary condensate using the {\em ansatz} in Eq.~({\ref{ansatz}}) is
\begin{eqnarray}
E & = &\frac{N_2 b^2}{\pi} \left(4 +\frac{3}{8} b^2 u_{22}\right)+
      \frac {N_1a^2\pi}{1024} \left\lbrace 32 \left[8 (11\right.\right.
      \nonumber\\
  &   & \left.+4 v_1^2-4 v_2^2+4 v_2^4)+b^2 u_{12} \left(15+8 v_1^2-12 v_2^2
        +8 v_2^4\right)\right]\nonumber\\
  &   &  +a^2 u_{11} \left[177-304 v_2^2+256 v_1^4+416 v_2^4-256 v_2^6
        \right.\nonumber\\
  &   & \left.\left.+256 v_2^8+32 v_1^2 \left(15-8 v_2^2+16 v_2^4\right)
        \right]\right\rbrace.
\label{energy}
\end{eqnarray}
In the same way, we can also determine the chemical potentials of the two
species in terms of the same parameters.
The stationary points of the energy function are the solutions of the coupled
equations $ \partial E/\partial v_1 =  0$ and $\partial E/ \partial v_2= 0$.
These equations are solvable analytically, but the solutions are lengthy and
cumbersome. The solution of particular interest is the vortex dipole located
along the $y$-axis ($v_1=0$). This is consistent with the observation that,
regardless of the orientation, from the symmetry the axis passing through
vortex dipole can be considered as the $y$-axis. A rotation
transformation is sufficient to achieve this. For the case considered, the
energy extrema conditions simplify to equations

\begin{eqnarray}
\frac{\partial E}{\partial v_1}
     & = & \frac{\pi a N_1}{256} \bigg\{16 \left[8
           \left(11-4 v_2^2+4 v_2^4\right)+b^2 u_{12} \left(15-12 v_2^2
           \right.\right.\nonumber\\
     &   & \left.\left.+8 v_2^4\right)\right] \frac{\partial a}{\partial v_1}
           +  a{}^2 u_{11} \left[177-304 v_2^2+416 v_2^4\right.\nonumber\\
     &   & \left.-256 v_2^6+256 v_2^8\right]
           \frac{\partial a}{\partial v_1}\bigg\},\nonumber\\
\frac{\partial E}{\partial v_2} & = &\frac{\pi a N_1}{256}
           \bigg\{64 a v_2 \left[8 \left(2 v_2^2 - 1\right)+b^2 u_{12}
           \left(4 v_2^2 - 3\right)\right]\nonumber\\
     &   & -8   a {}^3 u_{11} v_2 \left[19-52 v_2^2+48 v_2^4-64 v_2^6
            \left(4 v_2^2-1\right)\right]\nonumber\\
     &   & +16 \left[8 \left(11
           -4 v_2^2+4 v_2^4\right)+b^2 u_{12} \left(15-12 v_2^2
            \right.\right.\nonumber\\
     &   & \left.\left.+8 v_2^4\right)\right]
            \frac{\partial a}{\partial v_2}+a{}^2 u_{11}\left[177-304 v_2^2
            +416 v_2^4\right. \nonumber\\
     &   & \left.-256 v_2^6+256 v_2^8\right]
           \frac{\partial a}{\partial v_2}\bigg\}.
\end{eqnarray}
For the sake of illustration, consider, $N_1 = 10^5$, $N_2 = N_1/2$,
$a_{11} = 0.99a_0$, $a_{22} = 4.6a_0$, $a_{12} = 2.14a_0$, $\beta = 11.25$,
$\omega/2\pi=8$, and $V_0(t) = 0$. Call this set of parameters as parameters
{\em set b}. The stationary points corresponding to minimum energy for this
parameters set are $(v_1,v_2) = (0,\pm 0.9678)$.

\subsection{Coreless vortex dipole}

To compare the energy of this normal vortex dipole with that coreless
vortex dipole, consider that the cores of the vortices at $(v_1,v_2)$ and
$(v_1,-v_2)$ are filled with outer species. We approximate the cores as
circular with a radius of $\xi = 1/\sqrt{2\mu_1}$, the
coherence length of the inner species. We approximate the density profiles
of the outer species within the cores with the TF distribution
\begin{equation}
 \psi_2(x,y) = \left \{
      \begin{aligned}
         & \sqrt{\frac{\mu_2 - V(x,y)}{u_{22}}} & r_c^2 \leqslant\xi^2 \\
         & b\left(x^2+y^2\right)\exp\left(-\frac{x^2+y^2}{2}\right)
                 &  r_c^2>\xi^2,
      \end{aligned} \right .
\end{equation}
where $r_c = \sqrt{(x-v_1)^2+(y \pm v_2)^2}$ is the radial distance from the
core of each vortex and antivortex. The above expression for $\psi_2$ is
renormalized to obtain the modified value of $b$, which in turn enables us
to calculate the energy for the coreless vortex dipole. To obtain the
analytic expression for energy, we integrate $b\left(x^2+y^2\right)\exp
\left(-\frac{x^2+y^2}{2}\right)$ over the whole real space without excluding
the core regions. The modified value of $b$ after the normalization is
\begin{equation}
b \approx \left [ \frac{2u_{22}- \pi  \xi ^2 \left(\xi ^2+2 v_1^2+
               2 v_2^2-4 \mu _2\right)}{4\pi u_{22}}\right ]^{1/2}.
\label{modified_b}
\end{equation}
With this approximation, the number of atoms of outer species embedded within
the cores is
\begin{equation}
   \delta N_2 = N_2 \frac{\pi  \xi ^2 \left(\xi ^2 + 2 v_1^2 + 2 v_2^2
                - 4 \mu _2\right)} {2 u_{22}}.
\end{equation}
These contribute to the total energy through the potential and intra-species
interaction energies. The energy contribution from the cores is  then
\begin{eqnarray}
\delta E & = & -\frac{\pi  \xi ^2}{12 u_{22}} \left[\xi ^4+3 v_1^4+6 \xi ^2
               v_2^2+3 v_2^4+6 v_1^2\left(\xi ^2+v_2^2\right)\right.
               \nonumber\\
         &    &\left.-12 \mu _2^2\right].
\end{eqnarray}
The total energy of the binary condensate with coreless vortex is
\begin{equation}
 E_c = E + \delta E,
\end{equation}
where $E$ is defined by Eqs.~(\ref{energy}) and (\ref{modified_b}).
For the parameter {\em set b}, the energy per particle [$E_c/(N_1+N_2)$] is
$4.18$ which is smaller than the value of $4.26$ obtained from
Eqs.~(\ref{b}) and (\ref{energy}). This implies
that coreless vortex dipole is energetically more favorable than normal
vortex dipole with the percentage energy difference of $1.9\%$.

 The Gaussian form of the {\em ansatz} imposes limitations on the domain of
applicability. But, the ansatz provide qualitative information on the
stability coreless vortex dipoles. To relate to experimental realizations,
consider $^{85}$Rb-$^{87}$Rb binary condensate with $a_{11} = 460a_0$,
$a_{22} = 99a_0$, and $a_{12} = 214a_0$ as the scattering length values
and $2 N_1 = N_2 = 10^6$ as the number of atoms. Here $a_{11}$ is tunable
with magnetic Feshbach resonance \cite{Cornish}. With this set of parameters,
the stationary state of $^{85}$Rb-$^{87}$Rb binary condensate is just
phase-separated. The trapping potential and obstacle laser potential
parameters are same as those considered in
Ref.\cite{Neely}, that is $\omega/(2\pi) = 8$Hz, $\alpha = 1$, $\beta = 11.25$,
$V_0(0) = 93.0\hbar\omega$, and $w_0 = 10\mu$m. Hereafter we term this set of
scattering lengths, number of atoms, and trapping potentials as
{\em set a}.  For this parameter set, in absence of
the obstacle potential, the stationary point of minimum energy is
$(v_1,v_2) = (0,\pm 0.75275)$. And, the difference in the energies per
particle of the binary condensate with normal and coreless vortex dipoles is
$0.11$.

\subsection{TF approximation single vortex}

 As mentioned earlier, the applicability of the variational {\em antsatz} is
limited to the weakly interacting domain. Experiments, however, mostly explore
domains other than weakly interacting. To analyze stability in this domain, we
resort to the TF approximation.

