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--- | 
				
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title: "Paperlighter Template Implementation Example" | 
				
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author: | | 
				
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  - number: 1 | 
				
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    name: "Author One" | 
				
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    correspond: true | 
				
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    affiliation: "My City University" | 
				
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    address: "Orenomachi, Orenoshi, Orenoken, Japan" | 
				
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  - number: 2 | 
				
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    name: "Author Two" | 
				
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    affiliation: "My Other City University" | 
				
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    address: "Hokanomachi, Orenoshi, Orenoken, Japan" | 
				
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email: "xxx@myuni.ac.jp" | 
				
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titleSmall: "Paperlighter Example" | 
				
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authorSmall: "Author One et.al." | 
				
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abstract: | | 
				
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    Using \LaTeX\{\} to write papers is concise and convenient. However, for | 
				
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    writing in life, complicated \LaTeX\{\} style-files (e.g., elegantpaper) | 
				
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    are difficult to access, or submission style-files (e.g., journal or | 
				
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    conference) are not free indeed. To tackle these problems and satisfy an | 
				
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    elegant and straightforward scientific writing, | 
				
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    \textbf{paperlighter.sty}, a one-column style-file, is designed. This | 
				
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    document is edited from icml2022.sty and provides a basic paper | 
				
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    template. Compared to icml2022.sty, paperlighter.sty contain fewer | 
				
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    operations, reducing adjustment while keep graceful. | 
				
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    \textbf{\textit{Notably, the paper's main content only describes the format of icml2022.sty. We place the content to show the actual effect of paperlighter.sty.}} | 
				
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--- | 
				
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 | 
				
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 | 
				
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%\input{content/abstract} | 
				
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 | 
				
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 | 
				
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%\input{content/format} | 
				
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%\input{content/others} | 
				
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\hypertarget{format-of-the-paperlighter}{% | 
				
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\section{Format of the Paperlighter}\label{format-of-the-paperlighter}} | 
				
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 | 
				
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Format of paperlighter is defined in this section. | 
				
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 | 
				
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\hypertarget{dimensions}{% | 
				
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\subsection{Dimensions}\label{dimensions}} | 
				
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 | 
				
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The text of the paper has an overall width of | 
				
			||||
6.75\textasciitilde{}inches, and height of 9.0\textasciitilde{}inches. | 
				
			||||
The left margin should be 0.75\textasciitilde{}inches and the top margin | 
				
			||||
1.0\textasciitilde{}inch (2.54\textasciitilde{}cm). The right and bottom | 
				
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margins will depend on whether you print on US letter or A4 paper, but | 
				
			||||
all final versions must be produced for US letter size. | 
				
			||||
 | 
				
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The paper body should be set in 10\textasciitilde{}point type with a | 
				
			||||
vertical spacing of 11\textasciitilde{}points. Please use Times typeface | 
				
			||||
throughout the text. | 
				
			||||
 | 
				
			||||
\hypertarget{title}{% | 
				
			||||
\subsection{Title}\label{title}} | 
				
			||||
 | 
				
			||||
The paper title should be set in 14\textasciitilde{}point bold type and | 
				
			||||
centered between two horizontal rules that are 1\textasciitilde{}point | 
				
			||||
thick, with 1.0\textasciitilde{}inch between the top rule and the top | 
				
			||||
edge of the page. Capitalize the first letter of content words and put | 
				
			||||
the rest of the title in lower case. | 
				
			||||
 | 
				
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\hypertarget{author-information-for-submission}{% | 
				
			||||
\subsection{Author Information for | 
				
			||||
Submission}\label{author-information-for-submission}} | 
				
			||||
 | 
				
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\label{author info} | 
				
			||||
 | 
				
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Use \verb+\lighterauthor{...}+ to specify authors and | 
				
