Examples of embedding Sage in LaTeX with SageTeX
system:sage


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<h2 class="titleHead">Examples of embedding Sage in <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> with
Sage<span class="TEX">T<span class="E">E</span>X</span></h2>
<div class="author"><span class="cmr-12">Dan Drake and others</span></div><br/>
<div class="date"><span class="cmr-12">February 20, 2010</span></div>
   </div><h3 class="sectionHead"><span class="titlemark">1   </span> <a id="x1-10001"/>Inline Sage, code blocks</h3><p class="noindent">This is an example <!--l. 52--><span class="math">2 + 2 = 4</span>. If you
raise the current year mod <!--l. 53--><span class="math">100</span>
(which equals <!--l. 53--><span class="math">10</span>) to the
power of the current day (<!--l. 54--><span class="math">20</span>),
you get <!--l. 54--><span class="math">100000000000000000000</span>.
Also, <!--l. 55--><span class="math">2010</span>
modulo <!--l. 55--><span class="math">42</span>
is <!--l. 55--><span class="math">36</span>.
</p><p class="indent">   Code block which uses a variable <span class="cmtt-10">s </span>to store the solutions: </p>

{{{id=1|
1+1
var('a,b,c')
eqn = [a+b*c==1, b-a*c==0, a+b==5]
s = solve(eqn, a,b,c)
///
}}}

<p class="indent">   Solutions of <!--l. 66--><span class="math">\text{eqn} = \left [bc + a = 1,&#8722;ac + b = 0,a + b = 5\right ]</span>:
</p><div class="math-display"><!--l. 67--><div class="math">
           \left [a = {\left (\left (25I\right )\,\sqrt{79} + 25\right )\over  
 \left (\left (6I\right )\,\sqrt{79} &#8722; 34\right )} ,b = {\left (\left (5I\right )\,\sqrt{79} + 5\right )\over  
 \left (I\,\sqrt{79} + 11\right )}  ,c = \left ( {1\over 
10}I\right )\,\sqrt{79} +  {1\over  
10}\right ]
</div></div><p class="nopar">
</p><div class="math-display"><!--l. 70--><div class="math">
         \left [a = {\left (\left (25I\right )\,\sqrt{79} &#8722; 25\right )\over  
 \left (\left (6I\right )\,\sqrt{79} + 34\right )} ,b = {\left (\left (5I\right )\,\sqrt{79} &#8722; 5\right )\over  
 \left (I\,\sqrt{79} &#8722; 11\right )}  ,c = \left (&#8722;{1\over 
10}I\right )\,\sqrt{79} +  {1\over  
10}\right ]
</div></div><p class="nopar">
</p><p class="indent">   Now we evaluate the following block: </p>

{{{id=3|
E = EllipticCurve("37a")
///
}}}

<p class="indent">   You can&#8217;t do assignment inside <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sage</span></span></span> macros, since Sage doesn&#8217;t know how to
typeset the output of such a thing. So you have to use a code block. The elliptic curve
<!--l. 80--><span class="math">E</span> given by
<!--l. 80--><span class="math">{y}^{2} + y = {x}^{3} &#8722; x</span> has
discriminant <!--l. 81--><span class="math">37</span>.
</p><p class="indent">   You can do anything in a code block that you can do in
Sage and/or Python. Here we save an elliptic curve into a &#64257;le. </p>

{{{id=5|
try:
    E = load('E2')
except IOError:
    E = EllipticCurve([1,2,3,4,5])
    E.anlist(100000)
                                                                  

                                                                  
    E.save('E2')
///
}}}

<p class="indent">   The 9999th Fourier coe&#64259;cient of <!--l. 94--><span class="math">{y}^{2} + xy + 3y = {x}^{3} + 2{x}^{2} + 4x + 5</span>
is <!--l. 95--><span class="math"> &#8722; 27</span>.
</p><p class="indent">   The following code block doesn&#8217;t appear in the typeset &#64257;le&#8230;</p>

{{{id=7|
e = 2
e = 3*e + 1
///
}}}

<p class="indent">   but we can refer to whatever we did in that code block:
<!--l. 102--><span class="math">e = 7</span>.
</p>