\subsubsection{Normal vortex}

 To simplify the analysis, consider a normal vortex located at the
center of the inner species. For a normal vortex, the wave functions of
the first or inner species  in TF approximation is
\begin{equation}
 \psi_1(r)  =  \left \{
     \begin{aligned}
        & \sqrt{\frac{\mu _1-V(r)}{u_{11}}}\frac{r}{\sqrt{r^2+\xi^2}}
                 &r^2 \leqslant R_{\rm in}^2 , \\
        & 0      &r^2 > R_{\rm in}^2 .
      \end{aligned} \right .
\end{equation}
Similarly, the wave function of the second  or outer species is
\begin{equation}
 \psi_2(r)  =  \left \{
      \begin{aligned}
         & \sqrt{\frac{\mu _2-V(r)}{u_{22}}}
                 &R_{\rm in}^2\leq r^2\leq R_{\rm out}^2\\
         & 0     &r^2 > R_{\rm out}^2
      \end{aligned} \right .
\end{equation}
here $V(r)$, $R_{\rm in}$, $R_{\rm out}$, and $\xi$ are the trapping potential,
radius of the
interface between inner and outer species, radial extent of outer species, and
coherence length of inner species, respectively. The chemical potentials
$\mu_1$ and $\mu_2$ are parameters of the calculations and obtained from
inverting the normalization condition
\begin{eqnarray}
   N_1 & = & \frac{\pi}{8 u_{11} \mu _1^2}\left[2 R_{\rm in}^2 \left(\mu _1
             +4 \mu _1^3\right)- 2 R_{\rm in}^4 \mu _1^2+\ln \left(1
             +2 R_{\rm in}^2 \mu _1\right) \right.
                            \nonumber   \\
         & & \left. \times \left(1+4 \mu _1^2\right) \right]\label{N_1&N_2_1}
                                        \\
   N_2  & = & \frac{\pi  \left(R_{\rm in}^2-2 \mu _2\right){}^2}{4 u_{22}}.
\label{N_1&N_2_2}
\end{eqnarray}
The total energy of the vortex with a normal vortex located at origin is
\begin{eqnarray}
E & = &\int \left\lbrace \psi_1(r)\left[-\frac{1}{r}\frac{\partial}{\partial r}-
        \frac{\partial^2}{\partial r^2}+\frac{1}{r^2}+V(r) + \frac{u_{11}}{2}
        |\psi_1(r)|^2\right]\right.\nonumber\\
  &   & \left.\psi_1(r) + \psi_2(r) \left[V(r) + \frac{u_{22}}{2}|\psi_2(r)|^2 \right]
       \psi_2(r)\right\rbrace,
\end{eqnarray}
The energy can be minimized under the constraint that $\mu_1$ and $\mu_2$
satisfy Eqs.~(\ref{N_1&N_2_1}-\ref{N_1&N_2_2}) to completely determine the
stationary state of binary condensate with a normal vortex at the origin.

\subsubsection{Coreless Vortex}
For a coreless vortex, the wave functions of the two species in TF
approximation are
\begin{eqnarray}
 \psi_1(r)  &=& \left \{
     \begin{aligned}
        & \sqrt{\frac{\mu _1-V(r)}{u_{11}}}\frac{r}{\sqrt{r^2+\xi^2}}
                 &r^2 \leqslant R_{\rm in}^2\\
        & 0&r^2 > R_{\rm in}^2,
     \end{aligned} \right . \\
 \psi_2(r)  &=&  \left \{
     \begin{aligned}
        & \sqrt{\frac{\mu _2-V(r)-u_{12}\psi_1(r)^2}{u_{22}}}
                      && r^2 \leqslant \xi^2\\
        &  \sqrt{\frac{\mu _2-V(r)}{u_{22}}}
                      &&R_{\rm in}^2\leqslant r^2\leqslant R_{\rm out}^2\\
        &  0          &&r^2 > R_{\rm out}^2.
     \end{aligned} \right .
\end{eqnarray}
Normalizing the wave functions gives the constraint equations in terms
of $\mu_1$ and $\mu_2$ as
\begin{eqnarray}
N_1 & = & \frac{\pi}{8 u_{11} \mu _1^2}\left[2 R_{\rm in}^2 \left(\mu _1+4
          \mu _1^3\right)- 2 R_{\rm in}^4 \mu _1^2+\ln \left(1+2 R_{\rm in}^2
          \mu _1\right) \right.\nonumber\\
    &   & \left. \times\left(1+4 \mu _1^2\right) \right],\nonumber\\
N_2 & = & \frac{\pi}{16 u_{11} u_{22} \mu _1^2}  \left\{u_{21} \left[-1
          +\ln 4+8 (\ln 2-1) \mu _1^2\right]+u_{11}\right.\nonumber\\
    &   & \left.\times\left(4 R_{\rm in}^4 \mu _1^2+8 \mu _1 \mu _2
           -16 R_{\rm in}^2 \mu _1^2 \mu _2+16 \mu _1^2 \mu _2^2-1\right)
          \right\}.
\end{eqnarray}
From the expressions, we can calculate the total energy of the system. It is
a complicated expression which we refrain from writing. For the
present case, it is perhaps sufficient to say, it is possible to obtain an
analytical but cumbersome expression of total energy.

For the parameter {\em set a} with $V_0(t) = 0$, binary
condensate with the coreless vortex has lower energy than the one with normal
vortex. This is evident from Fig.\ref{fig_coreless_vortex} where the variation
of total energy is shown as a function of $R_{\rm in}$.
\begin{figure}[ht]
\includegraphics[width=8.5cm] {coreless_vortex}
\caption{The energy of the binary condensate with $V_0=0$ and rest of the
parameters same as those in the parameters {\em set a} as a function of
$R_{\rm in}$. The condensate has a vortex dipole located at origin.
Black and blue curves are for coreless and normal
vortex dipoles respectively.}
\label{fig_coreless_vortex}
\end{figure}
However, the stability
vanishes if one considers $N_1=N_2=10^6$, $a_{11}=51a_0$, $a_{22}=99a_0$ in
{\em set a}. It is then the normal vortex which is energetically stable. This
is evident form Fig.\ref{fig_normal_vortex}.
\begin{figure}[ht]
\includegraphics[width=8.5cm] {normal_vortex}
\caption{The energy of the binary condensate with $N_1=N_2=10^6$,
$a_{11}=51a_0$, $a_{22}=99a_0$, $V_0 = 0$, and rest of the parameters same as
those in parameters {\em set a} as a function of $R_{\rm in}$. The condensate
has a vortex located at the origin. Black and blue curves are for coreless and
normal vortex dipoles respectively.}
\label{fig_normal_vortex}
\end{figure}
 These results are in very good
agreement with numerical results. This TF analysis, albeit, is for single
vortex clearly illustrates that depending upon the interaction parameters, a
phase-separated binary condensate can either support coreless or normal
vortex. It must be emphasized, the energy, chemical potentials, and interface
radius obtained from the TF approximation are in very good agreement
with numerical solution of coupled GP equations. For example, the numerical
value of energy $66.32$ corresponding to Fig.\ref{fig_normal_vortex} is very
close to the value $66.06$ from TF approximation.

\subsection{TF approximation with vortex dipole}

 We use the TF approximation to examine the stability of coreless vortices
in the strongly interacting domain.

\subsubsection{Normal vortex dipole}
Assume the vortex affects the density of the condensate only  within the core
regions. We then adopt the following {\em ansatz} for binary condensate with a
normal vortex dipole
at $(v_1,\pm v_2)$.
\begin{eqnarray}
 \psi_1(r)  & = & \left \{
     \begin{aligned}
        & 0 \!&& x^2+y^2 > R_{\rm in}^2\\
        & 0 \!&& [(x-v_1)^2+(y\pm v_2)^2] \leqslant \xi^2\\
        & \sqrt{\frac{\mu_1 - V(x,y)}{u_{11}}}
              && \left \{
                   \begin{aligned}
                     & x^2+y^2 \leqslant R_{\rm in}^2~\&\\
                     &~[(x-v_1)^2+(y\pm v_2)^2] > \xi^2
                  \end{aligned} \right .
     \end{aligned} \right .   \\
 \psi_2(r)  &=&  \left \{
     \begin{aligned}
        & \sqrt{\frac{\mu _2-V(x,y)}{u_{22}}}
                 &&R_{\rm in}^2\leqslant (x^2+y^2) \leqslant R_{\rm out}^2\\
        & 0&&(x^2+y^2) > R_{\rm out}^2\\
        & 0&&(x^2+y^2) < R_{\rm in}^2.
     \end{aligned} \right .
\end{eqnarray}
The vortex dipole contributes mainly through the kinetic energy of $\psi_1$,
which may be approximated with the  value of single species condensate given
in Ref.\cite{Zhou}
\begin{equation}
   E_{\rm vd} = \frac{2\mu _1}{u_{11}}\ln\left(\frac{2v_2}{\xi}\right).
\end{equation}
Using these {\em ansatz} the number of atoms are
\begin{eqnarray}
N_1 & = & \frac{\pi  \left(1+4 v_1^2 \mu _1+4 v_2^2 \mu _1-8
          \mu _1^2-2 R_{\rm in}^4 \mu _1^2+8 R_{\rm in}^2
          \mu _1^3\right)}{8 u_{11} \mu _1^2},\nonumber \\
N_2 & = & \frac{\pi  \left(R_{\rm in}^2-2 \mu _2\right){}^2}{4 u_{22}}.
\end{eqnarray}
In a similar way, we can evaluate the energy of the entire condensate.