			||||
\verb+\lighteraddress{...}+ to specify affiliations. (Read the TeX code | 
				
			||||
used to produce this document for an example usage.) The author | 
				
			||||
information will not be printed unless \texttt{accepted} is passed as an | 
				
			||||
argument to the style file. | 
				
			||||
 | 
				
			||||
\hypertarget{abstract}{% | 
				
			||||
\subsection{Abstract}\label{abstract}} | 
				
			||||
 | 
				
			||||
The paper abstract should begin in the left column, | 
				
			||||
0.4\textasciitilde{}inches below the final address. The heading | 
				
			||||
`Abstract’ should be centered, bold, and in 11\textasciitilde{}point | 
				
			||||
type. The abstract body should use 10\textasciitilde{}point type, with a | 
				
			||||
vertical spacing of 11\textasciitilde{}points, and should be indented | 
				
			||||
0.25\textasciitilde{}inches more than normal on left-hand and right-hand | 
				
			||||
margins. Insert 0.4\textasciitilde{}inches of blank space after the | 
				
			||||
body. Keep your abstract brief and self-contained, limiting it to one | 
				
			||||
paragraph and roughly 4–6 sentences. Gross violations will require | 
				
			||||
correction at the camera-ready phase. | 
				
			||||
 | 
				
			||||
\hypertarget{partitioning-the-text}{% | 
				
			||||
\subsection{Partitioning the Text}\label{partitioning-the-text}} | 
				
			||||
 | 
				
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You should organize your paper into sections and paragraphs to help | 
				
			||||
readers place a structure on the material and understand its | 
				
			||||
contributions. | 
				
			||||
 | 
				
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\hypertarget{sections-and-subsections}{% | 
				
			||||
\subsubsection{Sections and | 
				
			||||
Subsections}\label{sections-and-subsections}} | 
				
			||||
 | 
				
			||||
Section headings should be numbered, flush left, and set in | 
				
			||||
11\textasciitilde{}pt bold type with the content words capitalized. | 
				
			||||
Leave 0.25\textasciitilde{}inches of space before the heading and | 
				
			||||
0.15\textasciitilde{}inches after the heading. | 
				
			||||
 | 
				
			||||
Similarly, subsection headings should be numbered, flush left, and set | 
				
			||||
in 10\textasciitilde{}pt bold type with the content words capitalized. | 
				
			||||
Leave 0.2\textasciitilde{}inches of space before the heading and | 
				
			||||
0.13\textasciitilde{}inches afterward. | 
				
			||||
 | 
				
			||||
Finally, subsubsection headings should be numbered, flush left, and set | 
				
			||||
in 10\textasciitilde{}pt small caps with the content words capitalized. | 
				
			||||
Leave 0.18\textasciitilde{}inches of space before the heading and | 
				
			||||
0.1\textasciitilde{}inches after the heading. | 
				
			||||
 | 
				
			||||
Please use no more than three levels of headings. | 
				
			||||
 | 
				
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\hypertarget{paragraphs-and-footnotes}{% | 
				
			||||
\subsubsection{Paragraphs and | 
				
			||||
Footnotes}\label{paragraphs-and-footnotes}} | 
				
			||||
 | 
				
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Within each section or subsection, you should further partition the | 
				
			||||
paper into paragraphs. Do not indent the first line of a given | 
				
			||||
paragraph, but insert a blank line between succeeding ones. | 
				
			||||
 | 
				
			||||
You can use footnotes\footnote{Footnotes | 
				
			||||
should be complete sentences.} to provide readers with additional | 
				
			||||
information about a topic without interrupting the flow of the paper. | 
				
			||||
Indicate footnotes with a number in the text where the point is most | 
				
			||||
relevant. Place the footnote in 9\textasciitilde{}point type at the | 
				
			||||
bottom of the column in which it appears. Precede the first footnote in | 
				
			||||
a column with a horizontal rule of | 
				
			||||
0.8\textasciitilde{}inches.\footnote{Multiple footnotes can | 
				
			||||
appear in each column, in the same order as they appear in the text, | 
				
			||||
but spread them across columns and pages if possible.} | 
				
			||||
 | 
				
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\begin{figure}[ht] | 
				
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\vskip 0.2in | 
				
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\begin{center} | 
				
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\centerline{\includegraphics[width=\columnwidth]{Figure/icml_numpapers.eps}} | 
				
