text:
\section{Introduction}\quad
Ever since the pioneering work of Lord Kelvin \cite{Kelvin}, there has been a fascination with the study of knotted and linked vortex filaments.
Recent research on this topic covers many areas including
hydrodynamics \cite{KI}, optics \cite{Den}, liquid crystals \cite{AS2}, relativistic field theory \cite{FN}, atomic condensates \cite{Bar} and ferromagnetic materials \cite{Sut}. The host system of interest in the present paper is that of an excitable medium, specifically the FitzHugh-Nagumo medium \cite{FH,Nag}, that is used to model electrical waves in cardiac tissue \cite{Kog} and neurons \cite{Izh}.

The FitzHugh-Nagumo equation is a nonlinear partial differential equation of reaction-diffusion type, so the evolution of
spiral wave vortex filaments takes place within the realm of non-equilibrium dynamics. It is therefore perhaps surprising that several decades ago it was hypothesized that knotted or linked vortex filaments in this excitable medium might preserve topology, as a result of short-range repulsive interactions between filaments \cite{Winfree:Nature:1984}.
Numerical evidence for this conjecture has been found by direct long-term evolution \cite{Sutcliffe:PRE:2003} and
by complex untangling dynamics of vortex filaments without untying \cite{Sutcliffe:PRL:2016}.

The preservation of topology alone is not sufficient to conclude that a steady regular conformation of the filament emerges at late times. Indeed the generic case for knots is a permanently irregular (though topology-preserving) evolution in the bulk of the medium, but it has been found that torus knot filaments can be stabilized by their interaction with the medium boundary, where they settle to particular conformations that appear to be stable attractors of the confined system \cite{Sutcliffe:PRE:2017}. The twist of a filament, which is an automatic consequence of filament linking or threading \cite{winfree:physD:1985}, plays a crucial role in the evolution.
It has been shown that linked vortex rings can have enhanced stability compared
to their unlinked counterparts, with the linking inherently yielding a non-vanishing total twist of the ring \cite{Winfree:SIAM:1990, Winfree:Nature:1994}.
Furthermore, twist can mediate long-range interactions between vortex rings \cite{Sutcliffe:JPhysA:2018}. This suggests that there might be differences in the dynamics of knots and links due to the fact that local twist can only be redistributed around each component of a link and cannot be shifted between different components.

In this paper, we systematically study the dynamics of filament links, including all torus links with no more than 12 crossings. First, we show that all these torus links can be stabilized by the boundary of the medium, in analogy to the situation for torus knots. We introduce new definitions of position and size for a filament link in an excitable medium via moments of a vorticity. Not only is our definition of size easier to compute than the length of a link, but we observe that torus links acquire a minimal size that has a simple dependence on the crossing number of the link and varies with the number of components. For torus links evolving in the bulk, we find that there is a time interval over which the filament motion is regular, in the sense that the size, velocity and rotation frequency all remain approximately constant (when averaged over a spiral vortex period). We find a simple fit for these quantities as a function of crossing number, but the result depends upon the number of components of the link.

The evolution of non-torus links is much more involved. The generic outcome is a complex irregular tumbling dynamics for filaments that typically leads to filaments breaking at the boundary of the medium.
However, there are examples where a confined medium can replace this type of dynamics by a regular motion that remains close to the medium boundary, as demonstrated by the example of the Borromean rings. Finally, we turn our attention to the collision of a pair of initially well-separated torus links. We consider the simplest case of a pair of Hopf links and show that if the impact parameter is zero then the pair settle into a persistent mutual locked configuration but otherwise the evolution is irregular as the pair wrestle and tumble around each other, leading to a variety of different final outcomes. The collision of a pair of links with different crossing numbers produces a wrestling between two unequal opponents and the link with the smallest crossing number can eventually dominate and push the other link into the boundary of the medium.

\section{The FitzHugh-Nagumo equation and filaments}\label{sec:FN}\quad
The FitzHugh-Nagumo equation \cite{FH,Nag} is the archetype of a mathematical model of an excitable medium and is used as an action potential description of cardiac dynamics \cite{Kog} and neuron activity \cite{Izh}. It is a two-component reaction diffusion equation given by
\begin{align}
\frac{\partial u}{\partial t}&=\frac{1}{\epsilon}(u-\frac{1}{3}u^3-v)+\nabla^2 u,\\
\quad
\frac{\partial v}{\partial t}&=\epsilon(u+\beta-\gamma v),
\label{eq:FHN}
\end{align}
where $u({\bf r},t)$, represents the electric potential and
$v({\bf r},t)$ is the recovery variable, both being real-valued physical fields defined throughout the three-dimensional medium with spatial coordinate ${\bf r}=(x,y,z)$ and time $t$. The constants $\epsilon,\beta,\gamma$ control the properties of the medium and the behaviour of the vortex filaments that it hosts. In this paper we choose the parameter set $\epsilon=0.3, \ \beta=0.7, \ \gamma=0.5$,
to ensure a positive filament tension for untwisted filaments with small curvature and to eliminate vortex meander \cite{Winfree}.

A two-dimensional medium with these parameter values supports a rotating spiral wave vortex \cite{Winfree} with a period $T=11.2$ and $u$ and $v$ wavefronts in the form of an involute spiral with a wavelength $\lambda=21.3$. The typical dimensions of our medium will be of the order of several wavelengths and when we refer to long-term dynamics we mean timescales of hundreds to thousands of periods. A vortex filament is a structure in a three-dimensional medium whose local two-dimensional cross-section normal to the filament contains a spiral wave vortex. A straight filament is the simplest example, in which the two-dimensional vortex is embedded by assuming independence in the third dimension, but then the filament must end on the boundary of the medium. The most elementary example of a vortex filament that is contained entirely within the medium is a vortex ring, where the filament is a circle. We shall briefly review this well-studied case in the following section, to introduce some useful concepts, before moving on to our main topic of multiple linked filaments.