\subsubsection{Coreless vortex dipole}
For coreless vortex dipole, we adopt the {\em ansatz}
\begin{eqnarray}
 \psi_1(r)  & = &  \left \{
      \begin{aligned}
         & 0 && x^2+y^2 > R_{\rm in}^2\\
         & 0 && [(x-v_1)^2+(y\pm v_2)^2] \leqslant \xi^2\\
         & \sqrt{\frac{\mu_1 - V(x,y)}{u_{11}}}
              && \left \{
                \begin{aligned}
                   & x^2+y^2 \leqslant R_{\rm in}^2~\&\\
                   &~[(x-v_1)^2+(y\pm v_2)^2] > \xi^2,
                \end{aligned} \right .
      \end{aligned} \right . \\
 \psi_2(r)  & = & \left \{
      \begin{aligned}
         & \sqrt{\frac{\mu _2-V(x,y)}{u_{22}}}
                &&\left \{
                   \begin{aligned}
                      & R_{\rm in}^2\leqslant (x^2+y^2) \leqslant R_{\rm out}^2~||~\\
                      & [(x-v_1)^2+(y\pm v_2)^2] \leqslant \xi^2
                   \end{aligned} \right . \\
         & 0&&(x^2+y^2) > R_{\rm out}^2\\
         & 0&& \left \{
                \begin{aligned}
                   & x^2+y^2 < R_{\rm in}^2~\&\\
                   &~[(x-v_1)^2+(y\pm v_2)^2] > \xi^2.
                \end{aligned} \right .
      \end{aligned} \right .
\end{eqnarray}
Using these {\em ansatz} the modified expressions for $N_2$ is
\begin{eqnarray}
N_2 &=& \frac{\pi}{8 u_{22} \mu _1^2}\left(2 R_{\rm in}^4 \mu _1^2
       + 8 \mu _1 \mu _2-8 R_{\rm in}^2 \mu _1^2 \mu _2+8 \mu _1^2 \mu _2^2 -1
         \right.       \nonumber \\
    & &\left . -4 v_1^2 \mu _1-4 v_2^2 \mu _1\right).
\end{eqnarray}
We can also calculate the total energy $ E$ of the system. The important
change in $E$ is the inclusion of interface interaction energy $E_{\rm int}$.
It arises from the interface interactions at the cores of the vortex and
antivortex. Based on Ref.~\cite{Timmermans},
\begin{equation}
 E_{\rm int} = \frac{8}{3}Pb\pi\xi\left (\frac{a_{12}}{\sqrt{a_{11}a_{22}}}-1
               \right ),
\end{equation}
where $P$ is the pressure on the circumference of the cores and
\begin{equation}
   b = 2\left [ \frac{3(\mu_1+\mu_2)\sqrt{a_{11}a_{22}}}{4\mu_1\mu_2
          (a_{12}-\sqrt{a_{11}a_{22}})} \right ] ^{1/2}.
\end{equation}
As in the previous section, energy can be minimized with the constraint
of the fixed number of atoms. For the parameters {\em set a}
without obstacle potential, the coreless vortex dipole has lower energy than
the normal vortex dipole and is shown in
Fig.~\ref{fig_coreless_vortex_dipole1} for the vortex dipole located
at $(0,\pm1)$.
\begin{center}
\begin{figure}[ht]
\includegraphics[width=8.5cm] {coreless_vortex_dipole1}
\caption{The energy of the binary condensate with $V_0=0$ and rest of the
parameters same as those in the parameters {\em set a} as a function of
$R_{\rm in}$. The condensate has a vortex dipole located at $(0,\pm 1)$.
Black and blue curves are for coreless and normal vortex dipoles
respectively.}
\label{fig_coreless_vortex_dipole1}
\end{figure}
\end{center}
 For $N_1=N_2=10^6$, $a_{11}=51a_0$, $a_{22} = 99a_0$ and rest
of the parameters same as in parameter {\em set a}, condensate with the normal
vortex dipole has lower energy than the one with coreless vortex dipole
(see Fig.~\ref{fig_coreless_vortex_dipole2}). These results are again in good
agreement with the numerical results.
\begin{figure}[ht]
\includegraphics[width=8.5cm] {coreless_vortex_dipole2}
\caption{The energy of the binary condensate with $N_1=N_2=10^6$,
$a_{11}=51a_0$, $a_{22}=99a_0$, $V_0 = 0$, and rest of the parameters same as
those in parameters {\em set a} as a function of $R_{\rm in}$. The condensate
has vortex dipole located at $(0,\pm 1)$. Black and blue curves are for
coreless and normal vortex dipoles respectively.}
\label{fig_coreless_vortex_dipole2}
\end{figure}

\section{Numerical results}

   For the numerical calculations, we again resort to experimentally realizable
system and paramteres. We again work with the $^{85}$Rb-$^{87}$Rb mixture and
parameter {\em set a} introduced earlier. Recall, the  values of paramteres in
this set are $a_{11} = 460a_0$, $a_{22} = 99a_0$, and $a_{12} = 214a_0$
$2 N_1 = N_2 = 10^6$, $\omega/(2\pi) = 8$Hz, $\alpha = 1$, $\beta = 11.25$, $
V_0(0) = 93.0\hbar\omega$, and $w_0 = 10\mu$m.
As mentioned earlier, $a_{11}$ is tunable with magnetic Feshback resonance
\cite{Cornish}.

   The transport of $n_2 $ within the region of influence of $V_{\rm obs}$
is crucial to generate coreless vortices. As
$V_{\rm obs}$ enters $n_1$, it nudges $n_1$ and creates an interstice, which
is occupied by $n_2$. This is an outcome of the difference in
coherence lengths ($\xi_i = 1/\sqrt{2n_iu_{ii}}$). The smaller $\xi$ of $n_2$
ensures that it recovers bulk value over shorter distances. It implies that it
is comparatively more difficult to displace the $n_2$ than $n_1$. As the
obstacle reaches the interface with diminished strength, it is unable to
displace $n_2$, but not for $n_1$ due larger $\xi$. This results in an inward
protrusion of the interface, which detaches from the interface at a later time.
For detachment to occur, it should be energetically favorable. The
difference in $\xi$ coupled with $a_{12}\gtrapprox\sqrt{a_1a_2}$ ensures that
the interstitial filling of $n_2$ is energetically lower, in other words
energetically favorable. This is easily confirmed with imaginary time
propagation of GP equation to obtain the stationary state geometry of the
binary condensate with $V_{\rm obs} $ located inside $n_1 $ as is shown in
Fig.~\ref{plot2}(c).
\begin{figure}
  \includegraphics[width=8.5cm] {plot0.png}
  \caption{Coreless vortex dipole shedding when the obstacle potential
           traverses from the edge to the inner component in a binary
           condensate with $V_0 = 125$, $x_0(0) = -15a_{\rm osc}$ and rest of
           the parameters same as those in {\em set a}. Figures a, b and c
           are the absolute values of wave functions at $0.21$s, $0.33$s,
           $0.56$s, respectively; while figures d, e and f
           are the corresponding phase plots.
          }
  \label{coreless_vdip}
\end{figure}

\subsection{Motion of $V_{\rm obs} $ towards interface}

   We first consider the motion of $V_{\rm obs} $, initially located in the
$n_2$ towards the interface. For this case, we consider $^{85}$Rb-$^{87}$Rb
binary condensate with parameter {\em set a} and $V_{\rm obs}$ of
$V_0(0) = 125.0$. The $V_{\rm obs} $ is initially located at
$x = -15 a_{\rm osc}$ and moves with the speed of $180\mu$m/s. It progressively
decreases in strength and vanishes at $x = 8a_{\rm osc}$. The obstacle creates
a normal vortex dipole as it traverses $n_2$. And, as $V_{\rm obs} $ approaches
the interface, it transports the vortex dipole. Further motion of $V_{\rm obs}$
in the $n_1$  generates coreless vortices in the inner component. The formation
of the coreless vortex dipole is shown in Fig. \ref{coreless_vdip}.
The plots in the figure show both the density and phase
distribution. From the phase plot, it is clear that the phase singularity of
the vortex is associated with $n_1$. And, from the density plot it is evident
that the vortex-antivortex cores are filled with $n_2$. One effect which
precedes the seeding of coreless vortex is the indentation of the inner
interface after it cleaves when the $V_{\rm obs} $ is fully
immersed in $n_1$. This may be explained in terms of the dynamical evolution
around $V_{\rm obs} $ and analyzed in two different ways.

