commit
This commit is contained in:
pkirwin
parent
bc826c8f86
commit
84cce75552
@ -7,4 +7,7 @@
|
||||
\newlabel{eq:analytical}{{1}{3}}
|
||||
\newlabel{eq:normalise}{{2}{3}}
|
||||
\newlabel{eq:integral_ident}{{3}{3}}
|
||||
\gdef \@abspage@last{8}
|
||||
\newlabel{eq:numerical_E}{{4}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:q1b_electrons}{{2}{4}}
|
||||
\gdef \@abspage@last{9}
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
# Fdb version 3
|
||||
["pdflatex"] 1614638543 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614638561
|
||||
["pdflatex"] 1614640686 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614640702
|
||||
"C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e ""
|
||||
"C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 ""
|
||||
"C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
|
||||
@ -29,7 +29,6 @@
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy6.tfm" 1136765053 1116 933a60c408fc0a863a92debe84b2d294 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy8.tfm" 1136765053 1120 8b7d695260f3cff42e636090a8002094 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy9.tfm" 1136765053 1116 25a7bf822c58caf309a702ef79f4afbb ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmti10.tfm" 1136765053 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1247593066 34811 78b52f49e893bcba91bd7581cdc144c0 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1247593066 32080 340ef9bf63678554ee606688e7b5339d ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247593067 30251 6afa5cb1d0204815a708a080681d4674 ""
|
||||
@ -44,8 +43,8 @@
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1247593066 33993 9b89b85fd2d9df0482bd47194d1d3bf3 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1247593073 34694 870c211f62cb72718a00e353f14f254d ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/iftex.sty" 1583523400 6501 4011d89d9621e0b0901138815ba5ff29 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/ifvtex.sty" 1583523400 1057 525c2192b5febbd8c1f662c9468335bb ""
|
||||
@ -96,15 +95,15 @@
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb ""
|
||||
"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/titlesec/titlesec.sty" 1571240058 48596 2b6a95da931c07a430b1a61904aaa42d ""
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 ""
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb ""
|
||||
"doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex"
|
||||
"doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb ""
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 ""
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b ""
|
||||
"doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "pdflatex"
|
||||
"doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b ""
|
||||
"q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b ""
|
||||
"q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a ""
|
||||
(generated)
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf"
|
||||
"doc.pdf"
|
||||
"doc.log"
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf"
|
||||
"doc.aux"
|
||||
"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
|
||||
|
||||
@ -515,7 +515,6 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\
|
||||
INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
|
||||
INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
|
||||
INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmti10.tfm
|
||||
INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux
|
||||
INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk
|
||||
INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk
|
||||
@ -547,7 +546,7 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\c
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb
|
||||
INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb
|
||||
|
||||
18
PS2/doc.log
18
PS2/doc.log
@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 15:42
|
||||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 16:18
|
||||
entering extended mode
|
||||
**d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
|
||||
(d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
|
||||
@ -332,19 +332,19 @@ File: q1a_eigenvals.png Graphic file (type png)
|
||||
<use q1a_eigenvals.png>
|
||||
Package pdftex.def Info: q1a_eigenvals.png used on input line 163.
|
||||
(pdftex.def) Requested size: 202.3529pt x 151.76526pt.
|
||||
[3 <d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/q1a_1and50.png (PNG copy)> <d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/q1a_eigenvals.png (PNG copy)>] [4] [5] [6] [7] [8] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) )
|
||||
[3 <d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/q1a_1and50.png (PNG copy)> <d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/q1a_eigenvals.png (PNG copy)>] [4] [5] [6] [7] [8] [9] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
8334 strings out of 479618
|
||||
125207 string characters out of 2865027
|
||||
8336 strings out of 479618
|
||||
125259 string characters out of 2865027
|
||||
981750 words of memory out of 3000000
|
||||
25398 multiletter control sequences out of 15000+200000
|
||||
413212 words of font info for 65 fonts, out of 3000000 for 9000
|
||||
25401 multiletter control sequences out of 15000+200000
|
||||
412866 words of font info for 64 fonts, out of 3000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s
|
||||
<C:\Users\speed\AppData\Local\MiKTeX\fonts/pk/ljfour/jknappen/ec/dpi600\tcrm1000.pk><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi9.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr12.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr17.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr5.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb>
|
||||
Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (8 pages, 248293 bytes).
|
||||
<C:\Users\speed\AppData\Local\MiKTeX\fonts/pk/ljfour/jknappen/ec/dpi600\tcrm1000.pk><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi9.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr12.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr17.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr5.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb><C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
|
||||
Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (9 pages, 244757 bytes).
|
||||
PDF statistics:
|
||||
101 PDF objects out of 1000 (max. 8388607)
|
||||
104 PDF objects out of 1000 (max. 8388607)
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
11 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
BIN
PS2/doc.pdf
BIN
PS2/doc.pdf
Binary file not shown.