{{{id=9|
var('x')
f(x) = log(sin(x)/x)
///
}}}

<p class="indent">   The Taylor Series of <!--l. 108--><span class="math">f</span>
begins: <!--l. 108--><span class="math">x\ \mapsto \  &#8722;  {1\over  
467775}\,{x}^{10} &#8722; {1\over  
37800}\,{x}^{8} &#8722; {1\over  
2835}\,{x}^{6} &#8722; {1\over  
180}\,{x}^{4} &#8722;{1\over  
6}\,{x}^{2}</span>.
</p><p class="noindent">
</p><h3 class="sectionHead"><span class="titlemark">2   </span> <a id="x1-20002"/>Plotting</h3><p class="noindent">Here&#8217;s a plot of the elliptic curve <!--l. 112--><span class="math">E</span>.
</p><p class="indent">   <img alt="PIC" class="graphics" height="168.30869pt" src="example0x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example0x.png" src="sage-plots-for-example.tex/plot-0.eps"  
-->
</p>

{{{id=11|
# the var line is unecessary unless you've defined x to be something
# other than a symbolic variable
var('x')
f(x) = -x^3+3*x^2+7*x-4
///
}}}

<p class="indent">   You can use variables to hold plot objects and do stu&#64256; with them. </p>

{{{id=13|
p = plot(f, x, -5, 5)
///
}}}

<p class="indent">   Here&#8217;s a small plot of <!--l. 128--><span class="math">f</span>
from <!--l. 128--><span class="math"> &#8722; 5</span>
to <!--l. 128--><span class="math">5</span>,
which I&#8217;ve centered:
</p><div class="center">
<!--l. 130--><p class="noindent">
</p><!--l. 130--><p class="noindent"><img alt="PIC" class="graphics" height="45.56955pt" src="example1x.png" width="70.06067pt"/><!--tex4ht:graphics  
name="example1x.png" src="sage-plots-for-example.tex/plot-1.eps"  
--> </p></div><p class="indent">   On second thought, use the default size of
<!--l. 132--><span class="math">3&#8725;4</span> the
<span class="obeylines-h"><span class="verb"><span class="cmtt-10">\textwidth</span></span></span> and don&#8217;t use axes:
</p><p class="indent">   <img alt="PIC" class="graphics" height="168.30869pt" src="example2x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example2x.png" src="sage-plots-for-example.tex/plot-2.eps"  
-->
</p><p class="indent">   Remember, you&#8217;re using Sage, and can therefore call
upon any of the software packages Sage is built out of. </p>

{{{id=15|
f = maxima('sin(x)^2*exp(x)')
g = f.integrate('x')
///
}}}

<p class="indent">   Plot <!--l. 143--><span class="math">g(x)</span>, but don&#8217;t typeset it. </p>

{{{id=17|
# g is a Maxima thingy, it needs to get converted into a Sage object
plot1 = plot(g.sage(),x,-1,2*pi)
///
}}}

<p class="indent">   You can specify a &#64257;le format and options for <span class="obeylines-h"><span class="verb"><span class="cmtt-10">includegraphics</span></span></span>. The default is for
EPS and PDF &#64257;les, which are the best choice in almost all situations. (Although see
the section on 3D plotting.)
                                                                  

                                                                  
</p><p class="indent">   <span class="framebox-c" style="width:5.69em;"><span class="cmbx-10">??</span></span>
</p><p class="indent">   If you use regular <span class="obeylines-h"><span class="verb"><span class="cmtt-10">latex</span></span></span> to make a DVI &#64257;le, you&#8217;ll see a box, because DVI &#64257;les
can&#8217;t include PNG &#64257;les. If you use <span class="obeylines-h"><span class="verb"><span class="cmtt-10">pdflatex</span></span></span> that will work. See the documentation
for details.
</p><p class="indent">   When using <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sageplot</span></span></span>, you can pass in just about anything that Sage can call
<span class="obeylines-h"><span class="verb"><span class="cmtt-10">.save()</span></span></span> on to produce a graphics &#64257;le:
</p><div class="center">
<!--l. 162--><p class="noindent">
                                                                  

                                                                  
</p><!--l. 163--><p class="noindent"><img alt="PIC" class="graphics" height="433.48326pt" src="example3x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example3x.png" src="sage-plots-for-example.tex/plot-4.eps"  
--> </p></div><p class="indent">   <img alt="PIC" class="graphics" height="257.28246pt" src="example4x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example4x.png" src="sage-plots-for-example.tex/plot-5.eps"  
-->
</p>