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\caption{Historical locations and number of accepted papers for International | 
				
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Machine Learning Conferences (ICML 1993 -- ICML 2008) and International | 
				
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Workshops on Machine Learning (ML 1988 -- ML 1992). At the time this figure was | 
				
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produced, the number of accepted papers for ICML 2008 was unknown and instead | 
				
			||||
estimated.} | 
				
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\label{icml-historical} | 
				
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\end{center} | 
				
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\vskip -0.2in | 
				
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\end{figure} | 
				
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 | 
				
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\hypertarget{figures}{% | 
				
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\subsection{Figures}\label{figures}} | 
				
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 | 
				
			||||
You may want to include figures in the paper to illustrate your approach | 
				
			||||
and results. Such artwork should be centered, legible, and separated | 
				
			||||
from the text. Lines should be dark and at least | 
				
			||||
0.5\textasciitilde{}points thick for purposes of reproduction, and text | 
				
			||||
should not appear on a gray background. | 
				
			||||
 | 
				
			||||
Label all distinct components of each figure. If the figure takes the | 
				
			||||
form of a graph, then give a name for each axis and include a legend | 
				
			||||
that briefly describes each curve. Do not include a title inside the | 
				
			||||
figure; instead, the caption should serve this function. | 
				
			||||
 | 
				
			||||
Number figures sequentially, placing the figure number and caption | 
				
			||||
\emph{after} the graphics, with at least 0.1\textasciitilde{}inches of | 
				
			||||
space before the caption and 0.1\textasciitilde{}inches after it, as in | 
				
			||||
\cref{icml-historical}. The figure caption should be set in | 
				
			||||
9\textasciitilde{}point type and centered unless it runs two or more | 
				
			||||
lines, in which case it should be flush left. You may float figures to | 
				
			||||
the top or bottom of a column, and you may set wide figures across both | 
				
			||||
columns (use the environment \texttt{figure*} in \LaTeX). Always place | 
				
			||||
two-column figures at the top or bottom of the page. | 
				
			||||
 | 
				
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\hypertarget{algorithms}{% | 
				
			||||
\subsection{Algorithms}\label{algorithms}} | 
				
			||||
 | 
				
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If you are using \LaTeX, please use the | 
				
			||||
\texttt{algorithm\textquotesingle{}\textquotesingle{}\ and}algorithmic’’ | 
				
			||||
environments to format pseudocode. These require the corresponding | 
				
			||||
stylefiles, algorithm.sty and algorithmic.sty, which are supplied with | 
				
			||||
this package. \cref{alg:example} shows an example. | 
				
			||||
 | 
				
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\begin{algorithm}[tb] | 
				
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   \caption{Bubble Sort} | 
				
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   \label{alg:example} | 
				
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\begin{algorithmic} | 
				
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   \STATE {\bfseries Input:} data $x_i$, size $m$ | 
				
			||||
   \REPEAT | 
				
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   \STATE Initialize $noChange = true$. | 
				
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   \FOR{$i=1$ {\bfseries to} $m-1$} | 
				
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   \IF{$x_i > x_{i+1}$} | 
				
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   \STATE Swap $x_i$ and $x_{i+1}$ | 
				
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   \STATE $noChange = false$ | 
				
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   \ENDIF | 
				
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   \ENDFOR | 
				
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   \UNTIL{$noChange$ is $true$} | 
				
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\end{algorithmic} | 
				
			||||
\end{algorithm} | 
				
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 | 
				
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\hypertarget{tables}{% | 
				
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\subsection{Tables}\label{tables}} | 
				