Binary file not shown.
65
PS2/doc.tex
65
PS2/doc.tex
@ -205,9 +205,20 @@ legend({'Numerical','Analytical'},'Location','northwest');
|
||||
|
||||
\item %1bii
|
||||
|
||||
{\em I really don't like this explanation}.
|
||||
|
||||
The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor if $\alpha$.
|
||||
Starting with the normalization condition for the numerical case:
|
||||
\begin{align*}
|
||||
a\sum_{l=1}^N|\phi_l|^2 = a \\
|
||||
a\sum_{l=1}^N|B\sin{(\frac{n\pi}{L}x_l)}|^2 = a,
|
||||
\end{align*}
|
||||
recalling that \(x = al\), and allowing \(a \to 0\) while holding \(L\)
|
||||
constant implies that \(N \to \infty\), since \(a=\frac{L}{N}\).
|
||||
An integral is defined as the limit of a Riemann sum as follows:
|
||||
\begin{equation*}
|
||||
\int_a^bf(x)\:dx = \lim_{n \to \infty}\sum_{i=1}^n\Delta x\cdot f(x_i).
|
||||
\end{equation*}
|
||||
|
||||
|
||||
The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor of $\alpha$.
|
||||
To correct for this, we need to factor the result of the summation by $\alpha$, which corresponds to
|
||||
|
||||
% Idk how to explain this at all, I need to chew on it internally and come back later.
|
||||
@ -226,18 +237,56 @@ legend({'Numerical','Analytical'},'Location','northwest');
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\item %1ci
|
||||
|
||||
From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that
|
||||
% TODO check that B is expressed correctly in all these equations (alpha, not ell, wait for pdf gen)
|
||||
From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} \alpha \ell \right)$, we can see that
|
||||
$\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities
|
||||
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and
|
||||
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where
|
||||
$\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$.
|
||||
$\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \alpha}{L}$.
|
||||
|
||||
Fro
|
||||
Plugging these identities into equation (7) from the assignment and simplifying, we get to this equation:
|
||||
|
||||
\begin{equation*}
|
||||
-t_0 B sin \left( \frac{n \pi \alpha \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi \alpha \ell}{L} \right)
|
||||
\end{equation*}
|
||||
|
||||
At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} \alpha \ell \right)$,
|
||||
so we can factor it out.
|
||||
|
||||
With some minor rearranging, this leaves us with the final expression for $E$:
|
||||
|
||||
\begin{equation}
|
||||
E = 2t_0 \left( 1 - cos \left( \frac{n \pi \alpha}{L} \right) \right)
|
||||
\label{eq:numerical_E}
|
||||
\end{equation}
|
||||
|
||||
\item %1cii
|
||||
\lipsum[1]
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{q1cii.png}
|
||||
\caption{Analytic solution to numeric system, plotted.}
|
||||
\label{fig:q1b_electrons}
|
||||
\end{figure}
|
||||
|
||||
We can see here that the "predicted" numerical response matches nearly exactly the actual
|
||||
calculated numerical solution.
|
||||
|
||||
\item %1ciii
|
||||
\lipsum[1]
|
||||
|
||||
Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$,
|
||||
on equation (\ref{eq:numerical_E}), we get the following expression
|
||||
|
||||
\begin{equation*}
|
||||
E = 2 t_0 \left( \frac{n^2 \pi^2 \alpha^2}{2 L^2} \right)
|
||||
\end{equation*}
|
||||
|
||||
Fully substituting the known value of $t_0$, and we can get our final expression for $E$:
|
||||
|
||||
\begin{equation*}
|
||||
E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 }
|
||||
\end{equation*}
|
||||
|
||||
\item %1civ
|
||||
\lipsum[1]
|
||||
\end{enumerate}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user