{{{id=19|
G4 = DiGraph({1:[2,2,3,5], 2:[3,4], 3:[4], 4:[5,7], 5:[6]},\
             multiedges=True)
G4plot = G4.plot(layout='circular')
///
}}}

<p class="indent">   <img alt="PIC" class="graphics" height="252.83377pt" src="example5x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example5x.png" src="sage-plots-for-example.tex/plot-6.eps"  
-->
</p><p class="indent">   To &#64257;ddle with aspect ratio, &#64257;rst save the plot object:
</p>

{{{id=21|
p = plot(x, 0, 1) + circle((0,0), 1)
p.set_aspect_ratio(1)
///
}}}

<p class="indent">   Now plot it and see the circular circle and nice 45 degree angle:
</p><p class="indent">   <img alt="PIC" class="graphics" height="115.60252pt" src="example6x.png" width="115.60252pt"/><!--tex4ht:graphics  
name="example6x.png" src="sage-plots-for-example.tex/plot-7.eps"  
-->
</p><p class="indent">   Indentation and so on works &#64257;ne. </p>

{{{id=23|
s     = 7
s2    = 2^s
P.<x> = GF(2)[]
M     = matrix(parent(x),s2)
for i in range(s2):
   p  = (1+x)^i
                                                                  

                                                                  
   pc = p.coeffs()
   a  = pc.count(1)
   for j in range(a):
       idx        = pc.index(1)
       M[i,idx+j] = pc.pop(idx)
matrixprogram = matrix_plot(M,cmap='Greys')
///
}}}

<p class="indent">   And here&#8217;s the picture:
</p><p class="indent">   <img alt="PIC" class="graphics" height="248.1979pt" src="example7x.png" width="258.75pt"/><!--tex4ht:graphics  
name="example7x.png" src="sage-plots-for-example.tex/plot-8.eps"  
-->
</p><p class="indent">   Reset <span class="cmtt-10">x </span>in Sage so that it&#8217;s not a generator for the polynomial ring: x
</p><p class="noindent">
</p><h4 class="subsectionHead"><span class="titlemark">2.1   </span> <a id="x1-30002.1"/>Plotting (combinatorial) graphs with TikZ</h4><p class="noindent">Sage now includes some nice support for plotting graphs using <a href="http://www.texample.net/tikz/">TikZ</a>. Here, we
mean things with vertices and edges, not graphs of a function of one or two
variables.
</p><p class="indent">   First de&#64257;ne our graph:
</p>

{{{id=25|
g = graphs.PetersenGraph()
g.set_latex_options(tkz_style='Art')
///
}}}

<p class="indent">   Now just do <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sage{}</span></span></span> on it to plot it. You&#8217;ll need to use the <a href="http://altermundus.com/pages/graphtheory.html"><span class="cmtt-10">tkz-berge</span></a> package
for this to work; that package in turn depends on <a href="http://altermundus.com/pages/graph.html"><span class="cmtt-10">tkz-graph</span></a> and TikZ.
See <a href="http://sagemath.org/doc/reference/sage/graphs/graph_latex.html">&#8220;<span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> Options for Graphs&#8221;</a> in the Sage reference manual for more
details.
</p><div class="center">
<!--l. 231--><p class="noindent">
</p><!--l. 232--><p class="noindent"><object data="example-1.svg" height="208.16475 " type="image/svg+xml" width="206.23886 "><p>SVG-Viewer needed.</p></object>
</p></div><p class="indent">   The above command just outputs a <span class="cmtt-10">tikzpicture </span>environment, and you can
control that environment using anything supported by TikZ&#8212;although the output of
<span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sage{g}</span></span></span> explicitly hard-codes a lot of things and cannot be &#64258;exibly controlled in its
current form.
</p><div class="center">
<!--l. 242--><p class="noindent">
                                                                  