\subsubsection{Inertia of $n_2$ }

 Just after the interface cleaves, when the $V_{\rm obs} $ is set
into motion at $t = t_0$, the $n_2$ due to inertia, does not follow
$V_{\rm obs}$ initially. As a result at $t_0+\delta t$, neglecting the back
flow of $n_2$, outer species (along $x$-axis) at $x' \in [x_{\rm min}(t_0),
x_{\rm min}(t_0+\delta t)]$ is approximately same
\begin{equation}
 n_2(x',0,t+\delta t) \approx n_2(x',0,t_0),
\end{equation}
here $\delta t$ is time taken by the outer species to redistribute itself
according to the displaced obstacle potential. As a result over a length
$ x_{\rm min}(t_0+\delta t) - x_{\rm min}(t_0)$, the density of inner species in
TF approximation is
\begin{equation}
 n_1(x',0,t_0+\delta t) =  \frac{\mu_1 - x^2/2 -n_2(x',t_0)u_{12}}{u_{11}}.
\end{equation}
Hence inner $n_1$ flows into the above domain and replaces $n_2$. The circular
symmetry of $V_{\rm obs}$ coupled with motion
along $x$-axis ensures that the aforementioned process starts along $x$-axis
earlier; also the replacement of $n_2$ by $n_1$ takes place at the much faster
speed as compared to the speed of the obstacle. This leads to creation of dimple
on the inner interface.

\subsubsection{Pressure balance}
 Let $x_0$ and $\Gamma_{x_0}$ be the location of the obstacle
just after the interface cleaves and approximately circular inner interface
centered around $x_0$ with radius $\tilde R_{\rm in}$, respectively. After
time $\delta t$, let $x_0'= x_0 + \delta x_0$ and $\Gamma_{x_0'}$ be the
location of the obstacle and inner interface, respectively. Assuming that
$\Gamma_{x_0'}$ has same radius as $\Gamma_{x_0}$, i.e. $\tilde{R}_{\rm in}$,
the points of intersection of $\Gamma_{x_0}$ and $\Gamma_{x_0'}$ are
\begin{equation}
(x_c,y_c) = \left(\frac{1}{2} (x_0+ x_0'),
            \pm \frac{1}{2} \sqrt{4 {\tilde{R}_{\rm in}}^2-
            \delta x_0^2}\right).
\label{cross_over}
\end{equation}
Thus the region $\delta \Gamma$ defined as
\begin{eqnarray}
 (x-x_0)^2+y^2  <  \tilde{R}_{\rm in}^2<(x-x_0')^2+y^2
\label{exp_region}
\end{eqnarray}
with area
\begin{equation}
A = \pi \tilde{R}_{\rm in}^2 - 2\tilde{R}_{\rm in}^2\cos^{-1}
     \frac{\delta x_0}{2\tilde{R}_{\rm in}}+ \frac{\delta x_0}{2}
     \sqrt{4\tilde{R}_{\rm in}^2-\delta x_0^2},
\end{equation}
lies within $\Gamma_{x_0}$ but outside $\Gamma_{x_0'}$.
Let $P_1 = n_1^2u_{11}/2$ and $P_2 = n_2^2u_{22}/2$ be the pressures exerted
by inner and outer species on the left arc of $\delta \Gamma$ region,
respectively. The potential experienced by $n_1$ along the left arc of $\delta
\Gamma$, which is approximately equal to harmonic trapping potential, does not
change as the obstacle is moving away from this region, and so does the $P_1$.
The potential experienced by $n_2$ over $\delta \Gamma$, which is equal to sum
total of harmonic trapping and obstacle potentials, decreases as the obstacle
moves from $x_0$  to $x_0+\delta x_0$. In fact, as one moves from $(x',0)$,
where $x'\in(x_0-\tilde{R}_{\rm in},x_0+\delta x_0-\tilde{R}_{\rm in})$, to
the $(x_c,y_c)$ along a curve with same sign of curvature as bounding arcs
of $\delta \Gamma$ region, the potential experienced by $n_2$ decreases.
This leads to the redistribution of $n_2$, i.e. it moves to the regions with
weaker trapping potential. The redistribution will lead to $\delta P=P_1-P_2>0$
which decreases as one moves along the aforementioned curve. This will lead to
indentation of the inner interface. In our simulations, when the obstacle
is moved, along $x$-axis, from a point near the center towards the outer
interface, indentation is suppressed. It confirms that this mechanism indeed
aids the indentation of the interface.

\subsection{Obstacle motion towards edge}
In this case, we start with the stationary state solution of
$^{85}$Rb-$^{87}$Rb binary condensate for the parameters {\em set a} with
obstacle laser potential maxima located at $x = -5.0a_{\rm osc}$. The static
solution in this case is shown in first column of
Fig.\ref{triply_charged_vdipole}, which has roughly doughnut shaped density
distribution of the outer species along with non-zero density in the circle of
influence of the obstacle.
\begin{center}
\begin{figure}[ht]
\includegraphics[width=8.5cm] {plot3}
\caption{The generation of triply charged vortex dipole in the
         $^{85}$Rb-$^{87}$Rb binary condensate with parameters {\em set a}.
         The obstacle potential, initially located at $x = -5.0 a_{\rm osc}$,
         is moved with a velocity of $220\mu$m/s up to $x = 5 a_{\rm osc}$.
         First, second, and third columns are the solutions at $t=0$s,
         $t=0.14$s and $t = 0.32$s, respectively.}
\label{triply_charged_vdipole}
\end{figure}
\end{center}

Such density distributions are prerequisite to create coreless vortices in the
inner species. The obstacle is moved with a uniform speed of $220 \mu$m/s,
progressively decreases in strength, and finally becomes zero at
$x = 5.0a_{\rm osc}$. With this velocity, we observe the creation of triply
charged coreless vortex dipole as is shown in last column of
Fig.\ref{triply_charged_vdipole}.
Singly and doubly charged vortex dipoles can be generated by moving the obstacle
potential at lower velocities. For example,
Fig.\ref{vdipole_trajectory} shows the singly charged coreless vortex dipole
and the trajectory followed by it for the $^{85}$Rb-$^{87}$Rb binary condensate
with parameters {\em set a}. The obstacle laser potential moves from
$x=0a_{\rm osc}$ at $t=0s$ to $x=8.0a_{\rm osc}$ with the speed of
$200\mu$m/s.
\begin{center}
\begin{figure}[ht]
\begin{tabular}{cc}
\resizebox{42mm}{!}
{\includegraphics[trim = 0mm 0mm 0mm 0mm,clip, angle=0,width=8cm]
                 {rb85-rb87-data33_25_2}}&
\resizebox{42mm}{!}
{\includegraphics[trim = 0mm 0mm 0mm 0mm,clip, angle=0,width=8cm]
                 {rb85-rb87-data33_dipole_traj_xy}}\\
\end{tabular}
\caption{The figure on the left side shows the vortex dipole in
$^{85}$Rb-$^{87}$Rb condensate with parameters {\em set a} at $t=0.12$s.
The white curve indicates the trajectory of lower vortex. The figure on
the right side shows the variation in position of the upper vortex with time.}
\label{vdipole_trajectory}
\end{figure}
\end{center}

\begin{center}
\begin{figure}[ht]
\includegraphics[width=8.5cm] {plot4}
\caption{The generation of normal vortex dipoles in the $^{85}$Rb-$^{87}$Rb
binary condensate with $a_{11} = 51a_0$, $N_1=N_2=10^6$, and the rest of the
parameters same as those defined in parameters {\em set a}. The obstacle
potential, initially located at $x = 0 a_{\rm osc}$, is moved with a velocity
of $200\mu$m/s up to $x = 5 a_{\rm osc}$. First, second, and third columns
are the absolute values of wave functions at $t=0$s, $t=0.01$s, and
$t = 0.14$s, respectively.}
\label{normal_vdipole}
\end{figure}
\end{center}

We also considered the $^{85}$Rb-$^{87}$Rb with $a_{11} = 51a_0$,
$N_1 = N_2 = 10^6$, and the rest of the parameters same as those in parameter
{\em set a}. The obstacle laser potential is initially located at origin
$x = 0 a_{\rm osc}$. The stationary solution in this case is strongly phase
separated ($a_{12}\gg\sqrt{a_1a_2}$), and the circular region of influence of
obstacle potential is devoid of atoms of outer component as is shown in first
column of Fig.\ref{normal_vdipole}. In our simulation, the obstacle potential
potential traverses a distance of $5a_{\rm osc}$ along x-axis with a speed
of $200\mu$m/s. We observe that the vortex dipole generated in the inner
component is a normal vortex dipole with empty core. This is
consistent with the TF results shown in Fig.~\ref{fig_coreless_vortex_dipole2}.