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 | 
				
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You may also want to include tables that summarize material. Like | 
				
			||||
figures, these should be centered, legible, and numbered consecutively. | 
				
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However, place the title \emph{above} the table with at least | 
				
			||||
0.1\textasciitilde{}inches of space before the title and the same after | 
				
			||||
it, as in \cref{sample-table}. The table title should be set in | 
				
			||||
9\textasciitilde{}point type and centered unless it runs two or more | 
				
			||||
lines, in which case it should be flush left. | 
				
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 | 
				
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\begin{table}[t] | 
				
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\caption{Classification accuracies for naive Bayes and flexible | 
				
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Bayes on various data sets.} | 
				
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\label{sample-table} | 
				
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\vskip 0.15in | 
				
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\begin{center} | 
				
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\begin{small} | 
				
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\begin{sc} | 
				
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\begin{tabular}{lcccr} | 
				
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\toprule | 
				
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Data set & Naive & Flexible & Better? \\ | 
				
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\midrule | 
				
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Breast    & 95.9$\pm$ 0.2& 96.7$\pm$ 0.2& $\surd$ \\ | 
				
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Cleveland & 83.3$\pm$ 0.6& 80.0$\pm$ 0.6& $\times$\\ | 
				
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Glass2    & 61.9$\pm$ 1.4& 83.8$\pm$ 0.7& $\surd$ \\ | 
				
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Credit    & 74.8$\pm$ 0.5& 78.3$\pm$ 0.6&         \\ | 
				
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Horse     & 73.3$\pm$ 0.9& 69.7$\pm$ 1.0& $\times$\\ | 
				
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Meta      & 67.1$\pm$ 0.6& 76.5$\pm$ 0.5& $\surd$ \\ | 
				
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Pima      & 75.1$\pm$ 0.6& 73.9$\pm$ 0.5&         \\ | 
				
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Vehicle   & 44.9$\pm$ 0.6& 61.5$\pm$ 0.4& $\surd$ \\ | 
				
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\bottomrule | 
				
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\end{tabular} | 
				
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\end{sc} | 
				
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\end{small} | 
				
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\end{center} | 
				
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\vskip -0.1in | 
				
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\end{table} | 
				
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 | 
				
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Tables contain textual material, whereas figures contain graphical | 
				
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material. Specify the contents of each row and column in the table’s | 
				
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topmost row. Again, you may float tables to a column’s top or bottom, | 
				
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and set wide tables across both columns. Place two-column tables at the | 
				
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top or bottom of the page. | 
				
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 | 
				
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\hypertarget{theorems-and-such}{% | 
				
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\subsection{Theorems and such}\label{theorems-and-such}} | 
				
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 | 
				
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The preferred way is to number definitions, propositions, lemmas, etc. | 
				
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consecutively, within sections, as shown below. | 
				
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 | 
				
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\begin{definition} | 
				
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\label{def:inj} | 
				
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A function $f:X \to Y$ is injective if for any $x,y\in X$ different, $f(x)\ne f(y)$. | 
				
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\end{definition} | 
				
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 | 
				
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Using \cref{def:inj} we immediate get the following result: | 
				
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 | 
				
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\begin{proposition} | 
				
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If $f$ is injective mapping a set $X$ to another set $Y$, | 
				
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the cardinality of $Y$ is at least as large as that of $X$ | 
				
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\end{proposition} | 
				
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\begin{proof} | 
				
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Left as an exercise to the reader. | 
				
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\end{proof} | 
				
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 | 
				
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\cref{lem:usefullemma} stated next will prove to be useful. | 
				
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 | 
				
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\begin{lemma} | 
				
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\label{lem:usefullemma} | 
				
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For any $f:X \to Y$ and $g:Y\to Z$ injective functions, $f \circ g$ is injective. | 
				
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\end{lemma} | 
				
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\begin{theorem} | 
				
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\label{thm:bigtheorem} | 
				
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If $f:X\to Y$ is bijective, the cardinality of $X$ and $Y$ are the same. | 
				