                                                                  
</p><!--l. 243--><p class="noindent"><object data="example-2.svg" height="114.18748 " type="image/svg+xml" width="112.26157 "><p>SVG-Viewer needed.</p></object>
</p></div><p class="noindent">
</p><h4 class="subsectionHead"><span class="titlemark">2.2   </span> <a id="x1-40002.2"/>3D plotting</h4><p class="noindent">3D plotting right now is problematic because there&#8217;s no convenient way to produce
vector graphics. We can make PNGs, though, and since the <span class="obeylines-h"><span class="verb"><span class="cmtt-10">sageplot</span></span></span> command
defaults to EPS and PDF, <span class="cmti-10">you must specify a valid format for 3D plotting</span>. Sage right
now (version 4.2.1) can&#8217;t produce EPS or PDF &#64257;les from <span class="cmtt-10">plot3d </span>objects, so if you
don&#8217;t specify a valid format, things will go badly. You can specify the &#8220;<span class="cmtt-10">imagemagick</span>&#8221;
option, which will use the Imagemagick <span class="cmtt-10">convert </span>utility to make EPS &#64257;les. See the
documentation for details.
</p><p class="indent">   Here&#8217;s the famous Sage cube graph in 3D.
</p>

{{{id=27|
G = graphs.CubeGraph(5)
///
}}}

<p class="indent">   <span class="framebox-c" style="width:5.69em;"><span class="cmbx-10">??</span></span>
</p><p class="indent">   And here&#8217;s a regular sort of 3D plot. Since <span class="cmtt-10">plot3d </span>objects don&#8217;t properly support
the kind of <span class="cmtt-10">.save() </span>method that we need, so we have to work around it a
bit and do things manually. Note that we can&#8217;t use <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\jobname</span></span></span> below. The
<span class="cmtt-10">sage.misc.viewer.BROWSER </span>bit tells Sage to not pop up a viewer program;
otherwise, when you run the <span class="cmtt-10">.sage </span>script, it will try to start a viewer program on the
resulting image, which we don&#8217;t want.
</p>

{{{id=29|
sage.misc.viewer.BROWSER=''
x, y = var('x y')
g = plot3d(sin(pi*(x^2+y^2))/2,(x,-1,1),(y,-1,1))
g.show(filename='sage-plots-for-example.tex/my-3d-plot', viewer='tachyon')
///
}}}

<p class="indent">   <img alt="PIC" src="sage-plots-for-example.tex/my-3d-plot.png"/>
</p><p class="noindent">
</p><h3 class="sectionHead"><span class="titlemark">3   </span> <a id="x1-50003"/>Pausing Sage<span class="TEX">T<span class="E">E</span>X</span></h3><p class="noindent">Sometimes you want to &#8220;pause&#8221; for a bit while writing your document if you have
embedded a long calculation or just want to concentrate on the <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> and ignore
any Sage stu&#64256;. You can use the <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sagetexpause</span></span></span> and <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\sagetexunpause</span></span></span> macros to do
that.
</p><p class="indent">   A calculation: <!--l. 300--><span class="math">\text{(Sage<mtext class="TeX">TeX</mtext> is paused)}</span>
and a code environment that simulates a time-consuming
calculation. While paused, this will get skipped over. </p>

{{{id=31|
import time
time.sleep(15)
///
}}}

<p class="indent">   Graphics are also skipped: <span class="fbox">Sage<span class="TEX">T<span class="E">E</span>X</span> is paused; no graphic</span>
</p><p class="noindent">
</p><h3 class="sectionHead"><span class="titlemark">4   </span> <a id="x1-60004"/>Make Sage write your <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> for you</h3><p class="noindent">With <span class="cmss-10">Sage</span><span class="TEX"><span class="cmss-10">T</span><span class="E"><span class="cmss-10">E</span></span><span class="cmss-10">X</span></span>, you can not only have Sage do your math for you, it can write parts
of your <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> document for you! For example, I hate writing <span class="cmtt-10">tabular</span>
environments; there&#8217;s too many &#64257;ddly little bits of punctuation and whatnot\ldots and
what if you want to add a column? It&#8217;s a pain&#8212;or rather, it <span class="cmti-10">was </span>a pain. Here&#8217;s how
to make Pascal&#8217;s triangle. It requires the <span class="cmtt-10">amsmath </span>package because of what Sage does
when producing a <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span class="E">E</span>X</span></span> representation of a string. (It puts it inside a <span class="obeylines-h"><span class="verb"><span class="cmtt-10">\text</span></span></span>
macro.)
</p>

{{{id=33|
def pascals_triangle(n):
    # start of the table
    s  = r"\begin{tabular}{cc|" + "r" * (n+1) + "}"
    s += r"  & & $k$: & \\"
    # second row, with k values:
    s += r"  & "
    for k in [0..n]:
        s += "& %d " % k
    s += r"\\"
                                                                  