\section{Conclusions}
We have studied the motion of the Gaussian obstacle potential through a
phase-separated binary condensate in pan-cake shaped traps. The motion of the
obstacle leads to the generation of either coreless or normal vortex dipole
depending upon the interaction parameters. The interaction parameters suitable
for coreless vortex dipole can also be employed to have obstacle assisted
transport of one species across another. We have used both a variational
{\em ansatz} based method, applicable when $N_ia_{ii}/a_{\rm osc}\sim 1$, and
TF approximation based method, applicable when $N_ia_{ii}/a_{\rm osc}\gg 1$,
to analyze the energetic stability of coreless versus normal vortices.
Our studies show that using Feshback resonances to tune one of the scattering
length, e.g. scattering length of $^{85}$Rb in $^{85}$Rb-$^{87}$Rb, it should
be experimentally possible to create coreless vortices in phase-separated
binary condensates.

\begin{acknowledgements}
We thank S. A. Silotri, B. K. Mani, and S. Chattopadhyay for very useful
discussions. The numerical computations reported in the paper were done on
the 3 TFLOPs cluster at PRL. The work of PM forms a part of Department of
Science and Technology (DST), Government of India sponsored research project.

\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
from __future__ import absolute_import, division, print_function
"""
Build a ply lexer, but without the implicit magic and global
state, just load prebuilt ply parser and lexers at roughly the
same cost of loading a Python module.
"""

import functools
from ply.lex import Lexer, LexerReflect
from ply.yacc import ParserReflect, LRTable, LRParser

def memoize(obj):
    cache = obj.cache = {}
    @functools.wraps(obj)
    def memoizer(*args, **kwargs):
        if args not in cache:
            cache[args] = obj(*args, **kwargs)
        return cache[args]
    return memoizer

# lexer is passed as argument to ensure that memoization is
# unique for parser/lexer pair.

@memoize
def yaccfrom(module, tabmodule, lexer):
    # Get the module dictionary used for the parser
    _items = [(k,getattr(module,k)) for k in dir(module)]
    pdict = dict(_items)

    # Collect parser information from the dictionary
    pinfo = ParserReflect(pdict)
    pinfo.get_all()

    # Read the tables
    lr = LRTable()
    lr.read_table(tabmodule)

    lr.bind_callables(pinfo.pdict)
    return LRParser(lr,pinfo.error_func)

@memoize
def lexfrom(module, lexmodule):
    lexobj = Lexer()
    lexobj.lexoptimize = 1

    _items = [(k,getattr(module,k)) for k in dir(module)]
    ldict = dict(_items)

    # Collect parser information from the dictionary
    linfo = LexerReflect(ldict)
    linfo.get_all()

    lexobj.readtab(lexmodule,ldict)
    return lexobj

redpajama_set_name: RedPajamaGithub


text:
Q: How to increase gaps between vertical gridlines in chartjs
I have drawn a chart using Chartjs for a large number of datasets. So, there are many vertical gridlines by default which looks pathetic. I want to increase the gap between the vertical lines.
Any help will be appriciated.Thanks

A: You can make use of the maxTicksLimit, this will make it so only that amount of ticks will get rendered which in turn makes the distance between ticks larger
code:
options: {
  scales: {
    xAxes: [
      {
        ticks: {
          maxTicksLimit: 10
        }
      }
    ]
  }
}

Source: https://www.chartjs.org/docs/latest/axes/cartesian/linear.html

A: Good answer from @LeeLenalee
I would like to add an additional option for fine-tuning in case of y-Axis (horizontal gridlines):
Chart.js v3.2.0
If you want to achieve a minimalistic design with even less gridlines (say only 2 or 3 gridlines for basic orientation), chart would tend to not take advantage of full chart height for little maxTicksLimits (no automatic scaling, see example with maxTicksLimit: 3)
Here are examples (y-Axis) to demonstrate:

*

*Normal (no ticks-costumization)
 options: {
    scales: {
      y: {
         ticks: {
            // no costumization
            }
         }
      }
   }

(Many gridlines and ticks, but automatic scaling working)

*maxTicksLimit: 10
 options: {
    scales: {
      y: {
         ticks: {
            maxTicksLimit: 10
            }
         }
      }
   }

(Less gridlines and ticks, automatic scaling still OK)

*maxTicksLimit: 3
  options: {
     scales: {
       y: {
          ticks: {
             maxTicksLimit: 3
             }
          }
       }
    }

(Does not take advantage of full chart height, no automatic scaling)

*callback
  options: {
     scales: {
       y: {
          ticks: {
             callback: function(value, index, values) {
                let step = 5;

                //values[0].value is first tick (tick_min)
                //values[values.length-1].value is last tick (tick_max)

                //tick_max - tick_min is range of values at y-Axis

                if (values[values.length-1].value - values[0].value > 15) {
                   //define condition (> 15) and
                   //step size (10) according to your estimated data range
                   step = 10;
                } else {
                   step = 5;
                };
                if (Number.isInteger(value/step)) {
                   return value;
                } else {
                   return '';
                }
               }
             },
             grid: {  //hide gridlines where there is no tick
                z: 1,
                color: function(context) {
                   if (context.tick.label != '')  {
                      return 'rgba(61,138,254,0.5)';
                   } else {
                      return 'rgba(61,138,254,0)'; //transparent
                   }
                }
             }
          }
       }
    }

(Few gridlines and automatic scaling OK)
Source: https://www.chartjs.org/docs/latest/axes/labelling.html#creating-custom-tick-formats

redpajama_set_name: RedPajamaStackExchange


What a great idea! You are very creative!
Your Easter Fairy for your daughter is BEAUTIFUL!!! Love her with her pretty beads!!
I saw the Peacock pillow on Jurga's site today! It's is sooo cute!!
This is such a beautiful egg for easter. You are so creative and so inspirational!
Праздничное настроение уже чувствуется! Отличные варианты декора!
You're such a creative person. I love these eggs.
А цвета какие весенние, нежные!
How beautiful! I love the colours and your creativity to make such a gorgeous egg. Lovely work!!
Лорета, какую красоту сделала! какая нежная цветовая гамма! а ты их приклееваешь наверное, да? как же они на кончиках-то держаться?
What a wonderful idea, they look lovely.