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\end{theorem} | 
				
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 | 
				
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An easy corollary of \cref{thm:bigtheorem} is the following: | 
				
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 | 
				
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\begin{corollary} | 
				
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If $f:X\to Y$ is bijective, | 
				
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the cardinality of $X$ is at least as large as that of $Y$. | 
				
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\end{corollary} | 
				
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\begin{assumption} | 
				
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The set $X$ is finite. | 
				
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\label{ass:xfinite} | 
				
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\end{assumption} | 
				
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\begin{remark} | 
				
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According to some, it is only the finite case (cf. \cref{ass:xfinite}) that is interesting. | 
				
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\end{remark} | 
				
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 | 
				
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\hypertarget{citations-and-references}{% | 
				
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\subsection{Citations and References}\label{citations-and-references}} | 
				
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 | 
				
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If you rely on the \LaTeX\{\} bibliographic facility, use | 
				
			||||
\texttt{natbib.sty} included in the style-file package to obtain | 
				
			||||
reference. | 
				
			||||
 | 
				
			||||
Citations within the text should include the authors’ last names and | 
				
			||||
year. If the authors’ names are included in the sentence, place only the | 
				
			||||
year in parentheses, for example when referencing Arthur Samuel’s | 
				
			||||
pioneering work \yrcite{Samuel59}. Otherwise place the entire reference | 
				
			||||
in parentheses with the authors and year separated by a comma | 
				
			||||
\cite{Samuel59}. List multiple references separated by semicolons | 
				
			||||
\cite{kearns89,Samuel59,mitchell80}. Use the `et\textasciitilde{}al.’ | 
				
			||||
construct only for citations with three or more authors or after listing | 
				
			||||
all authors to a publication in an earlier reference | 
				
			||||
\cite{MachineLearningI}. | 
				
			||||
 | 
				
			||||
Use an unnumbered first-level section heading for the references, and | 
				
			||||
use a hanging indent style, with the first line of the reference flush | 
				
			||||
against the left margin and subsequent lines indented by 10 points. The | 
				
			||||
references at the end of this document give examples for journal | 
				
			||||
articles \cite{Samuel59}, conference publications \cite{langley00}, book | 
				
			||||
chapters \cite{Newell81}, books \cite{DudaHart2nd}, edited volumes | 
				
			||||
\cite{MachineLearningI}, technical reports \cite{mitchell80}, and | 
				
			||||
dissertations \cite{kearns89}. | 
				
			||||
 | 
				
			||||
Alphabetize references by the surnames of the first authors, with single | 
				
			||||
author entries preceding multiple author entries. Order references for | 
				
			||||
the same authors by year of publication, with the earliest first. Make | 
				
			||||
sure that each reference includes all relevant information (e.g., page | 
				
			||||
numbers). | 
				
			||||
 | 
				
			||||
Please put some effort into making references complete, presentable, and | 
				
			||||
consistent, e.g.~use the actual current name of authors. If using | 
				
			||||
bibtex, please protect capital letters of names and abbreviations in | 
				
			||||
titles, for example, use \{B\}ayesian or \{L\}ipschitz in your .bib | 
				
			||||
file. | 
				
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 | 
				
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\hypertarget{acknowledgements}{% | 
				
			||||
\section{Acknowledgements}\label{acknowledgements}} | 
				
			||||
 | 
				
			||||
Acknowledgements is an unnumbered section at the end of the paper. | 
				
			||||
Typically, this will include thanks to colleagues who contributed to the | 
				
			||||
ideas, and to funding agencies and corporate sponsors that provided | 
				
			||||
financial support. | 
				
			||||
 | 
				
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%\bibliographystyle{plainnat} | 
				
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%\bibliography{ref} | 
				
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\bibliographystyle{plainnat}% | 
				
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\bibliography{manuscript}% | 
				
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 | 
				
			||||
%\newpage | 
				
			||||
%\appendix | 
				
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%\input{content/appendix} | 
				
			||||
 | 
				
			||||
\end{document} | 
				
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 | 
				
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# Basic properties of protein and nucleic acids | 
				