To analyze and visualize filaments it is helpful to introduce the quantity
\begin{equation}
 {\bm B} = \mbox{\boldmath $\nabla$} u \times \mbox{\boldmath $\nabla$} v,
\end{equation}
which we refer to as vorticity, because $|{\bm B}|$ is highly localized
around the vortex filament due to the fact that
in a two-dimensional vortex it
is maximal at the centre of the vortex \cite{Win3}.
Filaments are visualized by plotting tubes given by the isosurface $|{\bm B}|=0.1$, so that the centrelines of these tubes are the vortex filaments.
We illustrate filament twist, as in \cite{Sutcliffe:JPhysA:2018}, by colouring the isosurface by the value of the phase
\begin{equation}
 \varphi = \tan^{-1}\left(\frac{2(v+0.4)}{u+0.4}\right).
 \label{eq:angle}
\end{equation}

For filament links (or knots) we define the centre of mass and a concept of size by calculating moments of the vorticity, as follows. For any vector (or scalar) field ${\bf w}$, given throughout the medium, we define the filament averaged value to be
\begin{equation}
\langle{\bf w}\rangle=\bigg(\int_\Omega {\bf w} |{\bm B}|^4 \,d^3r\bigg)
\bigg(\int_\Omega |{\bm B}|^4 \,d^3r\bigg)^{-1},
\label{average}
\end{equation}
where $\Omega$ is the domain of the medium. The position of a link is defined to be ${\bf r}_o=\langle {\bf r} \rangle$ and hence its velocity is
$\dot{\bf r}_o$, where dot denotes differentiation with respect to time.
As all the links we consider have a non-zero velocity,
we can define the projector ${\cal P}$, that projects any vector field ${\bf w}$ onto the plane perpendicular to the link velocity, by
\begin{equation}
   {\cal P}{\bf w}={\bf w}-\frac{\dot {\bf r}_o}{|\dot {\bf r}_o|^2}({\bf w}\cdot \dot {\bf r}_o).
   \end{equation}
   We  define the size of a link, $s$, as a measure of the transverse extent of the link, using the formula
   \begin{equation}
   s=\sqrt{\langle |{\cal P}({\bf r}-{\bf r}_o)|^2\rangle}.
   \end{equation}
   In the simple case that the filament is a vortex ring, the size should be a good approximation to the radius of the ring, and indeed it is, as we demonstrate in the next section. In fact this requirement is the motivation for choosing the quartic average in (\ref{average}) rather than the more obvious quadratic average. Not only is the size easier to compute than the length of a link, but it will turn out that it correlates very well with the crossing number of the link, thereby providing a useful connection between topology and conformation.

 An important aspect of filament dynamics is that filaments are
 continuously emitting wavefronts that can slap filaments and drive their motion \cite{Winfree:book:2002}. Some insight into this phenomenon can be found by considering the time evolution of the vorticity,
\begin{equation}
  \frac{\partial {\bm B}}{\partial t}=\frac{(1-\epsilon^2\gamma-u^2)}{\epsilon}{\bm B}+(\nabla^2 \mbox{\boldmath $\nabla$} u)\times\mbox{\boldmath $\nabla$} v.
   \label{eq:B}
\end{equation}
Ignoring the higher derivative term, we see that the interaction between the
excitation wavefront in $u$ and the vorticity is generated by the nonlinear term  proportional to $u^2{\bm B}$.
Note that, in principle, the range of this interaction is limited only by the size of the medium. However, as excitation waves annihilate on collision with each other, in practice the range of the interaction depends in a complicated manner of the location of the collision interface. In the simplest case of a pair of parallel straight filaments that are close together, this interaction typically gives rise to the separation of the filaments to a certain distance, at which the the filaments then remain. However, even in the simplest linked case of vortex rings threaded by straight filaments, it has been found that twist mediated interactions can produce complicated dynamics that can even reverse the direction of motion of a ring \cite{Sutcliffe:JPhysA:2018}.
The dynamics of linked filaments, with multiple twisted components, is expected to be even more complicated.

Torus links are classified by a pair of integers $p\ge q>1$ that are not coprime. We denote such a torus link by the notation $\mathrm{TL}(p,q)$, where the two integers count the number of times that the link winds around the poloidal and toroidal directions of the torus on which it may be inscribed. The (minimal) crossing number of the link is $N=p(q-1)$ and the number of components is the greatest common divisor of $p$ and $q$. Although a knot is technically a one-component link, in this paper we shall reserve the use of the term link to refer to only those with multiple components, in order to distinguish filament links from the filament knots that have been studied previously \cite{Sutcliffe:PRE:2017}.
There are 9 torus links with crossing number $N\le 12$ and these are depicted
in~\reffig{fig:TL}, where they are ordered by increasing crossing number and colours are used to distinguish the different components of a given link. The link with two crossings, $\mathrm{TL}(2,2)$, is known as the Hopf link and the link with four crossings, $\mathrm{TL}(4,2)$, is confusingly known as Solomon's knot, despite the fact that it is a link and not a knot.
\begin{figure}[h!]
\begin{center}
 \includegraphics[width=1.0\columnwidth]{TLs.png}
\end{center}
 \caption{The 9 torus links with crossing number no greater than 12. The links are ordered by crossing number and the colours distinguish the different components of the link.
 }
\label{fig:TL}
\end{figure}

Initial conditions for torus links in the FitzHugh-Nagumo equation are obtained using Milnor maps \cite{Milnor},
as described in \cite{Sutcliffe:PRE:2003}.
Explicitly, the spatial coordinate ${\bf r}=(x,y,z)$ is mapped to the complex coordinates $(\eta,\zeta)$, that lie on a unit three-sphere in ${\mathbb C}^2$, via
\begin{equation}
(\eta,\zeta)=\frac{1}{x^2+y^2+z^2+L^2}\big(2L(x+iy),x^2+y^2+z^2-L^2+2iLz\big),
\end{equation}
where the real constant $L$ is positive and simply sets the length scale of the initial link. Initial fields for the torus link $\mathrm{TL}(p,q)$ are given by
\begin{equation}
u({\bf r},0)=2\Re(\eta^p+\zeta^q)-0.4,\qquad\qquad
v({\bf r},0)=\Im(\eta^p+\zeta^q)-0.4,
\end{equation}
where $\Re$ and $\Im$ denote the real and imaginary parts.
This creates a torus link with an initial position on the $z$-axis, ${\bf r}_o=(0,0,z_o)$, and an initial velocity aligned with this axis, $\dot{\bf r}_o=(0,0,\dot z_o)$.

To provide initial conditions for any non-torus link we apply the method introduced in \cite{Sutcliffe:PRL:2016}. This involves specifying discrete linear segments for each component of the link and computing the vector field defined by a
Biot-Savart integral along the discretized link. The initial fields $u({\bf r},0)$ and $v({\bf r},0)$ are then obtained from the value of the scalar potential of the vector field, calculated as a line integral. Note that the same method can also be used to provide initial conditions for torus links, but the approach using Milnor maps is more convenient.

We employ parallel numerical computations to calculate solutions of the FitzHugh-Nagumo equation (\ref{eq:FHN}) using standard methods, with time evolution performed using a fourth-order Runge-Kutta method with timestep $\Delta t=0.1$ and spatial derivatives calculated using the discrete cosine transform with a lattice spacing $\Delta x=0.5$. The shape of the medium is a cuboid, with faces normal to the coordinate axes, on which no-flux (Neumann) boundary conditions are imposed.