                                                                  
    # the n = 0 row:
    s += r"\hline" + "\n" + r"$n$: & 0 & 1 & \\"
    # now the rest of the rows
    for r in [1..n]:
        s += " & %d " % r
        for k in [0..r]:
            s += "& %d " % binomial(r, k)
        s += r"\\"
    # add the last line and return
    s += r"\end{tabular}"
    return s

# how big should the table be?
n = 8
///
}}}

<p class="indent">   Okay, now here&#8217;s the table. To change the size, edit <span class="cmtt-10">n </span>above. If you have several
tables, you can use this to get them all the same size, while changing only one
thing.
</p><div class="center">
<!--l. 353--><p class="noindent">
</p>
<div class="tabular"> <table cellpadding="0" cellspacing="0" class="tabular" id="TBL-2" rules="groups"><colgroup id="TBL-2-1g"><col id="TBL-2-1"/><col id="TBL-2-2"/></colgroup><colgroup id="TBL-2-3g"><col id="TBL-2-3"/><col id="TBL-2-4"/><col id="TBL-2-5"/><col id="TBL-2-6"/><col id="TBL-2-7"/><col id="TBL-2-8"/><col id="TBL-2-9"/><col id="TBL-2-10"/><col id="TBL-2-11"/></colgroup><tr id="TBL-2-1-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-1-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-1-2" style="text-align:center; white-space:nowrap;"> </td><td class="td11" id="TBL-2-1-3" style="text-align:right; white-space:nowrap;"><!--l. 354--><span class="math">k</span>:</td><td class="td11" id="TBL-2-1-4" style="text-align:right; white-space:nowrap;"> </td></tr><tr id="TBL-2-2-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-2-1" style="text-align:center; white-space:nowrap;"> </td><td class="td11" id="TBL-2-2-2" style="text-align:center; white-space:nowrap;"> </td><td class="td11" id="TBL-2-2-3" style="text-align:right; white-space:nowrap;"> 0</td><td class="td11" id="TBL-2-2-4" style="text-align:right; white-space:nowrap;">1</td><td class="td11" id="TBL-2-2-5" style="text-align:right; white-space:nowrap;"> 2</td><td class="td11" id="TBL-2-2-6" style="text-align:right; white-space:nowrap;"> 3</td><td class="td11" id="TBL-2-2-7" style="text-align:right; white-space:nowrap;"> 4</td><td class="td11" id="TBL-2-2-8" style="text-align:right; white-space:nowrap;"> 5</td><td class="td11" id="TBL-2-2-9" style="text-align:right; white-space:nowrap;"> 6</td><td class="td11" id="TBL-2-2-10" style="text-align:right; white-space:nowrap;">7</td><td class="td11" id="TBL-2-2-11" style="text-align:right; white-space:nowrap;">8</td>
</tr><tr class="hline"><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td><td><hr/></td></tr><tr id="TBL-2-3-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-3-1" style="text-align:center; white-space:nowrap;"><!--l. 354--><span class="math">n</span>:</td><td class="td11" id="TBL-2-3-2" style="text-align:center; white-space:nowrap;">0</td><td class="td11" id="TBL-2-3-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-3-4" style="text-align:right; white-space:nowrap;"> </td></tr><tr id="TBL-2-4-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-4-1" style="text-align:center; white-space:nowrap;"> </td><td class="td11" id="TBL-2-4-2" style="text-align:center; white-space:nowrap;">1</td><td class="td11" id="TBL-2-4-3" style="text-align:right; white-space:nowrap;"> 1</td><td class="td11" id="TBL-2-4-4" style="text-align:right; white-space:nowrap;">1</td>
</tr><tr id="TBL-2-5-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-5-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-5-2" style="text-align:center; white-space:nowrap;">2</td><td class="td11" id="TBL-2-5-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-5-4" style="text-align:right; white-space:nowrap;">2</td><td class="td11" id="TBL-2-5-5" style="text-align:right; white-space:nowrap;"> 1</td></tr><tr id="TBL-2-6-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-6-1" style="text-align:center; white-space:nowrap;"> </td><td class="td11" id="TBL-2-6-2" style="text-align:center; white-space:nowrap;">3</td><td class="td11" id="TBL-2-6-3" style="text-align:right; white-space:nowrap;"> 1</td><td class="td11" id="TBL-2-6-4" style="text-align:right; white-space:nowrap;">3</td><td class="td11" id="TBL-2-6-5" style="text-align:right; white-space:nowrap;"> 3</td><td class="td11" id="TBL-2-6-6" style="text-align:right; white-space:nowrap;"> 1</td>