text:
Team USA Hockey
Stan Bowman Named General Manager Of 2022 U.S. Olympic Men's Ice Hockey Team
By USA Hockey, 03/31/21, 12:00PM EDT
Bill Guerin To Serve As Assistant General Manager
COLORADO SPRINGS, Colo. – USA Hockey announced today the first two members of the 2022 U.S. Olympic Men's Ice Hockey Team in general manager Stan Bowman (Buffalo, N.Y./Chicago Blackhawks) and assistant general manager Bill Guerin (Wilbraham, Mass./Minnesota Wild).
"We could not be happier to have the combination of Stan and Bill, with the support of John Vanbiesbrouck, guiding our men's Olympic team," said Pat Kelleher, executive director of USA Hockey. "We have a deep talent pool thanks to so many people, including volunteers at the grassroots level across the country, and it's great to have the leadership and expertise that Stan and Bill bring in building our team."
Bowman, who is the president of hockey operations and general manager of the NHL's Chicago Blackhawks, has guided the franchise to three Stanley Cup titles during his time in the Windy City. He has served as a member of the U.S. Men's National Team Advisory Group since 2012 to help select players and staff for U.S. Men's National Teams. Guerin, a three-time Olympian as a player who had an 18-year Hall of Fame NHL playing career, is in his second year as general manager of the Minnesota Wild after serving in a management capacity for the Pittsburgh Penguins since 2014.
"The combination of a Stanley Cup-winning GM and a Hall of Fame player who has competed in the Olympics in addition to his experience as an NHL GM, is certainly something we're excited about," said Vanbiesbrouck, assistant executive director of hockey operations for USA Hockey, who will work with Bowman and Guerin and the U.S. Men's National Team Advisory Group in building the 2022 U.S. Olympic Men's Ice Hockey Team. "Stan and Bill bring varied experiences over their highly successful careers, but the thing that stands out to me is that they are both winners."
"I am extremely proud and honored to serve such a prestigious role for USA Hockey," said Bowman. "It will be exciting to work alongside Bill, John and the rest of our advisory group as we build a group of American hockey players and add to the storied tradition and legacy of Team USA. I look forward carrying the Olympic spirit through the game of hockey as we select a team for the 2022 Olympic Winter Games in Beijing."
"I know how excited I was to play in the Olympics and I'm equally as excited to work with Stan, John and the other GMs part of our advisory group in building our 2022 Olympic Team," said Guerin. "It's amazing to see the continued growth and skill level of players across the country within USA Hockey. The good news is that we'll have an extremely difficult task in settling on a final roster as we have so many great players."
2022 U.S. Olympic Men's Team
About Bowman
Stan Bowman is the president of hockey operations and general manager of the Chicago Blackhawks and is in his 19th season with the organization. He spent five seasons as senior vice president/general manager prior to beginning in his current role in December of 2020. Bowman was originally named the ninth general manager in Blackhawks history on July 14, 2009, after eight years in the Blackhawks hockey operations department.
Bowman led the Blackhawks to their sixth Stanley Cup championship on June 15, 2015, becoming the first GM to win three titles in the salary cap era. In 2010, Bowman became the youngest GM in NHL history to guide an organization to the championship, as the team ended a 49-year title drought. During his time at the helm of the hockey operations department, Bowman has secured the team's core talent with long-term contract extensions, rebuilt the club's prospect depth with multiple acclaimed draft classes and acquired many key roster pieces through trades and free-agent signings.
Bowman originally joined the Blackhawks in 2000 and spent four seasons as special assistant to the general manager before being promoted to director of hockey operations, a role he served in for two years (2005-07). As assistant general manager, Bowman managed the day-to-day administration of the hockey operations department, including all CBA-related matters, such as contract negotiations, free agency, salary arbitration and player movement and assignment. He also tracked the progress of Blackhawks prospects by working closely with the staff of the club's minor-league affiliate in Rockford while assisting with player evaluation, prospect development and professional and amateur scouting.
Since 2012, Bowman has served on U.S. Men's National Team Advisory Group to help select players and staff for U.S. Men's National Teams. In addition, he was part of the management for Team North America at the 2016 World Cup of Hockey alongside Peter Chiarelli.
Bowman graduated from the University of Notre Dame in 1995 with degrees in finance and computer applications. He was named to Crain's Chicago Business "40 under 40" Class of 2010 and inducted into the Chicagoland Sports Hall of Fame in 2015.
About Guerin
Bill Guerin is in his second season as general manager of the Minnesota Wild and has quickly helped shaped the Wild into a playoff contender.
Guerin spent the eight seasons prior to arriving in Minnesota with the Pittsburgh Penguins, including the last five as assistant general manager. He served as the Penguins player development coach for three seasons (2011-14), helping mold the organization's minor league, junior and college prospects, many who were key contributors on Pittsburgh's 2016 and 2017 Stanley Cup teams. Guerin was promoted to assistant general manager by the organization on June 6, 2014, and served as general manager of Wilkes-Barre/Scranton, overseeing the day-to-day operations of Pittsburgh's American Hockey League affiliate, the last two seasons.
A three-time Olympian (1998, 2002, 2006) in his Hall of Fame playing career, Guerin starred in the NHL for 18 seasons with seven different teams (New Jersey, Edmonton, Boston, Dallas, St. Louis, San Jose, New York Islanders). The two-time Stanley Cup champion (1995, New Jersey; 2009, Pittsburgh) amassed 856 career points (429 goals, 427 assists) and helped his teams to the Stanley Cup Playoffs 15 times, with 74 points (39 goals, 35 assists) in 140 career playoff contests. A four-time NHL All-Star Game selection (2001, 2003, 2004, 2007), Guerin is one of just nine American players ever to record 400 career goals and 400 career assists in the NHL.
In addition to playing in three Olympic Games, including a silver medal finish in 2002, Guerin also helped the U.S. to one of the seminal moments in U.S. hockey history when Team USA captured the 1996 World Cup of Hockey championship. Guerin also played for Team USA in the 2004 World Cup of Hockey and in the 1989 and 1990 IIHF World Junior Championship.
Guerin, who played collegiately at Boston College, was inducted into the U.S. Hockey Hall of Fame in 2013.
About Vanbiesbrouck
John Vanbiesbrouck was named assistant executive director of hockey operations at USA Hockey in June of 2018 and has made a significant impact in a short time. Under his leadership overseeing the sled, women's and men's international programs, the U.S. has enhanced its standing as a world power in the sport.
As part of his responsibilities, Vanbiesbrouck serves as general manager of the U.S. National Junior Team and he's crafted teams that have played in the gold-medal game in two of the past three IIHF World Junior Championships, a feat never before accomplished in U.S. history. Team USA captured gold in 2021 and earned a silver medal in 2019.
While the global pandemic has resulted in the cancellation of world championships at various levels, during his tenure, the U.S. has earned gold at the 2019 IIHF Women's World Championship, gold (2020) and silver (2019) at the IIHF U18 Women's World Championship; bronze at the IIHF U18 Men's World Championship (2019) and gold at the 2019 IPC Sled Hockey World Championship.
Vanbiesbrouck had a remarkable NHL playing career that spanned parts of 20 seasons and included stops with the New York Rangers, Florida Panthers, Philadelphia Flyers, New York Islanders and New Jersey Devils. The five-time NHL All-Star won 374 games, the second most by any American netminder in NHL history. He earned the Vezina Trophy as the top goaltender in the NHL in 1986 as a member of the New York Rangers and 10 years later helped the Florida Panthers advance to the Stanley Cup Final.
Inducted into the U.S. Hockey Hall of Fame in 2007 and Florida Sports Hall of Fame in 2015, Vanbiesbrouck has represented the United States on the international stage as a player on nine occasions, including the Olympic Winter Games in 1998; two Canada Cups (1987, 1991), four men's world championships (1985, 1987, 1989, 1991) and two world junior championships (1982, 1983).
NOTES: The selection of the general manager and assistant general manager of the 2022 U.S. Olympic Men's Ice Hockey Team was made in consultation with Jim Smith, president of USA Hockey; Gavin Regan, vice president of USA Hockey and chair of the organization's international council; Pat Kelleher, executive director of USA Hockey; and John Vanbiesbrouck, assistant executive director of USA Hockey … The U.S. has earned 13 total medals in men's Olympic ice hockey competition, including two gold (1980, 1960), eight silver (2010, 2002, 1972, 1956, 1952, 1932, 1924, 1920) and one bronze (1936) … With NHL players in the Olympics, the U.S. has finished: 1998-6th; 2002-Silver; 2006-8th; 2010-Silver; 2014-4th … Previous general managers of U.S. Olympic Men's Ice Hockey Teams with NHL players: 1998-Lou Lamoriello; 2002-Craig Patrick; 2006-Don Waddell; 2010-Brian Burke; 2014-David Poile.
Men's Olympic History
More Team USA News
2022 Women's Olympic Notebook | Let The Journey Begin
By USA Hockey 01/27/2022, 1:45pm EST
Follow along on Team USA's journey during the 2022 Olympic Winter Games
Kendall Coyne Schofield Named Captain of 2022 U.S. Olympic Women's Ice Hockey Team
By USA Hockey 01/24/2022, 10:00am EST
Brianna Decker, Hilary Knight and Lee Stecklein to Serve as Alternates
U.S. Paralympic Sled Hockey Team Unveiled
17 Players to Represent Team USA at the 2022 Paralympic Winter Games
U.S. Sled National Team Begins Residency in Nashville
By Dan Scifo 01/18/2022, 12:45pm EST
Paralympic prep ramps up in pursuit of a fourth consecutive gold medal
Paralympic Hockey Schedule Announced
Team will open play on March 5
Tag(s): Men's Olympics/Paralympics Men's Olympics Home Spotlight 2022 - Beijing
This website is powered by SportsEngine's Sports Relationship Management (SRM) software, but is owned by and subject to the Team USA Hockey privacy policy. ©2022 SportsEngine, Inc.

redpajama_set_name: RedPajamaCommonCrawl


text:
Time to rethink local government and RMA reform
The Local Government Commissions announcement that it would move to strengthen local government in the Hawkes Bay at the same time as backing down on recommendations for Wellington and Northland demonstrates the need for a first principles rethink of what local government does, why and how, something a Royal Commission is best placed to investigate and which must include stalled discussions on the role of the Resource Management Act and other planning laws, says Stephen Selwood CEO of the NZ Council for Infrastructure Development.
The Commissions decisions are set to entrench different governance structures, roles and activities in different regions with net costs to New Zealand as a whole.
The Local Government Commission was at risk of having its decisions reversed by local referenda driven by local politicians. Had the Commission been in a position to recommend strengthened governance for Wellington and Northland, New Zealand could have begun the transition to a consistent national and regional spatial planning and implementation framework supported by local boards at the grass roots level.
This would be consistent with international best practice and would help align central and local government on major planning and investment decision making.
It would also facilitate private activity and investment while enhancing community participation and decision making through local boards.
It would have delivered a superior outcome for NZ Inc. Instead, much more narrow interests have been allowed to prevail.
The result is an unfolding piecemeal governance scenario with large regional councils like Auckland providing clear direction and a single voice on major issues, whilst small councils like Kaipara next door struggle to sustain the expertise necessary to provide modern public services.
A Royal Commission is now required into local government in New Zealand.
Only a Royal Commission can access the resources and maintains the mana necessary to complete a full first principles review of local government across New Zealand.
Such a review would align well with another stalled reform process, that of the Resource Management Act 1991.
Currently, we have councils making plans under the Local Government Act, large aspects of which are actually implemented under the RMA and transport Act. We have rules under the RMA which impede the plans of councils developed under the LGA and proposals under the RMA which are not funded because council spending falls under the jurisdiction of the LGA.
The whole thing is a complete mess.
Until we achieve alignment over what it is we want local government to deliver and how, it will be very difficult for the Local Government Commission to make sound recommendations which mobilise the support of communities.
That process cannot happen without the LGA, RMA and Land Transport Management Act being part of the same discussion, Selwood says.