\section{Dynamics of torus links}\quad
Before we turn our attention to torus links, we first illustrate our use of vorticity moments to determine size and position by considering the well-studied case of a single vortex ring.
A motivation for introducing our new definition of the size of a filament link is to provide a simple method to interpret the data, without the need to resort to more involved computations, such as computing the length of a filament via a discrete Fourier transform and interpolation \cite{Sutcliffe:PRE:2017}.
\begin{figure}[ht!]
\begin{center}
 \includegraphics[width=0.6\columnwidth]{unknot_radius_and_z_other_coarse.pdf}
\end{center}
 \caption{The dynamics of the square of the radius (upper image) and the position along the symmetry axis (lower image) of a single vortex ring.
   The black dots represent numerical data computed using the more traditional method of interpolation and the red circles use the new method of vorticity moments. It can be seen that the results from these two methods agree and match very well to the oscillating curves that are fits to the numerical data.
The lines are the analytic approximations for the period-averaged values. There are two regimes, one where the
 radius shrinks and one where the radius remains constant. The speed of the ring is constant in the regime where the radius is constant. Insets show magnified regions of the data and fits.
  }
\label{fig:unknots}
\end{figure}

 \begin{figure}[h!]
  \begin{center}
 \includegraphics[width=0.9\columnwidth]{hopf_link.png}
 \caption{The dynamics of a Hopf link. The upper images display the evolution of the filament of the link, coloured by the phase $\varphi$ to illustrate twist.
The lower images are reproductions of the upper images but also show wavefronts given by $u=0$ isosurfaces coloured by the value of the phase, with a quarter of each plot removed to aid visualization.
The waves emitted from both components of the link yields a mutual slapping
and the overall result is a rotation and translation of the link.
 }
 \label{fig:Hopf_self_propelled}
 \end{center}
 \end{figure}
 \begin{figure}[h!]
\begin{center}
 \includegraphics[width=1.0\columnwidth]{torus_links}
\end{center}
\caption{The size and motion of torus links. (a) the position $z_o$ along the symmetry axis as a function of time for all 6 of the 2-component torus links with crossing number $N\le 12.$ \quad
  (b) the size $s$ as a function of time for all 9 torus links with crossing number $N\le 12.$ Black, blue and red curves denote links with 2, 3 and 4-components.
  (c) the speed $\dot z_o$, during regular bulk motion, as a function of crossing number $N$ for all 6 of the 2-component torus links with $N\le 12$, plus $N=20$ (black crosses) and the vortex ring (red circle). The curve is the fit (\ref{fitspeed}).
  (d) The final size of the links at the medium boundary as a function of crossing number $N$. Black crosses correspond to all 6 of the 2-component torus links with $N\le 12$ plus $N=20$. Blue and red crosses denote torus links with 3 and 4 components and the red circle is the vortex ring. The curve is the fit (\ref{fitsize}) for the 2-component links.
  }
\label{fig:torus_links}
 \end{figure}
\begin{figure}[h!]
\begin{center}
 \includegraphics[width=0.7\columnwidth]{period}
\end{center}
 \caption{The rotation period for torus links at the medium boundary, (a) as a function of the crossing number $N$, (b) as a function of the final size $s_{\rm f}$ of the link. Black, blue and red crosses correspond to links with 2, 3 and 4 components respectively. The curve in (a) is the fit (\ref{fitperiod}) and the the curve in (b) is obtained by combining the fit (\ref{fitperiod}) with the fit (\ref{fitsize}).
 }
\label{fig:period}
\end{figure}

It is been derived analytically and demonstrated numerically that in the case of positive filament tension the period-averaged radius $R(t)$ of an initially large circular vortex ring shrinks according to the formula $R^2(t)=R^2(0)-2\tau t$,
where the positive constant $\tau$ is by definition the filament tension
\cite{Keener:PhysicaD:1988,Biktashev:1994}. As it shrinks, the vortex ring propagates along its symmetry axis. Once the ring attains its minimal radius the evolution enters a second regime where both the
period-averaged radius and the period-averaged speed remain constant.
The black dots in \reffig{fig:unknots} reproduce the data from \cite{Sutcliffe:JPhysA:2018}, that uses a standard interpolation method to locate the filament and compute its position and length. The oscillating curves are numerical fits to this data and the lines are the analytic approximations for the period-averaged values. The vortex ring is initialized so that the $z$-axis is the axis of symmetry and the position refers to the centre of mass along this axis.  The red circles in \reffig{fig:unknots} denote the position ${\bf r}_o=(0,0,z_o)$ and square of the size $s^2$ computed using the definitions in the previous section from the moments of the vorticity. It is clear that the new method produces results that are in excellent agreement with the more traditional computation. Given the ease of calculating vorticity moments, we shall use this method in our analysis of links. As mentioned earlier, the excellent agreement between the results for the two methods is the motivation for choosing the quartic average in (\ref{average}).

 Using the initial conditions described in the previous section, we make a systematic study of the dynamics of all torus links with crossing number no greater than 12. The simplest example, the Hopf link, is presented in \reffig{fig:Hopf_self_propelled}, where the upper images display the evolution of the link filament and the lower images also include wavefronts given by $u=0$ isosurfaces.
 Filaments and wavefronts are coloured according to the value of the phase $\varphi$ in order to illustrate twist. This colouring confirms that each component of the link is twisted through one total revolution, in agreement with the general theorem \cite{winfree:physD:1985} that for each component of a link the total twist (defined with an appropriate sign convention) is equal to the sum of its linking numbers with all components. The twisting of each filament results in a time lag for wave emission at different points along the filament and the overall motion of the link is a translation along the $z$-axis accompanied by a rotation around this axis.

We have found a similar evolution for all torus links with crossing number up to 12, with all displaying a rigid rotation and translation with a well-defined minimal size, in analogy to previous results on torus knots \cite{Sutcliffe:PRE:2017}. A priori, the same behaviour for knots and links is not guaranteed because links have additional degrees of freedom. In particular, an important difference for links is that once a component develops a local excess of twist, other components cannot participate in reducing and redistributing this twist. This contrasts with the situation for a knot, where the whole length of the filament
can participate in reducing the local twist rate to a minimum.

The results presented in \reffig{fig:torus_links} display detailed information about the motion and size of torus links and how these properties relate to the crossing number $N$ of the link. In \reffig{fig:torus_links}(a) the position $z_o$ along the $z$-axis is shown as a function of time for all 2-component links with $N\le 12$. Similar data has been obtained for the torus links with 3 and 4 components but for clarity these results are not shown on this figure. The dynamics can be described by three different stages. For each link there is an initial transient stage as the filaments of the link are formed and the link can move either up or down the axis in a manner that depends on the details of the initial condition. In the second stage the link has attained its minimal size and travels within the bulk of the medium at an approximately constant speed $\dot z_o$. This stage of the motion emulates the behaviour found in the second regime of motion for a vortex ring, where the ring travels at a constant speed once it has acquired its minimal size. In the third stage the link has settled snugly at the boundary of the medium, given by $z=50$ for $N=2$, $z=40$ for $N=4$, and $z=25$ for $N\ge 6$, to reflect the different speeds of these links. Note that the torus link $\mathrm{TL}(12,2)$ has not quite reached the boundary in the time displayed in \reffig{fig:torus_links}(a). This third stage of the motion has no analogue for the vortex ring, as the ring annihilates with its mirror image on reaching the boundary of the medium.