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## Basic principle of protein | 
				
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### Unit structure of protein | 
				
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There are 4 levels of protein organization: | 
				
			||||
 | 
				
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* Primary structure \ | 
				
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  It consists of a chain arrangement/sequence of amino acids that are joined together to make protein. It usually uses abbreviations for the amino and residues. The example in figure (1) is Polypeptide amino. | 
				
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 | 
				
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* Secondary structure \ | 
				
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  This is the region within the long protein chains organized into regular structures known as alpha-helices ($\alpha$-helices) and beta sheet. | 
				
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 | 
				
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* Tertiary structure \ | 
				
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  Tertiary structure is a description of the way that whole chain from one or several units folds and forms 3-dimensional shape, called domain. | 
				
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 | 
				
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* Fourth order structure \ | 
				
			||||
  This structure is formed from several tertiary structure proteins or domains with more complex shape. | 
				
			||||
 | 
				
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 | 
				
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 | 
				
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<!-- | 
				
			||||
\begin{figure}[h!] | 
				
			||||
  \centering | 
				
			||||
  \includegraphics[width=0.7\linewidth]{Figure1} | 
				
			||||
  \caption{Structure of protein} | 
				
			||||
  \label{fig:figure1} | 
				
			||||
\end{figure} | 
				
			||||
--> | 
				
			||||
 | 
				
			||||
### Basic motif of protein | 
				
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There are two types of secondary structure for hydrophobic type: | 
				
			||||
 | 
				
			||||
* $\alpha$-helix\ | 
				
			||||
  This is one of common motif in secondary structure of protein. It consists of hydrogen bond with conformation of N-H group donates a hydrogen bond to the C=O group of amino acid. This structure has small Coulomb interaction in hydrogen bond and has dipole moment. | 
				
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 | 
				
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* $\beta$ sheet\ | 
				
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  Beta sheet is also one of common motif in protein, consist of $\beta$ strands connected hydrogen bonds, forming a generally twisted and pleated sheet. | 
				
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 | 
				
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 | 
				
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<!-- | 
				
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\begin{figure}[h!] | 
				
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  \centering | 
				
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  \includegraphics[width=0.7\linewidth]{Alpha_beta_structure_(full)} | 
				
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  \caption{Secondary alpha-helix and beta sheet structures} | 
				
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  \label{fig:alphabetastructurefull} | 
				
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\end{figure} | 
				
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--> | 
				
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 | 
				
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## Structure of nucleic acid, DNA and protein | 
				
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 | 
				
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###  3 structures of DNA: | 
				
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  DNA has 3 conformation that include A-DNA, B-DNA, and Z-DNA forms | 
				
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 | 
				
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* A-DNA, has same periodic double helical structure\ | 
				
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* B-DNA, slightly similar to A-DNA, but with longer and less compact than A-DNA\ | 
				
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* Z-DNA, has double helical structure which the helix winds to the left in a zigzag pattern. | 
				
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 | 
				
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 | 
				
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<!-- | 
				
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\begin{figure}[h!] | 
				
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  \centering | 
				
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  \includegraphics[width=0.4\linewidth]{Dnaconformations} | 
				
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  \caption{3 types of DNA conformations: A-DNA, B-DNA, and Z-DNA} | 
				
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  \label{fig:dnaconformations} | 
				
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\end{figure} | 
				
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--> | 
				
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 | 
				
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# Lipid and membrane protein | 
				
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  Membrane is a function of protein that divides the biological cell into inside and outside part. | 
				
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  Functions of lipid bilayer are formation of boundary, permeability, and domain formation. | 
				
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  Lipids are divided into 2 types, hydrophobic and amphiphilic molecule. \ | 
				
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 | 
				
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##  Phospholipid | 
				
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  This is one kind of lipid which is a major component in all cell membranes. | 
				