</tr><tr id="TBL-2-7-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-7-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-7-2" style="text-align:center; white-space:nowrap;">4</td><td class="td11" id="TBL-2-7-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-7-4" style="text-align:right; white-space:nowrap;">4</td><td class="td11" id="TBL-2-7-5" style="text-align:right; white-space:nowrap;"> 6</td><td class="td11" id="TBL-2-7-6" style="text-align:right; white-space:nowrap;"> 4</td><td class="td11" id="TBL-2-7-7" style="text-align:right; white-space:nowrap;"> 1</td>
</tr><tr id="TBL-2-8-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-8-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-8-2" style="text-align:center; white-space:nowrap;">5</td><td class="td11" id="TBL-2-8-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-8-4" style="text-align:right; white-space:nowrap;">5</td><td class="td11" id="TBL-2-8-5" style="text-align:right; white-space:nowrap;">10</td><td class="td11" id="TBL-2-8-6" style="text-align:right; white-space:nowrap;">10</td><td class="td11" id="TBL-2-8-7" style="text-align:right; white-space:nowrap;"> 5</td><td class="td11" id="TBL-2-8-8" style="text-align:right; white-space:nowrap;"> 1</td>
</tr><tr id="TBL-2-9-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-9-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-9-2" style="text-align:center; white-space:nowrap;">6</td><td class="td11" id="TBL-2-9-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-9-4" style="text-align:right; white-space:nowrap;">6</td><td class="td11" id="TBL-2-9-5" style="text-align:right; white-space:nowrap;">15</td><td class="td11" id="TBL-2-9-6" style="text-align:right; white-space:nowrap;">20</td><td class="td11" id="TBL-2-9-7" style="text-align:right; white-space:nowrap;">15</td><td class="td11" id="TBL-2-9-8" style="text-align:right; white-space:nowrap;"> 6</td><td class="td11" id="TBL-2-9-9" style="text-align:right; white-space:nowrap;"> 1</td>
</tr><tr id="TBL-2-10-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-10-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-10-2" style="text-align:center; white-space:nowrap;">7</td><td class="td11" id="TBL-2-10-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-10-4" style="text-align:right; white-space:nowrap;">7</td><td class="td11" id="TBL-2-10-5" style="text-align:right; white-space:nowrap;">21</td><td class="td11" id="TBL-2-10-6" style="text-align:right; white-space:nowrap;">35</td><td class="td11" id="TBL-2-10-7" style="text-align:right; white-space:nowrap;">35</td><td class="td11" id="TBL-2-10-8" style="text-align:right; white-space:nowrap;">21</td><td class="td11" id="TBL-2-10-9" style="text-align:right; white-space:nowrap;"> 7</td><td class="td11" id="TBL-2-10-10" style="text-align:right; white-space:nowrap;">1</td>
</tr><tr id="TBL-2-11-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-11-1" style="text-align:center; white-space:nowrap;">                                                                  </td><td class="td11" id="TBL-2-11-2" style="text-align:center; white-space:nowrap;">8</td><td class="td11" id="TBL-2-11-3" style="text-align:right; white-space:nowrap;">                                                                 1</td><td class="td11" id="TBL-2-11-4" style="text-align:right; white-space:nowrap;">8</td><td class="td11" id="TBL-2-11-5" style="text-align:right; white-space:nowrap;">28</td><td class="td11" id="TBL-2-11-6" style="text-align:right; white-space:nowrap;">56</td><td class="td11" id="TBL-2-11-7" style="text-align:right; white-space:nowrap;">70</td><td class="td11" id="TBL-2-11-8" style="text-align:right; white-space:nowrap;">56</td><td class="td11" id="TBL-2-11-9" style="text-align:right; white-space:nowrap;">28</td><td class="td11" id="TBL-2-11-10" style="text-align:right; white-space:nowrap;">8</td><td class="td11" id="TBL-2-11-11" style="text-align:right; white-space:nowrap;">1</td>
</tr><tr id="TBL-2-12-" style="vertical-align:baseline;"><td class="td11" id="TBL-2-12-1" style="text-align:center; white-space:nowrap;">                                                                  </td></tr></table>
</div>
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