The evolution of the size $s$ of the link is shown in \reffig{fig:torus_links}(b) for all 9 torus links with $N\le 12,$ with colour coding for the number of components of the link. After the transient first stage, there is a rapid decrease in size to an asymptotic minimal size, $s_{\rm f}$, and this final size is maintained when the link settles at the boundary in the third stage of the motion. The speed during the second phase of the motion is plotted as a function of crossing number $N$ in \reffig{fig:torus_links}(c) for all 6 of the 2-component torus links with $N\le 12$, plus $N=20$ (black crosses) and the vortex ring (red circle).
The curve is a reciprocal linear fit to the data given by
\begin{equation}
\dot z_o=\frac{0.088}{N+0.785}.
\label{fitspeed}
\end{equation}
Note that setting $N=0$ in the above formula does not provide a good approximation for the speed of the vortex ring, suggesting that this formula only applies to 2-component torus links.

The final size $s_{\rm f}$ of each torus link, computed once it has settled at the boundary, is plotted in \reffig{fig:torus_links}(d), as a function of the crossing number $N$. The crosses are colour coded, black for 2-component links, blue for 3-component links and red for the 4-component link. The red circle is the size of the vortex ring. The final size of 2-component torus links displays an almost linear dependence on the crossing number, with the optimal quadratic fit to the data given by
\begin{equation}
s_{\rm f}=4.249+3.151\,N+0.017\,N^2,
\label{fitsize}
\end{equation}
where we note the small coefficient of the quadratic term. This fit is included as the curve in \reffig{fig:torus_links}(d), where we see that it provides an excellent estimate for the final size of 2-component links, but is not applicable to links with more than 2 components. Setting $N=0$ in formula (\ref{fitsize}) does provide a reasonable estimate for the size of the vortex ring, suggesting that the first term in (\ref{fitsize}) might be universal but that the remaining coefficients depend on the number of components of the link.

Once a torus link settles at the boundary of the medium then obviously its speed is zero, however, the link continues to rotate about an axis normal to the boundary and therefore we can compute the period $P$ of this rotation. The results of these calculations are shown by the crosses in \reffig{fig:period}, where we use the same colour coded scheme as earlier to indicate the number of components of the link.  A quadratic fit to the data for 2-component links gives the following approximate formula for the period $P$ as a function of crossing number $N$
\begin{equation}
P= 643+143\,N+3.7\,N^2.
\label{fitperiod}
\end{equation}
In \reffig{fig:period}(a) it can be seen that this approximation also provides a reasonable estimate of the period of the 3-component and 4-component links with $N\le 12$, suggesting that the rotation period may have a universal behaviour that is independent of the number of components of the link. As shown earlier, the size of the link does depend on the number of components, so this implies that there cannot be a component-independent relation between the rotation period and the final size of the link. This is confirmed by the plot in \reffig{fig:period}(b), where the rotation period is shown as a function of the final size and the curve obtained by combining the expressions (\ref{fitsize}) and (\ref{fitperiod}) provides a good estimate for 2-component links, but not for links with more components.

As the rotation period increases with crossing number $N$, we expect that the rotation frequency tends to zero in the asymptotic limit of large $N$.
This can be understood intuitively by observing the dynamics with a fine time resolution, where it becomes clear that the rotation is not smooth but rather can be described as a two steps forward, one step back type of motion, similar to the escapement mechanism of a clock. In other words, the link first rotates in one direction and then rotates in the opposite direction by a smaller amount.
As the crossing number increases the forward and backward steps become more comparable, presumably because there is less difference between the inner and outer parts of the filament as it winds around the torus, and the result is a decrease in the net rotation frequency.

\section{Dynamics of the Borromean rings}\quad
The generic dynamics for links that begin their evolution in the bulk of a medium is an irregular tumbling dynamics that preserves topology but not conformation and eventually leads to filaments breaking at the boundary of the medium. The regular dynamics of torus links presented in the previous section is exceptional behaviour and mirrors that found previously for torus knots \cite{Sutcliffe:PRE:2017}. For both torus knots and torus links, sufficiently symmetric initial conditions produce translational motion and if the knot or link can reach the boundary of the medium before any instabilities have developed then a boundary stabilized configuration with a periodic evolution is obtained.
\begin{figure}[h!]
\begin{center}
 \includegraphics[width=0.9\columnwidth]{boromean_rings.png}
\end{center}
\caption{The same symmetric initial condition for Borromean rings leads to irregular tumbling dynamics in a sufficiently large medium (upper panel) but regular dynamics near the boundary of a more tightly confined medium (lower panel). Note that the total simulation time is much greater for the lower panel and that the scale is slightly different in the two panels because the medium in the upper panel is slightly larger. Eventually the tumbling dynamics leads to the filament breaking (not shown) at the boundary of the medium.
 }
\label{fig:Borromean}
\end{figure}

As we have seen, as the crossing number of a torus link increases then its speed decreases, hence it takes longer to reach the medium boundary and this gives more time for any instability to develop. The computations of the speed of torus links presented in the previous section require a sufficient period of regular
translational motion within the bulk. Even for the torus links with crossing number no greater than 12, we found that the link $\rm{TL}(4,4)$  did not travel a sufficient distance during regular motion to be able to reliably assign a speed. However, links with this type of behaviour can be stabilized by creating the link sufficiently close to the boundary of the medium and this allows the size of the link to be computed.

A striking demonstration of the difference between bulk evolution and boundary stabilized dynamics is provided by considering the Borromean rings: a 3-component non-torus link with crossing number $N=6$. This link is a particularly interesting example because the three components are linked, but not pairwise, thus all the component filaments have zero total twist, unlike any of the torus links.
The initial condition for the Borromean rings is obtained using the method for non-torus links discussed in section \ref{sec:FN}, and the associated filament is shown in the first image in both the upper and lower panels of \reffig{fig:Borromean}.

The evolution shown in the upper panel in \reffig{fig:Borromean} is for a medium with the dimensions $-90\le x,y \le 90$ and $-50\le z\le 80.$ This medium is sufficiently large that bulk evolution ensues. The length of the link initially decreases and although each component has zero total twist, regions of locally large twist rate develop, as evidenced by the colouring of the filament. The filament dynamics becomes increasingly irregular and the link tumbles through the medium until eventually it approaches a boundary of the medium and a filament breaks at this boundary (not shown). Note that up until the point at which the filament breaks at the boundary, the topology is preserved as there are no filament reconnections, despite the irregular motion of each component.