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The Houthis accuse Saudi Arabia of airstrikes that killed 30 people, including civilians, in Al Jawf Governorate. The Houthis say women and children were among those killed in the strike, which was said to be a retaliation for the downing of a coalition plane the previous day.
Sources: Reuters
The Houthis accuse Saudi Arabia of bombings that killed 30, including civilians in Al Jawf Region. The Houthis say women and children were killed in the strike, which was said to be a retaliation for the downing of a coalition plane the previous day.
2020-02-15 13:03:21 (First version)
The Houthis accuse Saudi Arabia of bombings that killed 30, including civilians in Al Jawf Governorate. The Houthis say women and children were killed in the strike, which was said to be a retaliation for the downing of a coalition plane the previous day.
2020-02-15 17:03:08 (Update)
(Current version)
Saudi-led coalition airstrikes kill at least seven civilians when they hit houses in the Al Jawf Governorate of Yemen. The Houthis report nine killed, two of them children. The coalition did not immediately comment on the airstrike.
A Saudi-led coalition begins a military offensive against the Houthis, carrying out airstrikes against the capital Sanaa, as well as the provinces of Ma'rib, al-Jawf, al-Bayda, Hajjah and Saada. The local news describe the airstrikes as "violent" and reports scores of people injured. This is said to be the response to missile and drone attacks against Saudi Arabia from last week to last night.
Saudi Arabian-led coalition troops seize control of Hodeida International Airport in Al Hudaydah after driving out Houthi forces. Around 280 people have been reportedly killed in the last four days inside the Yemeni port city.
Saudi-led coalition forces begin an operation to take control of the Yemeni port city of Al Hudaydah, which has been held by the Houthis since 2015.
Houthi shelling kills seven in Saudi Arabia; nine Yemenis from one family die from an airstrike by the Saudi-led alliance.
Two senior army commanders from Saudi Arabia and the United Arab Emirates are killed, including the head of the Saudi Special forces in Yemen, Colonel Abdullah al-Sahyan, along with dozens of Saudi, Emirati, Sudanese and Yemeni coalition troops in an OTR-21 Tochka missile attack by Houthi forces on their base near the city of Taiz in southwestern Yemen.
Yemeni military forces backed by Houthi militants reportedly destroy a Saudi warship off the coast of Yemen in the strategic Bab-el-Mandeb strait.
Houthi militants backed by allied Yemeni troops seize control of four Saudi military bases in Saudi Arabia's southern 'Asir Region according to Yemeni state TV.
Cross-border fire from Yemen kills a Saudi general making him the highest-ranking officer to be killed in border attacks by Houthi militants since March.
Heavy Saudi Arabian air strikes hit southern Yemen resulting in at least 20 deaths of Houthis soldiers and two members of rival militias. Saudi Arabia claims to have killed 500 rebels since the start of military operations in March.


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Educated
A gripping, heartbreaking memoir of a woman who, against all odds, overcomes immense family obstacles to gain an education, opening her eyes to a world she never knew existed.
⭐⭐⭐⭐
SUMMARY TARA WESTOVER never went to school, never saw a doctor and did not have a birth certificate. Her parents were Idaho survivalists, and wanted nothing to do with the government, schools or hospitals. She and her six brothers and one sister lived off the land. Her mother was a midwife and healer and treated every family malady—cuts, burns, broken bones, and head trauma— with herbs and oils. At age 10, Tara is put to work savaging scrap metal from her father's junkyard, a dangerous job with no consideration to safety. When one of Tara's older brother becomes physical and mentally abusive to her, her parents turn a blind eye. At fifteen, Tara begins educating herself. She learned enough math and grammar to pass the ACT and be admitted to Brigham Young University at the age of 17. There she studied history and learned of events such as the civil right movement and the Holocaust for the first time. From Brigham Young her quest continued at Cambridge and Harvard, ultimately earning a PhD at the age of 27. Throughout her education Tara Westover experiences tremendous conflict between the awareness she gained from her education and her loyalty to her family.
REVIEW My experience in reading EDUCATED was not without its own conflicts. This hard to forget story is both maddening and heartbreaking. It is both gripping and difficult to read. I wanted to reach out and shake Tara out of her silence of the torment and abuse she suffered. I wanted to put my arm around her and give her the confidence to yell and scream at those holding her down. I wanted to tell her to get out, and not to go back home again. She touched me with this book, and I hope it will be the salve she needs to heal. Perhaps she has finally found her voice. It's truly amazing what Tara has been able to accomplish. My hat is off to her. I hope that someday she realizes the fault is not and was never with her.
A father is suppose to protect and keep his children out of harms way, Tara's did not. A mother is supposed to love and educate her children, Tara's did not. A big brother is suppose to look out for his little sister. Tara's did not. But now she's educated and hopefully will break the cycle of abuse, denial and most of all, silence.
Thanks to Netgalley, Random House and Tara Westover for a copy of this book in exchange for an honest review. Book Published February 20, 2018.
#netgalley,#randomhouse#2018 netgalley#2018#4 star review#tarawestover#bookreviews#books#bluestockingreviews#bluestocking woman#times up#silence breakers
Reef Road
The Silent Sister


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Proposed Duboce Triangle Historic District: FAQ
Would the historic district come with extra cost or requirements for property owners who wish to renovate or expand their properties?
No. Age-eligible historic properties already undergo a certain level of scrutiny when changes are proposed (for example, street-facing windows must be wood, and must look similar to historic windows) and these requirements do not change for properties in a state historic district. No additional requirements or costs are added when a property is recognized as historic by the state.
In fact, the survey process will result in savings of both time and money for property owners pursuing major upgrades or renovations. Age-eligible properties are required to undergo historic surveying before major work can be approved, and so the survey work DTNA proposes to do would eliminate the time and money which property owners would otherwise need to spend on this work.
Would there be any impact on property taxes?
No. The type of historic district being proposed (listing in the California Registry of Historic Places) does not raise or lower property taxes. Owners of historic properties who agree to maintain them in historic condition may be eligible to enter a Mills Act contract with the city in exchange for reduced property taxes, however this would be something individual property owners would need to pursue, and is not part of what DTNA is proposing.
Would a historic district guarantee that the neighborhood would never change?
No. But it's likely that development projects which propose to demolish state-recognized historic properties would undergo additional scrutiny, and approval for demolition of historic properties would be more difficult to obtain. New development would be more likely to occur at sites where no historic buildings are present.
Is DTNA trying to prevent new development in Duboce Triangle?
Not at all! DTNA has supported hundreds of new residential units in our neighborhood, and will continue to support good development projects at appropriate sites, and will continue to push for maximum affordability on-site. But there are plenty of opportunities for new development in SF which do not require destruction of our historic resources. Our neighborhood's unique history should be preserved and treasured.
How likely is it that a Duboce Triangle Historic District will be approved by the state of California?
Assuming we are able to fund the survey process and completion of the historic context statement, it's almost certain that we will gain state recognition of our historic district. The neighborhood was already identified as an eligible district, so it's just a matter of doing (and paying) for the work to take it over the finish line.