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  Phospholipids can form lipid bilayers because of their amphiphilic characteristic. | 
				
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  There are 3 kinds of self-assemble and conformation of phospholipid: | 
				
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 | 
				
			||||
* Spherical liposome\ | 
				
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* Bilayer\ | 
				
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* Micelle\ | 
				
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  Micelles and bilayers form in the polar medium by a process of hydrophobic effect. | 
				
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 | 
				
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 | 
				
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<!-- | 
				
			||||
  \begin{figure}[h!] | 
				
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    \centering | 
				
			||||
    \includegraphics[width=0.5\linewidth]{phospholipid} | 
				
			||||
    \caption{Self-organization of phospholipids: a spherical liposome, a micelle, and a lipid bilayer.} | 
				
			||||
    \label{fig:phospholipid} | 
				
			||||
  \end{figure} | 
				
			||||
--> | 
				
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 | 
				
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 | 
				
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# Molecular dynamics simulation | 
				
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## Simulation (Computational science) | 
				
			||||
  Simulation is the one that connect the theory and experiment in science. | 
				
			||||
  When the theory contributes in analysis and prediction, and experiments are contributing in confirming the theory and developing, simulation takes part in calculation in advanced theoretical and virtual simulation experiment. | 
				
			||||
  Simulation is using the computer to do things, such as reproduce, understand, and predict the theory and also the experiment. | 
				
			||||
  Simulation is applied in many fields, such as science, pharmacy, climate, earthquake, finance and social, and many other things. | 
				
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 | 
				
			||||
## Supercomputer\ | 
				
			||||
  Supercomputer is a computer with high level of performance. | 
				
			||||
  One of the biggest supercomputer in Japan is K Computer, located in Kobe, is having 705,024 cores and 10.510 PFlops (floating-point operations per second). | 
				
			||||
  Supercomputer is practically used in computational science, such as biological, material, earth, and fundamental physical sciences. | 
				
			||||
  Also, supercomputer is used in simulation of next-generation technology. | 
				
			||||
 | 
				
			||||
## Computational science in biology\ | 
				
			||||
  Computational science is usually used in exploring phenomena in biological system; substances, structure and functions of proteins, membranes, and enzymes, energy and chemical reaction, and signal transduction. | 
				
			||||
  These can be modeled using molecular dynamics (MD) simulation. | 
				
			||||
  The fundamental parts in MD are analytical mechanics (Newton's equation), Schrodinger's equation in quantum mechanics, statistical mechanics, and electromagnetization principle. | 
				
			||||
 | 
				
			||||
 | 
				
			||||
###  _Brownian Dynamics_ | 
				
			||||
  Brownian dynamics describes the physical phenomena of zig-zag motion of particle. The motion is arising from collision between the particles. This motion resulted diffusion equation that obeys the law of mass conservation, formulated in the equation: | 
				
			||||
  \begin{equation} | 
				
			||||
    \frac{\partial \rho (x,t)}{\partial t} = -\frac{\partial J(x,t)}{\partial x} | 
				
			||||
  \end{equation} | 
				
			||||
  where $\rho (x,t)$ is density of material and $J (x,t)$ is flux, the rate of flow. | 
				
			||||
 | 
				
			||||
###  _Langevin Equation_ | 
				
			||||
  This equation describes the \textit{Brownian motion} that uses Newton's equation of motion and degree of freedom. $\varrho$ is denoted as the friction coefficient. | 
				
			||||
  \begin{equation} | 
				
			||||
    m\frac{d^2 x}{d t^2} = F - \varrho \frac{dx}{dt} + \xi (t) | 
				
			||||
  \end{equation} | 
				
			||||
 | 
				
			||||
###  Einstein's relational expression: | 
				
			||||
  \begin{equation} | 
				
			||||
    D=\frac{k_BT}{\varrho} | 
				
			||||
  \end{equation} | 
				
			||||
 | 
				
			||||
 | 
				
			||||
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		Reference in new issue