The lower panel in \reffig{fig:Borromean} displays the contrasting evolution using the same initial condition in a slightly smaller and much thinner medium with dimensions
$-80\le x,y \le 80$ and $0\le z\le 30.$ In this case the motion is regular with the link maintaining its size and a conformation that is not quite as symmetric as the initial condition but is similar. This boundary stabilized link rotates in the same manner found earlier for torus links that sit snuggly at the boundary of the medium, and its stability has been tested over thousands of rotation periods of the underlying vortex that forms the filaments.

The boundary stabilization of vortex rings has recently been demonstrated in experiments on chemical excitable media \cite{ATE,TES} that are modelled by reaction diffusion equations with a very similar structure to the FitzHugh-Nagumo equation. It therefore appears that boundary stabilization is a generic phenomenon that applies to vortex rings, knots and links across a range of excitable media.

\section{Colliding links}\quad
In this section we consider the collision of a pair of initially well-separated links moving in opposite directions. As we have already seen, a link can snuggly sit indefinitely at a no-flux boundary of the medium and this is equivalent to the collision of a link in the bulk with its mirror image.
The mirror image of a link in the boundary $z=0$ is obtained by making the replacement $(x,y,z)\mapsto(x,y,-z)$, and this reverses the direction of motion of the link. A less symmetric initial condition for a collision can be initiated by applying the transformation $(x,y,z)\mapsto(x,-y,-z)$ to obtain the second link, rather than using the mirror image. This transformation also reverses the direction of motion of the link but yields a pair of contra-rotating links, instead of the co-rotation that follows from using the mirror image. We shall consider contra-rotating links in all the collisions presented in this section, as this is the less symmetric situation.
This simplest example, a pair of Hopf links, is displayed in \reffig{fig:no_impact_collision}, where the first image shows the links just before the collision and the second image demonstrates the stalemate of mutual blocking that persists after thousands of rotation periods of the constituent vortex cores. Despite the contra-rotation, the links sit snuggly together in a comparable manner to a single link at a no-flux boundary. Similar results are obtained for other torus links and knots in the head-on collision of an identical pair.
\begin{figure}[h!]
\begin{center}
 \includegraphics[width=0.5\columnwidth]{hopf_collision_com}
\end{center}
\caption{The head-on collision of a pair of contra-rotating Hopf links results in a stalemate of mutual blocking.}
\label{fig:no_impact_collision}
\end{figure}
\begin{figure}[h!]
\begin{center}
 \includegraphics[width=\columnwidth]{hopf_collision_impact.png}
 \end{center}
 \caption{The collision of a pair of Hopf links with a non-zero impact parameter produces a complex irregular tumbling evolution, where
 the two links wrestle each other until the filament of one link breaks at the boundary. The upper and lower panels show the filaments at four different times from two different viewing angles.
 }
\label{fig:impact_collision}
\end{figure}
\begin{figure}[h!]
 \includegraphics[width=\columnwidth]{TL22vsTL42.png}
 \caption{The head-on collision of a Hopf link with Solomon's knot.
   The Hopf link pushes Solomon's knot to the boundary and permanently traps it there.
 }
\label{fig:TL22vsTL42}
\end{figure}
\begin{figure}[h!]
 \includegraphics[width=\columnwidth]{hopf_trefoil.png}
 \caption{
   A head-on collision of a Hopf link with a trefoil knot. The upper and lower panels show the filaments at four different times from two different viewing angles. The Hopf link pushes the trefoil to the bottom boundary but the asymmetry of the pair allows the trefoil to escape to the side. After an irregular tumbling motion, where the link and knot wrestle each other and change conformation, the knot is pushed into the boundary and its filament breaks, leaving a Hopf link that settles snugly at the boundary.
 }
\label{fig:hopf_trefoil_collision}
\end{figure}

The introduction of a sufficiently large impact parameter for the collision resolves the stalemate of a head-on collision. The upshot is an irregular tumbling dynamics, where the links wrestle each other until one link dominates by pushing the other into the boundary, where the filament breaks. An example is presented in \reffig{fig:impact_collision}, where the upper and lower panels show the filaments at four different times from two different viewing angles. Eventually, the surviving link comes to rest snuggly at the boundary, forming the boundary stabilized Hopf link that we have seen earlier. The precise details of the evolution depend upon the value of the impact parameter and the survival of only a single link is not always the eventual outcome. Another possible outcome that we have observed is the formation of a threaded ring by the breaking of the filament of one link at the boundary, which coexists with the surviving Hopf link settled at a different region of the boundary.

Next we consider the collision of two links of different types.
The example of the head-on collision of a Hopf link with Solomon's knot is displayed in \reffig{fig:TL22vsTL42}. The Hopf link pushes Solomon's knot to the boundary, where the whole structure permanently resides with an unexpected stability. We attribute the dominance of the Hopf link to its speed advantage over Solomon's knot. A similar phenomenon has recently been observed in the dynamics of threaded vortex rings \cite{Sutcliffe:JPhysA:2018}, where smaller rings, with a higher twist rate and frequency of wave emission, travel faster and can push larger rings, even changing their direction of motion.

Finally, we consider the head-on collision between a link and a knot.
The example with lowest total crossing number is the collision between a Hopf link and a trefoil knot, presented in \reffig{fig:hopf_trefoil_collision}, where the upper and lower panels show the filaments at four different times from two different viewing angles. Once again, the faster moving Hopf link dominates as it pushes the trefoil knot to the bottom boundary. However, even in a head-on collision the asymmetry between the knot and the link  allows the trefoil a means of escape to the side. This generates a period of wrestling between unequal opponents, with irregular tumbling dynamics that results in considerable changes in conformation to both the knot and the link. Eventually, the attempted escape ends in failure as the Hopf link pushes the trefoil knot into a side boundary where its filament breaks. The Hopf link then settles at the boundary, forming the usual boundary stabilized configuration.

\section{Conclusion}\quad
In this paper we have presented the results of parallel numerical computations to systematically investigate the dynamics of torus links in the FitzHugh-Nagumo excitable medium. The results are displayed by using vorticity isosurfaces to identify filaments that are phase-coloured to illustrate twist. Furthermore, by calculating moments of the vorticity, we have analyzed the properties, such as speed, size and rotation frequency, of all torus links with crossing number no greater than 12 and a torus link with crossing number 20 to check asymptotics. We found that these properties fit a simple dependence on crossing number for a fixed number of components of the link.

We have demonstrated that instabilities exist over long timescales for the evolution of links in the bulk, but that these instabilities can be removed by boundary interactions to yield stable torus links, in a manner similar to that found earlier for torus knots \cite{Sutcliffe:PRE:2017}. As an example of a non-torus link, we studied the dynamics of the Borromean rings, which is novel in that the filaments are linked but each has zero total twist. The complexity of the dynamics of bulk evolution of the Borromean rings is beautifully complex, and yet can be regularized by boundary interactions through confinement in a sufficiently tight medium. The fact that filament dynamics is dramatically influenced by the proximity of the medium boundary is of significant relevance to cardiac applications of the FitzHugh-Nagumo model, because in this context the depth of the medium is comparable to a spiral wavelength, so it is certainly evolution within a tight confining medium.

