From 066f7ebd44e714d9320ae2da6924e331b7091a50 Mon Sep 17 00:00:00 2001 From: diener Date: Wed, 13 Nov 2002 16:46:11 +0000 Subject: Finally the documentation for the AHFinder converted from HTML into LaTeX. Any errors in the conversion is entirely my fault so you shouldn't complain to Miguel. The eps figures are of pretty low quality, but the have been converted from gif files. When I find some time, I'll try and generate some new plots of higher quality, but don't hold your breath. git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinAnalysis/AHFinder/trunk@318 89daf98e-ef62-4674-b946-b8ff9de2216c --- doc/documentation.tex | 345 ++- doc/hori_bl.eps | 5440 +++++++++++++++++++++++++++++++++++++++++++++++ doc/p2_areacomp.eps | 5558 +++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 11326 insertions(+), 17 deletions(-) create mode 100644 doc/hori_bl.eps create mode 100644 doc/p2_areacomp.eps (limited to 'doc') diff --git a/doc/documentation.tex b/doc/documentation.tex index b097bbe..d8bb9a6 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -74,15 +74,15 @@ \begin{document} % The author of the documentation -\author{} +\author{Miguel Alcubierre/Latex version by Peter Diener} % The title of the document (not necessarily the name of the Thorn) -\title{} +\title{AHFinder} % the date your document was last changed, if your document is in CVS, % please us: % \date{$ $Date$ $} -\date{} +\date{$ $Date$ $} \maketitle @@ -94,38 +94,349 @@ % Add an abstract for this thorn's documentation \begin{abstract} - +Finding Apparent Horizons in a numerical spacetime \end{abstract} % The following sections are suggestive only. % Remove them or add your own. -\section{Introduction} +\section{Purpose} + +Thorn {\tt AHFinder} finds Apparent Horizons (AHs) in numerical +spacetimes. It calulates various quantities like horizon area and +its corresponding mass. + +\section{Using AHFinder} -\section{Physical System} +Thorn AHFinder can be used either with a minimization or a flow algorithm. +\subsection{Parameters} +\subsubsection*{General parameters} +\begin{itemize} +\item {\tt ahfinder::ahf\_active} (steerable) \\ + To activate the thorn set ahf\_active = "yes". This parameter is set + by default to "no". \\ +\item {\tt ahfinder::ahf\_flow} (steerable) \\ + By default the minimization algorithm is used. To switch to the + flow algorithm one has to set ahf\_flow = "yes" +\end{itemize} +\subsubsection*{Parameters used in evolutions} +\begin{itemize} +\item {\tt ahfinder::ahf\_findevery} (steerable) \\ + Specifies how often the finder is called. The default is to find + horizons at every iteration. +\item {\tt ahfinder::ahf\_findafter} (steerable) \\ + The number of iterations after which the thorn is called the + first time can be specified by this parameter. Here the default is 0, + calling the thorn also on the initial time slice. +\item {\tt ahfinder::ahf\_findaftertime} \\ + Instead of specifying the number of iterations, one can specify + after how much coordinate time the thorn is called the first time. + When different from zero, this parameter overrides the value of + ahf\_findafter. Default here is also 0. +\end{itemize} +\subsubsection*{Parameters specifying the expansion of the surface in sperical + harmonics.} +\begin{itemize} +\item {\tt ahfinder::ahf\_lmax} \\ + The maximal number of terms in the expansion in $\theta$. The default + value is 8. The maximal value is 20. +\item {\tt ahfinder::ahf\_phi} (steerable) \\ + If axisymmetry is expected the surface does not need to be + expanded in phi. This is the default. To look for non-axisymmetric + surface use ahf\_phi = "yes". +\item {\tt ahfinder::ahf\_[xyz]c} (steerable) \\ + Sets the x-, y-, and z-coordinates of the center of the expansion. + The default is the origin (ahf\_xc = 0, ahf\_yc = 0, ahf\_zc = 0). + The center of the expansion should be set inside the expected + apparent horizon, otherwise the algorithm will fail. +\item {\tt ahfinder::ahf\_wander} \\ + The center of the expansion can also be allowed to move. To do this + use ahf\_wander = "yes". However, this only works with the + minimization algorithm. The default is for the center not to move. +\item {\tt ahfinder::ahf\_r0} (steerable) \\ + Sets the radius of the initial sphere. The default is 0.0, forcing + the largest sphere possible in the grid. +\end{itemize} +\subsubsection*{Looking for three horizons} +The finder can also be used to look for three horizons each time. +This is done by just running the algorithm three consecutive times +with different initial guesses, and is useful for simulations of +black hole collisions. +\begin{itemize} +\item {\tt ahfinder::ahf\_find3} \\ + Set ahf\_find3 = "yes" to search for three horizons. The default is + to look for only one horizon. +\item {\tt ahfinder::ahf\_[xyz]\_[012]} (steerable) \\ + Sets the x-, y-, and z-coordinates of the center of the expansion for + horizon 0, 1 and 2. The default in each case is the origin. +\item {\tt ahfinder::ahf\_r0\_[0-2]} (steerable) \\ + Sets the radius of the initial spheres for horizon 0, 1 and 2. + The default in all cases is 0.0, forcing the largest sphere possible + in the grid. +\end{itemize} +\subsubsection*{Further parameters for the initial guess} +The initial guess can be furthermore controlled by some parameters +which are set to "no" by default. +\begin{itemize} +\item {\tt ahfinder::ahf\_guessold} (steerable) \\ + To use on old horizon as initial guess set ahf\_guessold = "yes". + However, if during the evolution the apparent horizon jumps + discontinuously it might be lost by using this option. +\item {\tt ahfinder::ahf\_nn0, ahfinder::ahf\_nn2} \\ + If no old horizon is used the inital guess can be specified + further for the minimization algorithm. This algorithm is + sensitive to the initial guess, so this is important. The initial + guess is set up by an expansion in spherical harmonics in the + first two coefficients (l=0,l=2). The default for both these + parameters are 10, in which case the algorithm tests 100 different + combinations to find the best initial guess. +\item {\tt ahfinder::ahf\_sloppyguess} \\ + It is also possible to use only a sphere as initial guess. This + is much faster and is done by using ahf\_sloppyguess = "yes". In this + case a number of spheres (specified by ahf\_nn0) with different radii + are tested for the initial guess. +\item {\tt ahfinder::ahf\_inner} \\ + If one wants to look for an inner horizon instead of an outer + one, use ahfinder::ahf\_inner = "yes". This only works with the + minimization algorithm. +\end{itemize} +\subsubsection*{Parameters for surface intergrals} +\begin{itemize} +\item {\tt ahfinder::ahf\_ntheta} (steerable) \\ + The number of subdivisions in $\theta$. Default is 200. +\item {\tt ahfinder::ahf\_nphi} (steerable) \\ + The number of subdivisions in $\phi$. Default is 200. +\end{itemize} +\subsubsection*{Parameters indicating symmetries} +\begin{itemize} +\item {\tt ahfinder::ahf\_ref[xyz]} (steerable) \\ + Specifies the existence of reflection symmetry on the yz-plane, + xz-plane and xy-plane respectively. By default all are set to "no". +\item {\tt ahfinder::ahf\_octant} (steerable) \\ + Octant symmetry is specified by using ahf\_octant = "yes". This is set + to "no" by default. Possible parmeter settings are "yes" for reflection + symmetries on all three coordinate planes and "high" for an additional + rotational symmetry of $\pi/2$ around the z axis. +\item {\tt ahfinder::ahf\_cartoon} (steerable) \\ + A further symmetry can (must) be specified when running with the + "axisymmetric" mode given by the Cartoon method by using + ahf\_cartoon = "yes". This is set to "no" by default. +\end{itemize} +\subsubsection*{Parameters for minimization algorithm} +\begin{itemize} +\item {\tt ahfinder::ahf\_tol} \\ + Sets the tolerance for the minimization algorithm. The default + value is 0.1. +\item {\tt ahfinder::ahf\_maxiter} \\ + Sets the maximum number if iterations for each step of the POWELL + algorithm. The default value is 10. +\item {\tt ahfinder::ahf\_minarea} \\ + Usually the square of the expansion is minimized. To switch to + minimization of the area one can use ahf\_minarea = "yes" + (default is "no"). Notice that only for time symmetric data the + surface found by area minimization will correspond to an apparent + horizon. +\end{itemize} +\subsubsection*{Parameters for the flow algorithm} +\begin{itemize} +\item {\tt ahfinder::ahf\_flow} (steerable) \\ + The fow algotithm is used by setting ahf\_flow = "yes". The default + is "no" +\item {\tt ahfinder::ahf\_flowiter} (steerable) \\ + Sets the maximum number of iterations for the flow algorithm. The + default value is 200. +\item {\tt ahfinder::ahf\_flowtol} \\ + Sets the tolerance for the flow. The default value is 0.0001. +\item {\tt ahfinder::ahf\_flowa} \\ + Sets the $\alpha$ parameter for the flow. The default value is 0.01. +\item {\tt ahfinder::ahf\_flowb} \\ + Sets the $\beta$ parameter for the flow. The default value is 0.5. +\item {\tt ahfinder::ahf\_flowh} \\ + Sets the weight of $H$ flow. The default value is 0.0. +\item {\tt ahfinder::ahf\_flowc} \\ + Sets the weight of $C$ flow. The default value is 1.0. +\item {\tt ahfinder::ahf\_flown} \\ + Sets the weight of $N$ flow (not yet implemented). The default + value is 0.0. +\end{itemize} +The character of the different flows and the $\alpha$ and $\beta$ +parameters are described in Carsten Gundlach's paper on his +pseudo-spectral apparent horizon finder (gr-qc/9707050). +\subsubsection*{Parameters for output} +\begin{itemize} +\item {\tt ahfinder::ahf\_logfile} \\ + By default no logfile for