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- - https://cultureoflife.org -
What Constitutes an "Undue Burden" on a Woman's Right to Abortion?
Posted By Margaret Datiles Watts, J.D. On September 24, 2015 @ 9:31 am In Issue Briefs,Life | No Comments
The most significant abortion case in a quarter of a century will likely be heard by the Supreme Court in the coming year. The case, Whole Woman's Health, et al. v. Cole, et al., would potentially set forth with greater specificity than ever before what kinds of abortion laws constitute an "undue burden" on a woman's right to abortion. The case is an appeal of a Fifth Circuit Court of Appeals' decision upholding the constitutionality of a Texas abortion clinic regulation which requires abortion clinics to meet basic medical standards for surgical centers. In June, by a slim 5-4 vote, the Supreme Court issued a temporary stay on the law, barring its enforcement while it deliberates on the next course of action. The Justices are scheduled to decide whether to take on the case on October 9th.
The portions of the Texas law being challenged are: (1) a requirement that abortion providers have admitting privileges at a hospital within 30 miles, in case of complications requiring emergency hospital care; and (2) a requirement that abortion clinics comply with the same minimal health and safety standards as other outpatient surgical centers. Examples of these basic standards are requirements for:
Appropriate physician qualification;
Immediate transfer of a patient to a hospital in cases of emergency complications;
Fire alarm systems and elevator doors large enough to accommodate a stretcher; and
Obtaining the informed consent of the patient.
(For a full summary of the Texas law, click here [1].)
An "Undue Burden" on a Woman's Right to Abortion
In Planned Parenthood v. Casey (1992), the Supreme Court reaffirmed the right to abortion established in Roe v. Wade, and it also created a new standard for analyzing abortion laws. Casey ruled that an abortion law is invalid if it constitutes an "undue burden" on a woman's right to abortion: "An undue burden exists, and therefore a provision of law is invalid, if its purpose or effect is to place a substantial obstacle in the path of a woman seeking an abortion before the fetus attains viability."
However, the Supreme Court in Casey also held that state legislatures may regulate abortion clinics to protect maternal health, and that certain such protective laws are constitutional and valid: "As with any medical procedure, the State may enact regulations to further the health or safety of a woman seeking an abortion."
Casey also reaffirmed Roe's recognition of the State's "important and legitimate interests in preserving and protecting the health of the pregnant woman [and] in protecting the potentiality of human life." Indeed, Roe specifically stated that the State's authority to regulate abortion clinics to protect women's health during an abortion "obviously extends at least to the performing physician and his staff, to the facilities involved, to the availability of after-care, and to adequate provision for any complication or emergency that may arise."
Considering these statements in Roe and Casey, it appears that the Texas law falls squarely into the parameters set by the Supreme Court for valid, constitutional abortion laws protecting women's health, and does not constitute an undue burden. This is why the Fifth Circuit upheld the Texas law.
The point of fetal viability is also part of the undue burden analysis. Under Casey, the undue burden analysis is implicated prior to viability. Thus, any future conversation by the Supreme Court on the undue burden standard should involve some kind of discussion on the point of viability, especially considering recent medical advances which have revealed more about human development in utero than was known 23 years ago, when Casey was decided in 1992. That said, it is not unreasonable to predict that the Court might re-consider the point of legal fetal viability.
A "Due" Burden?
Casey distinguished that, although an abortion law may be a burden, it may be a due burden: "Not all burdens on the right to decide whether to terminate a pregnancy will be undue."
For example, the Texas law requiring a physician to be trained in the abortion procedures he/she performs on women is a "burden" on abortion, but it is a due burden. The fact that something is a burden does not in itself make it illegal or unnecessary.
The Texas law does not impose impossible, unheard-of standards unprecedented in medicine; rather, it simply extends existing regulations for outpatient surgical clinics to abortion clinics. The law does not require clinics to be turned into "mini-hospitals" as some have argued. Texas abortion clinics were not previously required to meet medical industry safety standards, and the new law simply brings them under the umbrella of existing clinic regulations appropriate for the type of procedure performed. The laws regulating out-patient surgical centers are specifically tailored to serve the needs of patients undergoing same-day procedures. Abortion is an out-patient, same-day procedure. Therefore, it is reasonable for abortion clinics to be held to the same basic medical standards as other outpatient surgical centers.
If the Supreme Court decides not to hear the case, the Fifth Circuit decision upholding the Texas law will stand, and other states can confidently enact similar laws protecting women's health. However, a Supreme Court ruling overturning the Texas law could drastically alter the landscape of American abortion jurisprudence. In short, a case of this potential magnitude is worth watching closely.
Article printed from : https://cultureoflife.org
URL to article: https://cultureoflife.org/2015/09/24/what-constitutes-an-undue-burden-on-a-womans-right-to-abortion/
[1] click here: http://www.culture-of-life.org/2015/07/28/should-abortion-clinics-be-held-to-basic-surgical-center-regulations/
Copyright © 2015 Culture of Life Foundation. All rights reserved.


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Asbestos Disposal and Removal
Asbestos is a mineral that was commonly used as a building material due to its structural advantages. Its use was phased out in building products through the 1980s and completely banned in 2003 due to its link to disease.
Some structures and much of the housing stock within the Shire was built during a time where asbestos was readily available and commonly used.
Important note: If you are buying a pre-1990 house, it is advised you ask that asbestos containing products be assessed as part of the building inspection report.
Structures containing asbestos are aged and reaching the end of their useful life, complete removal is preferred over maintenance and repair.
Asbestos health risks
In most cases, the risk of developing disease from exposure to asbestos products while in the community is very low. Health effects normally only result from extended exposures through work activities. Asbestos-related diseases are increasingly occurring among building maintenance workers such as carpenters, electricians and plumbers, and also members of the public undertaking home renovations personally.
For further information about asbestos disease risk refer to the Department of Health website.
Important note: Removal of over 10 square metres of asbestos must be done by a licensed and trained individual or business.
It is safe for residents to undertake light removal and disposal work so long as proper precautions are followed.
Where asbestos removal proves difficult or risky, it is highly recommended that you contact a professional with an appropriate licence.
If you decide to remove asbestos from your home yourself it is important that you comply with the Health (Asbestos) Regulations 1992.
Safe asbestos removal
Wet everything down to reduce the number of airborne fibres
Use manual tools or tools with dust suppression and extraction capabilities
Use vacuum equipment designed to extract asbestos fibres or sweep up any dust residues after thoroughly wetting down
Wear protective equipment including a respirator (P1 or P2 type), disposable coveralls, safety glasses and disposable gloves
Inform neighbouring properties on the scope of the works.
Unsafe asbestos removal – what to avoid
Use of high-pressure water hoses or compressed air to clean asbestos sheets
Breaking or damaging asbestos materials
Using power tools on asbestos materials.
All asbestos cement products can be removed safely without causing a risk to the public or workers, provided safe work procedures are followed.
When preparing asbestos for disposal:
Separate asbestos-containing products from other material for disposal
Wrap or contain asbestos-containing products with heavy duty plastic so that fibres are not released when being transported
Label or mark the package with the words 'CAUTION ASBESTOS' (or similar) in letters no less than 50 millimetres high
Dispose of the material at a site licensed to accept asbestos waste, ensure you inform the facility before delivery.
The closest facility that can accept asbestos waste is Red Hill Waste Management Facility. Ensure you contact the waste facility prior to visit.
Asbestos soil contamination
Soil contamination of asbestos has become a growing issue throughout Western Australia as a result of inappropriate removal, structural degradation, legacy dumping or the use of contaminated soil.
The presence of undisturbed asbestos material in soil does not directly translate to an increased risk to health. Remediation of the soil contamination should have an appropriate consideration to air borne fibres.
Low-risk soil remediation
Where the asbestos cement material is sitting on the soil surface as pieces or sheets and not crumbling, it is safe and acceptable for individuals to hand pick the material in line with the measures previously described.
Reporting asbestos soil contamination
In circumstances where there is a significant amount of buried asbestos, contact the Shire's Health Services if there is a significant amount of material that:
Is or may become buried
May become further damaged.
In more complicated cases the site may need to be reported to Department of Environment Regulation under the Contaminated Sites Act 2003.
More information and contacts
For more information about asbestos, please contact the Shire's Health Services on 9290 6742 or email shire@mundaring.wa.gov.au.


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using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace _17.Currency_Converter
{
    class Currency_Converter
    {
        static void Currency_Converter(double value, string InputCurrency, string outpurCureency)
        {
            var lev = 1.0;
            var dollar = 1.79549;
            var euro = 2.53405;

        }
    }
}


In 1997 NEB was gaining a reputation for quality construction in the Grand Prix race boat market. At the same time, American designer, Bill Tripp was making his mark under the IMS handicap rating rule in a fast developing group of competitive yachts from 40-60 ft. NEB and Tripp joined forces to produce a custom fractional rig 43 footer for a New England based client. Now after 16 years of racing on the US East coast and some years of fresh water sailing on the Great Lakes, RAVEN has returned to her birthplace in Portsmouth, RI. The Kevlar composite hull has stood the test of time so with a fresh coat of paint the yacht will be racing again in the summer of 2013 on New England waters.
Plans for this first season do not yet include any modification or upgrades by NEB. Other than a new topsides paint job, the only change will be the boats new name, SETTLER. The previous SETTLER, a Peterson 42, underwent several successful modifications over the years by NEB. This latest SETTLER will likely undergo race mod's or upgrades in the future but for now plans are to sail the boat as is in order to get a feel for where she stands with her competition and how NEB enhancements might improve performance going forward.
Follow NEB News updates to hear more about NEB upgrades or modifications to help SETTLER and other race boats looking for an edge in performance!