Finally, we considered the collision of initially well-separated links. Head-on collisions lead to regular dynamics with a mutual blocking or trapping of links. However, a non-zero impact parameter produces irregular tumbling dynamics in which links wrestle with each other, while substantially varying their conformation, until one link eventually dominates by pushing the other into the boundary of the medium, where its filament breaks. These results illustrate the remarkable complexity of filament dynamics that occurs on long timescales, with bulk evolution preserving filament topology despite considerable changes in conformation and exotic dynamics. Eventually, regular dynamics prevails once the surviving torus link has settled at the boundary of the medium.

\section*{Acknowledgements}
\noindent
This work is funded by the
Leverhulme Trust Research Programme Grant RP2013-K-009, SPOCK: Scientific Properties Of Complex Knots.

redpajama_set_name: RedPajamaArXiv


According to Soccer Laduma report, Messi needs only 14 more goals to become the all-time record scorer.
Messi netted a hat-trick in his side's 4-0 friendly win over Haiti on Wednesday morning, taking his overall tally for the Albiceleste to 64 goals.
He has now overtake the record of Selecao forward Ronaldo, who retired with 62 strikes for Brazil.
Three-time World Cup Winner, Pele, who sits in seventh spot in the worldwide all-time ranking, is the only South American player to have scored more than Messi. Pele retired with 77 goals for Brazil, which means Messi needs to score 14 more times to overtake the legendary striker.


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text:
Home India Mysuru's GRAAM, ISEC Bengaluru join hands
Mysuru's GRAAM, ISEC Bengaluru join hands
Mysuru's GRAAM, ISEC Bengaluru sign up with hands
MoU signed to work jointly on research study, professors training, jobs and internships for trainees.
MoU signed to work collectively on research study, faculty training, projects and internships for trainees.
Grassroots Research and Advocacy Movement (GRAAM), a public law research study and advocacy institute, in Mysuru has inked a Memorandum of Understanding with the Bengaluru-based Institute of Social and Economic Modification (ISEC), India's premier institute for Interdisciplinary Research and Training in the Social Sciences.
The MoU was signed to work collectively on research, faculty training and research study cooperation, tasks and internships for trainees.
Under the terms of the MoU, both the research organisations have agreed to undertake partnerships in research and development, including exchange of details, publications and training.
Thinking about the growing requirement for inter-institutional collaborations in grassroots research study and abject necessity for enhanced policy-level choice making, both the celebrations involved have accepted deal with joint jobs, provide support in implementing tasks and look for collective funding, encompassing research on all elements of an establishing society within the interdisciplinary structure, a news release stated here.
This partnership is likewise carried out with the intention of constituting faculty training, providing a chance for faculty of both GRAAM and ISEC to go to workshops and management training constituted jointly by both the organisations.
They have actually likewise settled on making up inter-institution access to faculties and trainees to release academic and research study work for mutual knowing and to share worths and contribute towards the general development of society; jointly establish short-term and distance knowing programs for the consumption of public apart from the short-term training through workshops, lecture series, workshops and conferences, the release added.
R. Balasubramaniam, Creator and Chairman, GRAAM; Basavaraju R. Shrestha, Executive Director, GRAAM; S. Madheswaran, Director, ISEC; and Krishnaraj existed throughout the finalizing of the MoU. Others present consisted of K.V. Raju, Economic Consultant to Chief Minister of Uttar Pradesh; M.K. Sridhar (previous Member Secretary of Karnataka Knowledge Commission); P.V. Krishna Bhat, Chancellor, Central University of Odisha, Koraput; and M.V. Nadakarni, Honorary Checking Out Professor, ISEC.
CLICK ON THIS LINK TO READ MORE
( the headline, this story has actually not been released by Essential India News staff and is published from a syndicated feed.).
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Actor Sapna Pabbi on Friday dismissed the reports that she left the country after she was summoned by the Narcotics Control Bureau (NCB), which is probing the drugs case linked to Sushant Singh Rajput's death. There were reports that Pabbi's name cropped up during the interrogation of Arjun Rampal's girlfriend Gabriella Demetriades' brother Agisilaos, who…
Delhi CM Kejriwal To Get In Touch With Union Env Minister On Friday Over New Tech Established By IARI: Officials
Delhi Chief Minister Arvind Kejriwal will meet Union Environment Minister Prakash Javadekar on Friday to discuss a low-cost technology developed by the Indian Agricultural Research Institute here to manage crop residue, officials said. Scientists at the institute have developed "decomposer capsules". Four capsules, some jaggery and chickpea flour are enough to prepare 25 litres of…
Coronavirus India lockdown Day 185 updates | Delhi's COVID-19 fatality rate less than 1% in last 10 days: Satyendar Jain
The Union Health Ministry said on Saturday, India has conducted a total of seven crore COVID-19 tests till now, with the country's testing capacity having crossed 14 lakh tests daily. Those unable to afford a probable COVID-19 vaccine will get it for free and the government is considering a large immunisation programme to make the…

redpajama_set_name: RedPajamaCommonCrawl


In this three-part video podcast, ASHP/CML's Donna Thompson Ray shares the benefit of her area of expertise with New York City Department of Education teachers in a discussion about the work of artist Jacob Lawrence. Eliciting contributions from teachers, Thompson Ray leads a conversation about Lawrence's Migration Series and how educators can, and do, use the images in the series to teach students about the Great Migration. She discusses Lawrence's own experiences with migration and the art of the Harlem Renaissance, how the Great Migration was experienced by those who took part in it, conditions that caused it, and how it affected people's lives. In the process, Thompson Ray models a close reading of select images from the series, demonstrating how teachers might use the work to deepen their students' understanding of the era.
The images shown here are from The Phillips Collection. For more on the Great Migration go to Up South: African-American Migration in the Era of the Great War .