AHFinder is written. To obtain a log + file one must set ahf\_logfile = "yes". +\item {\tt ahfinder::ahf\_verbose} \\ + if ahf\_verbose = "yes" messages are printed to screen at the + beginning and the end of the algorithm. This is the default. +\item {\tt ahfinder::ahf\_veryverbose} \\ + if ahf\_veryverbose = "yes" messages are also printed to screen + during the iteration process. The default is "no". +\item {\tt ahfinder::ahf\_2Doutput} (steerable) \\ + To get 2D output of grid functions use ahf\_2Doutput = "yes" (default + is "no"). The output is controlled be the thorn itself, not by cactus + standard output. +\item {\tt ahfinder::ahf\_3Doutput} (steerable) \\ + In future versions to get 3D output of grid functions use + ahf\_3Doutput = "yes" (default is "no"). +\item {\tt ahfinder::ahf\_areamap} \\ + To get an area map use ahf\_areamap = "yes" (default is "no"). + This is useful for looking at the behaviour of the area for + surfaces close to the horizon. +\end{itemize} +\subsubsection*{Parameters for mask} +\begin{itemize} +\item {\tt ahfinder::ahf\_mask} (steerable) \\ + The mask is 0 inside the horizon and 1 outside, and is used in + black-hole excision techniques. By default the mask is off. It + can be enabled by setting ahf\_mask = "strong", which sets the mask + only if the finder is sure that a horizon was found, or by setting + ahf\_mask = "weak", which makes the finder set the mask also if a + horizon is probably there but either the resolution or lmax are to + low to really resolve it. +\item {\tt ahfinder::ahf\_masktype} (steerable) \\ + The mask can be of two types. Either ahf\_masktype = "simple", + which masks a cube contained inside the horizons, or ahf\_masktype + = "lego", which masks a lego-sphere. +\item {\tt ahfinder::ahf\_maskshrink} (steerable) \\ + Sets a buffer zone between the region inside the horizon and the + region where the mask is 0, by limiting the mask to a region smaller + than ahf\_maskshrink times the horizon radius. The default value + is 0.8. +\item {\tt ahfinder::ahf\_maskbuffer} (steerable) \\ + Sets a buffer zone between the region inside the horizon and the + region where the mask is 0 with a width of at least ahf\_maskbuffer + grid points. The default value is 5. +\end{itemize} +\subsection{Minimal parameter settings} +Usually only a few of the parameters described above are needed in the +parameter file. The simplest parameter settings for using the flow +algorithm for a full 3D horizon with a large sphere as initial guess is +{\tt +\begin{verbatim} +interpolation_order = 2 # Second order interpolation +ahf_active = "yes" +ahf_flow = "yes" +ahf_phi = "yes" +\end{verbatim} +} +This looks for a non-axisymmetric horizon around the origin with lmax = 8 +and using the flow algorithm. It starts with the largest sphere that fits +in the 3D grid and outputs 2D grid functions. The other parameters can be +used if needed. +\subsection{Hints for parameter settings} +In full 3D the flow algorithm is faster than the minimization algorithm. +However, in cases when there are very few terms in the expansion in +spherical harmonics the minimization can be faster. In axisymmetry this +typically happens for lmax$\leq 10$. -\section{Numerical Implementation} +While the default settings usually work fine, they can be changed to meet +special purposes: +\begin{itemize} +\item If the horizon is expected to be far from spherical the parameter + ahf\_lmax can be set to a higher value. 12 should be high enough. + However, only values up to 20 are supported. +\item If the latter parameter is set to a values higher then 8 the + parameter ahf\_maxiter can be raised to e.g. 14. This can be useful + since more iterations can be necessary for higher coefficients of the + expansion. +\end{itemize} +\subsection{Output to Files} +The output of the thorn consists of two gridfunctions and several one +dimensional output files. +\begin{itemize} +\item To depict the position of the horizon, the most important files are + ahfgrid\_2d\_...ieee. These files contain a 2D gridfunction whose zero + level locates the horizon. +\item The files ahf\_exp\_2d\_...ieee show the expansion of outgoing photons + on the level set of the gridfunction ahfgrid. The horizon coincides + with zeros of the expansion. +\item The surface area of the horizon is given in ahf\_area.tl +\item The mass of the horizon is given in ahf\_mass.tl +\item The coefficients of the expansion in spherical harmonics are given in + ahf\_coeff.alm. +\item The files ahf\_circ\_eq.tl, ahf\_meri\_p1.tl and ahf\_meri\_p2.tl contain + the equatorial circumference of the surface, the length of the + meridian at $\phi=0$, and the length of the meridian at $\phi=\pi/2$ of the + surface, respectively. +\item If an output of a logfile is set in the parameters, the log file for + the