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Lexus has always been at the forefront of luxury but with the LS, they give luxury a warm and welcoming feel.
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The 2018 model of the XC60 is undergoing a major redesign since the model launched 9 years ago. Volvo claims that this model will be one of the safest models they have ever built. One of the groundbreaking safety features included in this vehicle is Oncoming Lane Mitigation that can help prevent head-on collisions. The car also comes with a 360-degree camera that can give drivers a more complete look at their surrounding areas. This will help drivers when they find themselves in locations with impaired visibility.
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EL2014 | EtherCAT Terminal, 4-channel digital output, 24 V DC, 0.5 A, with diagnostics
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EL2014 | EtherCAT Terminal, 4-channel digital output, 24 V DC, 0.5 A, with diagnostics
The EL2014)
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- Related productsThe 2021 Ford Bronco Sport gets EPA ratings up to 25 mpg city, 28 highway, 26 combined. The off-road ready compact crossover shares many of the same parts as the 2020 Ford Escape, including the 181-horsepower 1.5-liter turbo-3 base engine that earns those ratings. The larger 245-hp 2.0-liter turbo-4 standard in the top Badlands trim gets 21/26/23...\section{INTRODUCTION}\label{intro}

Numerical studies of the Einstein equations promise to deepen dramatically our
knowledge of general relativity, astrophysics, and cosmology.
Shapiro and Teukolsky~\cite{ST86} suggested that the ``Holy Grail of
numerical relativity'' is a code that (i) avoids singularities,
(ii)  handles black holes, (iii)  maintains high accuracy, and
(iv)  is capable of running forever.  These four difficulties in
numerical relativity are closely tied to each other.  Black holes, and within
them spacetime singularities, may be formed after some
evolution, even if none exists initially.  The presence of such objects implies
that
the
dynamical range of the calculation is large, with very
different length and time scales involved.  This makes it
difficult to maintain accuracy or even to keep the code from
crashing in numerical evolution.

The traditional way to deal with this problem is to use the
coordinate degrees of freedom to avoid the ``extreme'' regions.
With ``the many fingers of time'' in relativity, one may evolve other regions
in
space without evolving the region in
which a singularity is about to form.  Many
different types of singularity-avoiding slicings have been
proposed and studied in detail (see, e.g.~\cite{SY78,ES79,BP83}).
This idea of using the freedom in slicing the spacetime to avoid
singularities is ingenious but not perfect.  In the vicinity of the singularity
these
slicings
inevitably contain a region of abrupt change near the horizon,
and a region in which the constant time slices dip back deep into
the past in some sense.  Depending on the details of the choices
of the spacetime coordinates, the code will sooner
or later crash due to these pathological properties of the
slicing.  The problem can appear as the development of spikes in
the spatial metric functions~\cite{SY78}, the steepening of spatial
gradients~\cite{ST86}, ``grid stretching''~\cite{ST86} or large
coordinate shift~\cite{BHS} on the black hole throat, etc.

Hence it is important to investigate other ways to handle
singularities and black holes in numerical relativity.  Cosmic
censorship suggests that in physical situations, singularities
are hidden inside black hole horizons.
With the problematic region of numerical evolution
mostly
inside the horizon, it is tempting to cut away this region from the numerical
calculation by imposing a boundary
condition on or slightly inside the horizon.  To an outside observer no
information is lost since the
region cut away is unobservable.
There is then no singularity to
crash the code, the observable region can be
evolved, the dynamic range is drastically reduced so
accuracy is easier to maintain, and there is in principle no
physical reason that the code cannot run forever.

Using a horizon boundary condition by itself is not a new
idea~\cite{BP83,Unruh,York89,Choptuik92,Thornburgh91}.  However, it is non-
trivial to implement a
horizon boundary condition in dynamical evolution~\cite{Thornburgh91}.  The
boundary
condition to be imposed on a black hole horizon, which is a one
way membrane~\cite{Membrane}, should presumably be some sort of out-going
(into the hole) boundary condition.  However, except for the
case of linear, non-dispersive fields propagating in a flat
spacetime, we are not aware of any satisfactory out-going wave
boundary condition in numerical calculation~\cite{Orzsag}, let alone
for waves in relativity, which can be non-linear, dispersive, and
have tails and other complications~\cite{Waves}.

In this Letter we demonstrate that the idea of a
horizon boundary condition in numerical relativity can indeed be
realized.  In the next section, we discuss the two major
ingredients for our successful implementation of it:  (i)
a ``horizon locking coordinate'' (HLC) system which ties the spatial
coordinates to the spatial geometry,
(ii) a finite differencing scheme that we call causal differencing (CD) which
respects the
causal structure of the spacetime.  CD is
not only essential for the stability of the code using the
HLC, but also eliminates the need of explicitly
imposing boundary conditions on the horizon.  It is, in a sense,
the horizon boundary condition without a boundary condition:
since the horizon is a one way membrane, the quantities on the
horizon can be affected only by quantities outside but not inside
the horizon.  Hence, in CD, all quantities on the boundary can
be updated explicitly in terms of known quantities residing on or
outside the boundary, provided the boundary is inside the horizon.  There is no
need to impose boundary
conditions to account for information not covered by the
numerical evolution.  Such a horizon boundary
condition is particularly
desirable since it is general to all kinds of source terms in the Einstein
equations.

\subsection{Horizon Locking Coordinate}

We use the numerical construction of the Schwarzschild spacetime
to illustrate the idea of locking the horizon.  As shown in~\cite{BHS},
the numerical construction of a Schwarzschild spacetime
is non-trivial.  In all nine differencing schemes and for all
choices of coordinate conditions
studied in ~\cite{BHS}, problems arise after
an evolution in time of about $100 M$ with reasonable choices of grid