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text:
Dictionaries thesauruses pictures and press releases
Harmonious Blacksmith, The
Updated About encyclopedia.com content Print Article Share Article
Harmonious Blacksmith, The. Nickname for air and variations in Handel's 5th hpd. suite in E in the 1st set of 8 suites (1720). The name was bestowed after Handel's death and has no connection with the circumstances of the work's comp.
The Concise Oxford Dictionary of Music
"Harmonious Blacksmith, The ." The Concise Oxford Dictionary of Music . . Encyclopedia.com. 17 Jan. 2020 <https://www.encyclopedia.com>.
"Harmonious Blacksmith, The ." The Concise Oxford Dictionary of Music . . Encyclopedia.com. (January 17, 2020). https://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/harmonious-blacksmith
"Harmonious Blacksmith, The ." The Concise Oxford Dictionary of Music . . Retrieved January 17, 2020 from Encyclopedia.com: https://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/harmonious-blacksmith


A tax loophole exempting tar sands pipeline operators from paying an eight-cent tax per barrel of oil they transport in the US is costing the federal Oil Spill Liability Trust Fund millions of dollars every year. With expected increases in tar sands oil production over the next five years, this loophole may have deprived US citizens of $400-million dollars worth of critical oil-spill protection funds come 2017.
According to a report by the US Natural Resources Committee the federal government pays for immediate oil-spill response from the Liability Trust Fund which is supported by an excise tax on all crude oil and gas products in the US.
But in 2011 the Internal Revenue Service exempted tar sands oil from the tax, saying the substance did not fit the characterization of crude oil.
This exemption has come under scrutiny this week after Exxon Mobil's Pegasus pipeline ruptured in Mayflower, Arkansas, releasing 300,000 litres of tar sands oil and water into a residential neighbourhood and surrounding wetlands. Because the line carried tar sands-derived oil from Alberta, Exxon was exempt from paying into the spill liability fund for the corrosive fuel's potential cleanup.
The Pegasus pipeline was built in the 1940s to carry regular crude north from the Gulf Coast. In 2006, Exxon reversed the flow of the 1300 kilometre line in order to transport tar sands diluted bitumen from Illinois to the coast.
The Enbridge disaster in Michigan's Kalamazoo river has cost over $820 million, making it the most expensive onshore cleanup in US history.
Enbridge, the company currently vying to build a 1172 kilometre-long pipeline from Alberta to the British Columbia coast, currently has no spill-response plan – for either onshore or offshore spills – prepared for the project.
During cross-examination in the Northern Gateway Pipeline hearings in BC, Enbridge admitted they will have no spill response plan until six months before the proposed tar sands line will begin operation.
An investigation in the Enbridge Kalamazoo disaster found the company – due to "pervasive organizational failures" – improperly respond to the pipeline breach. The US National Transportation Safety Board likened Enbridge employees to the Keystone Kops – a clumsy, incompetent troups of cops from the silent films of the early 20th century.
The estimated cleanup cost for conventional oil run at about $2000 per barrel of oil. Tar sands diluted bitumen cleanup is estimated to cost an average $29,000 per barrel of spilled oil.
The Oil Spill Liability Trust Fund was established in the wake of the Exxon Valdez oil spill, when the enormous costs of oil spill recovery were first understood on a grand scale, to ensure adequate cleanup funds were available to protect local residents and ecosystems.
As Climate Progress reported in 2010, Exxon refused to shoulder the cost of cleanup in the Valdez, where more than 11 million gallons of crude oil contaminated almost 3000 kilometres of shoreline.
"Exxon fought paying damages and appealed court decisions multiple times, and they have still not paid in full. Years of fighting and court appeals on Exxon's part finally concluded with a U.S. Supreme Court decision in 2008 that found that Exxon only had to pay $507.5 million of the original 1994 court decree for $5 billion in punitive damages. And as of 2009, Exxon had paid only $383 million of this $507.5 million to those who sued, stalling on the rest and fighting the $500 million in interest owed to fishermen and other small businesses from more than 12 years of litigation.
And as DeSmogBlog's Ben Jervey reports, Exxon has a long and litigious history of evading cleanup costs.
The US is facing a dramatic increase of tar sands oil imports. As the National Resource Committee estimates, production in the Alberta tar sands is projected to rise to over 2.7 million barrels per day in 2017. The tar sands industry projects figures as high as 5 million barrels per day in 2030 and 6 million per day in 2035. Currently approved projects have the capacity to produce 5.2 million barrels per day.
Regardless of how many barrels of tar sands oil will be traversing US soil, none of them should be exempt from spill liability taxes. If anything, corrosive diluted bitumen should be taxed more for the inherent dangers it present to local ecologies and communities during its production, refining, and transport.


In stock. Order in the next 3 hours, 19 minutes and it will go out today!
Scott Thomson, one of the most respected names in the ever more popular underground world of Cardistry, presents nine of his signature flourishes! From the flowing, pristine Golden Gate to the tight twists & turns of Monaco II to the frankly jaw dropping Merengue, this is contemporary flourishing at it's best.
Heavily influenced by the rhythmic pulse of Scott's other 'life' as a respected member of the UK underground music scenes, the flourishes featured on AURORA are animated, fluid and visually direct - all keystone's of Scott's renowned style.
'Follow along' explanations are provided in four angles. Pick the one that works for you - straight on / over the shoulder / over the shoulder (slo-mo) / over the shoulder & straight on combined.
Scott Thomson is a well known face in the Cardistry world, having spent the last several years producing hit videos for his Profoundmagic Youtube channel. This has proved to be a vibrant hub for the Cardistry community, as demonstrated by the fact that the site is closing in on a million views!!! Scott also created, and continues to run, the ever popular Essential Cardistry channel - Youtube's #1 Cardistry hotspot.


Bradley Johnson Design designed the offices for media company Time Inc located in London, England.
BJD transformed what was a standard cat A office space into a dynamic and quirky workplace for 600 staff, mostly creative content editors for Time Inc͛s lifestyle and home brands. This 45,000 square foot workplace fit-out for the UK's leading publisher of print and online content was completed in December 2016.
The transition from assigned desks on a 1:1 ratio to desk sharing at a 3:4 ratio is a challenge which was met by creating a new level of options in work setting and carefully zoning the space to accommodate different sized groups, various levels of activity and noise, and plenty of space for heads down individual work we were able to create a dynamic new workplace which has had nothing but positive comments from staff and stakeholders alike. There are a multitude of work-settings to inspire and support various activities. Each of the 450 desks are only steps from one of the six unique coffee bars, and a mix of formal and informal meeting areas as well as heads-down quiet work zones. The first-floor features access to a 100 square meter atrium used for everything from town hall meetings to morning yoga classes and lunchtime cafe seating.
We were asked to create a ͚Brooklyn Loft͛ feel in a 1980͛s Canary Wharf office building. We began by stripping back the Cat A finishes; popping out the mineral fibre ceiling tiles but leaving the grid, pulling up the carpet tiles to expose the patinated access floor, and spraying the grid and tiles matte black in the cellular spaces. We layered a variety of finishes over this stripped back base to add color, texture and pattern. The project was completed from brief to build in only three months to the great satisfaction of staff and stakeholders alike.