last time the horizon was called is ahf\_logfile. -\section{Using This Thorn} +\end{itemize} +\subsection{Some results with the finder} +The finder has been examined with puncture initial data for single and +binary-black hole scenarios. -\subsection{Obtaining This Thorn} +Calculations with different grid spacings but constant grid size show +convergence of the horizon area. -\subsection{Basic Usage} +This has been checked with different linear momenta in the z direction +$P_{z}=(0M,2M,5M)$ and vanishing spin. Also for $P_{z}=2M$ and a spin of $5M$ +in the x direction the horizon converges. +Figure~\ref{CactusEistein_AHFinder_p2_conv} shows the case with +$P_{z}=2M$ and vanishing spin. -\subsection{Special Behaviour} +\begin{figure}[ht] +\begin{center} +\ifpdf +\else +\includegraphics[angle=0,width=8cm]{p2_areacomp.eps} +\fi +\end{center} +\caption{Convergence of the horizon area for $P_{z}=2M$} +\label{CactusEistein_AHFinder_p2_conv} +\end{figure} -\subsection{Interaction With Other Thorns} +Further, not only the area converges but also the shape of the horizon. +For both the minimization and the flow algorithm the horizon converges to +the same shape, as can be seen from the coefficients fo the expansion. +The order of convergence for the coefficients is between 1.4 and 1.7. -\subsection{Support and Feedback} +By using the parameters ahf\_xc, ahf\_yc, ahf\_zc it can also be shown that +the finder also locates horizons which are not centered. This works in +general as long as the surface can be expanded in spherical harmonics +around this point, but the error increases with the off-centering. -\section{History} +The parameter ahf\_r0 can be used e.g. when dealing with two black holes. +If one searches for separate horizons one can center the finder on one of +the locations of the holes and use an initial radius ahf\_r0 smaller than +the coordinate distance of the holes. With this parameter settings the +single horizon can be found faster. But also a setup with an initial +sphere of maximum radius should work at least for the flow algorithm. +This has been checked with puncture data for two holes with vanishing +linear and angular momentum for each hole (equivalent to Brill-Lindquist +data) and is shown in Figure~\ref{CactusEinstein_AHFinder_p2_conv}. +Here for a coordinate distance of the holes of 1.6M the separated +horizons for the holes are found but no common horizon. For a coordinate +distance of 1.5M a common horizon is found and also single ones, which +are inner surfaces in this case. This coincides with other work where the +critical coordinate distance for a single horizon is between 1.53M and +1.56M (gr-qc/9809004). -\subsection{Thorn Source Code} +\begin{figure}[ht] +\begin{center} +\ifpdf +\else +\includegraphics[angle=0,width=10cm]{hori_bl.eps} +\fi +\end{center} +\caption{Horizon positions for BL data} +\label{CactusEinstein_AHFinder_p2_conv} +\end{figure} -\subsection{Thorn Documentation} +The dashed lines show inner trapped surfaces in the left figure and the +surface where the algorithm stopped without finding a horizon in the +right figure. -\subsection{Acknowledgements} +Also the Misner case was checked. Here for $\mu$ = 1.35 a common horizon is +found. For $\mu$ = 1.37 separated horizons are found. From the literature we +know that (e.g. gr-qc/9809004) the critical value of $\mu$ is 1.36. This is +confirmed by the horizon finder. +The information of when a horizon was found can be seen in the +cactus-logfile. There will be output from the thorn even if no horizon +was found. \begin{thebibliography}{9} diff --git a/doc/hori_bl.eps b/doc/hori_bl.eps new file mode 100644 index 0000000..03fc551 --- /dev/null +++ b/doc/hori_bl.eps @@ -0,0 +1,5440 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (hori_bl.eps.0) +%%CreationDate: (Fri May 10 14:10:14 2002) +%%BoundingBox: 0 0 640 290 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%PageOrientation: Portrait +%%EndDefaults + +%%BeginProlog +% +% Display a color image. 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EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (p2_areacomp.eps) +%%CreationDate: (Fri May 10 14:01:38 2002) +%%BoundingBox: 0 0 500 380 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%PageOrientation: Portrait +%%EndDefaults + +%%BeginProlog +% +% Display a color image. The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + /Times-Roman findfont pointsize scalefont setfont + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt { PseudoClassImage } { DirectClassImage } ifelse + grestore +} bind def +%%EndProlog +%%Page: 1 1 +%%PageBoundingBox: 0 0 500 380 +userdict begin +DisplayImage +0 0 +500 380 +1000.000000 +500 380 +1 +1 +1 +8 +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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