text:
Album: The Civil Wars – Barton Hollow
by Ian Parker • 7 April 2011
Some voices were meant to go together. And it will only take a few moments of Barton Hollow opener 20 Years for the uninitiated to realise that John Paul White and Joy Williams were born to sing together. Like the modern heirs to Gram Parsons and Emmylou Harris, they play off each other wonderfully, and sound better than any current male/female pairing you can think of. And that silky-sweet harmonising is the perfect sound to convey the simply beauty of their soft alt-country.
Barton Hollow is the result of a long wait, but sounds like perfectly aged wine tastes. Back in 2009, performing only their second ever show, the pair recorded the results and made it available as a free live album via myspace. The result, Live At Eddie's Attic, made it apparent that some very special seeds were instantly bearing fruit. But that does not mean they were about to rush things. There was another year's wait until the Poison & Wine EP and I was beginning to worry they'd split up or something as almost two years passed until Barton Hollow appeared.
But it is here now, and let us simply be grateful it is. This is an album to wrap yourself in. Either close yourself in a dark room with this one, or get the best quality headphones you can lay your hands on and turn it up. The exquisite vocals convey emotionally fragile lyrics, with the highlights the dark, brooding 'Poison & Wine' with it's refrain, "I don't love you but I always will". It's been jazzed up a little from how it first sounded in Eddie's Attic, with a powerful bass and drumming kicking in, albeit turned down in the mix, midway through, but the soaring beauty of the song is unchanged.
The stomping, bluesy title track is a change of pace from what we've heard before, but is not out of place amongst the softer love songs and laments. Instead it forms the centrepiece of a debut from a band who could go far.
Album | Joy Williams – Front Porch
#633 John Paul White - The Martyr
Record Store Day | Shopping Guide Part One
Album | The Civil Wars - The Civil Wars
News | Civil Wars return with new record
Live | The Civil Wars at The Roundhouse, London
News | Civil Wars halt tour
Album | The Chieftains - Voice of Ages
Record Store Day | The FFS shopping guide
News | More Record Store Day releases announced
Tags: The Civil Wars
← Album: Four Quartets – Paragon of Animals
EP: Marcus Foster – Tumble Down →


text:
Inniss: Focus On The Way Forward
by Theresa Blackman | Oct 11, 2016 | Top Stories
Industry Minister, Donville Inniss greets Director General, Office of Trade Negotiations, Ambassador Gail Mathurin at the opening of a two-day Regional Consultation at the Radisson Aquatica Resort while Senior Private Sector Specialist, Trade and Competitiveness Global Practice, World Bank Group, Alicia Stephens looks on. (C.Pitt/BGIS)
Industry Minister, Donville Inniss is urging his regional counterparts to focus on making a difference in driving forward the process for better facilitation of business within the region.
He made these comments this morning at the opening of a two-day Regional Consultation on the CARICOM Strategy for Regional Implementation of the World Trade Organization Agreement on Trade Facilitation, at the Radisson Aquatica Resort.
During his address, Mr. Inniss told attendees that far too often he has heard the cries of legitimate business persons who were frustrated with the tardiness of agencies responsible for the smooth movement of goods.
"This region really needs to adopt an attitude that no one owes us a favour and that we have to get up and earn our way in life in a sustainable manner. Hence, inordinate delays at the airports and sea ports not only frustrate buyers and sellers, but certainly add to the cost of living in this region."
When goods are not released from the Bridgetown Port in an adequate time, he added, someone had to pay the price for such delays.
"When importers have to pay high fees to have containers inspected outside of "normal" business hours, someone has to pay that additional cost. I look forward to the day in this region when normal work hours means that critical businesses operate on a 24/7 basis with reduced overtimes at exorbitant rates and a true shift system. I am afraid that most of our ports in this region are controlled by an antiquated labour group and a style that continues to inject fear sometimes without being reasonable," he charged.
Mr. Inniss contended that "we frustrate, not facilitate, trade", when exporters have to pay workers to stand up with folded arms and watch others load vessels.
He said: "When officers take their cool time to inspect items as a means of protesting new work arrangements, I say we frustrate, not facilitate, trade. When importers follow all known rules and regulations and yet weeks or months later cannot get an explanation as to why goods are being held up, we frustrate, not facilitate, trade.
"When we spend valuable time arguing for greater protectionist measures such as more licences, higher rates of duties and quotas, as opposed to focusing on greater efficiencies, higher standards and areas of comparative advantage, we really are not doing our economies any favours – we are really just legitimising our inefficiencies and incompetence," Mr. Inniss underscored.
The Minister stated that Barbados could sign on to all of the WTO agreements, get World Bank funding and technical support and create new laws and entities.
However, he pointed out that if the country did not adopt a more positive and welcoming attitude in respect of the trade of goods and services in this region, then it would have wasted valuable resources.
Tags: Donville Inniss, Minister of Industry International Business Commerce and Small Business Development, Trade, World Trade Organization, WTO


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An e-mail from the county's Republican Party suggests Bremerton's Keli Carender (Liberty Belle) will be there, along with radio talkers Dori Monson, Kirby Wilbur and David Boze.
There will be chartered bus rides available for $15. The buses leave from Poulsbo, Silverdale and Port Orchard and getting a ride requires a reservation.


text:
Click to copyhttps://apnews.com/a2228f8ea6994e938dd8bc5335239135
Report: $15M sought after body used for medical practice
BELLINGHAM, Wash. (AP) — Relatives of a man whose body was used by the Bellingham Fire Department for intubation practice have filed claims against the city totaling more than $15 million, a newspaper reported.
Eleven fire department employees — including two office workers — acknowledged practicing inserting and removing breathing tubes on the body of Bradley Ginn Sr. while waiting for it to be transported to a funeral home on July 31, The Bellingham Herald reported .
A review by a Seattle attorney determined it had been an accepted practice for department paramedics to meet certification requirements by practicing on patients who had just died, the newspaper said.
The fire department said in October that such "tube checks" were not to be performed without the consent of the patient's next-of-kin.
Bellingham Mayor Kelli Linville declined to comment on the litigation.
Twelve employees were disciplined. One paramedic was suspended for a week without pay. A captain with nearly three decades of experience with the department retired rather than face demotion, and a division chief resigned, the department said.
The state Department of Health is investigating the nine members of the department who participated and were credentialed health care providers.
Records from the Bellingham City Attorney's Office show that Ginn's wife, Jai Ginn, filed a $15 million claim for damages on Oct. 4, saying she had suffered physical and mental pain from the "intentional and negligent care of a body and abuse by city of Bellingham employees."
She also made claims for the unauthorized invasion and desecration of a dead body and possible wrongful death.
One of the claims was filed by Bradley Ginn Jr. and seeks $200,000, the records show. Ginn Jr.'s lawyer said the intubations took place as family members were trying to locate Ginn Sr.'s body.
Ginn Sr. had a do-not-resuscitate directive, which prohibited the use of invasive procedures such as intubation, the records state.
Information from: The Bellingham Herald, http://www.bellinghamherald.com


I recently installed Windows to a partition on an imac. I have to say, it's fairly painless if you just follow the directions provided by the Boot Camp Assistant on OSX. That said, I tend to complicate things by not following the directions explicitly.
The circumstances that lead to this condition are: I ended up missing the time for the keystroke (must be held when apple sound is made) and instead of waiting for OSX to load and restart I pressed the power button. If you do this, there is a chance these keystrokes will stop working. The solution is simple: just turn off the computer and unplug for a while (seconds to minutes). That seemed to fix the problem for me.


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