@ -6,10 +6,12 @@ author:
correspond: true
affiliation: "My City University"
address: "Orenomachi, Orenoshi, Orenoken, Japan"
email: "one@myuni.ac.jp"
- number: 2
name: "Author Two"
affiliation: "My Other City University"
address: "Hokanomachi, Orenoshi, Orenoken, Japan"
email: "two@myuni.ac.jp"
email: "xxx@myuni.ac.jp"
titleshort: "Paperlighter Example"
authorshort: "Author One et.al."
@ -19,302 +21,321 @@ linkDir:
appendix:
- appendix/1
- appendix/2
abstract:
Using \LaTeX{} to write papers is concise and convenient. However, for
writing in life, complicated \LaTeX{} style-files (e.g., elegantpaper)
are difficult to access, or submission style-files (e.g., journal or
conference) are not free indeed. To tackle these problems and satisfy an
elegant and straightforward scientific writing,
\textbf{paperlighter.sty}, a one-column style-file, is designed. This
document is edited from icml2022.sty and provides a basic paper
template. Compared to icml2022.sty, paperlighter.sty contain fewer
operations, reducing adjustment while keep graceful.
\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.}}
abstractTex:
\abstract{To investigate the physical nature of the `nuc\-leated instability' of
proto giant planets, the stability of layers
in static, radiative gas spheres is analysed on the basis of Baker's
standard one-zone model.}
{To investigate the physical nature of the `nuc\-leated instability' of
proto giant planets, the stability of layers
in static, radiative gas spheres is analysed on the basis of Baker's
standard one-zone model.}
{It is shown that stability depends only upon the equations of state, the opacities and the local
thermodynamic state in the layer. Stability and instability can
therefore be expressed in the form of stability equations of state
which are universal for a given composition.}
{The stability equations of state are
calculated for solar composition and are displayed in the domain
$-14 \leq \lg \rho / \mathrm{[g\, cm^{-3}]} \leq 0 $,
$ 8.8 \leq \lg e / \mathrm{[erg\, g^{-1}]} \leq 17.7$. These displays
may be
used to determine the one-zone stability of layers in stellar
or planetary structure models by directly reading off the value of
the stability equations for the thermodynamic state of these layers,
specified
by state quantities as density $\rho$, temperature $T$ or
specific internal energy $e$.
Regions of instability in the $(\rho,e)$-plane are described
and related to the underlying microphysical processes.}
{Vibrational instability is found to be a common phenomenon
at temperatures lower than the second He ionisation
zone. The $\kappa$-mechanism is widespread under `cool'
conditions.}
{}
keywords: giant planet formation -- $\kappa$-mechanism -- stability of gas spheres
acknowledgements:
Part of this work was supported by the German
\emph{Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} project
number Ts~17/2--1.
---
# Introduction
In the \emph{nucleated instability\/} (also called core
instability) hypothesis of giant planet
formation, a critical mass for static core envelope
protoplanets has been found. \citet{langley00} determined
the critical mass of the core to be about $12 \,M_\oplus$
($M_\oplus=5.975 \times 10^{27}\,\mathrm{g}$ is the Earth mass), which
is independent of the outer boundary
conditions and therefore independent of the location in the
solar nebula. This critical value for the core mass corresponds
closely to the cores of today's giant planets.
Although no hydrodynamical study has been available many workers
conjectured that a collapse or rapid contraction will ensue
after accumulating the critical mass. The main motivation for
this article
is to investigate the stability of the static envelope at the
critical mass. With this aim the local, linear stability of static
radiative gas spheres is investigated on the basis of Baker's
(\citeyear{mitchell80}) standard one-zone model.
Phenomena similar to the ones described above for giant planet
formation have been found in hydrodynamical models concerning
star formation where protostellar cores explode
(Tscharnuter \citeyear{kearns89}, Balluch \citeyear{MachineLearningI}),
whereas earlier studies found quasi-steady collapse flows. The
similarities in the (micro)physics, i.e., constitutive relations of
protostellar cores and protogiant planets serve as a further
motivation for this study.
# Baker's standard one-zone model
\begin{figure*}
\centering
\caption{Adiabatic exponent $\Gamma_1$.
$\Gamma_1$ is plotted as a function of
$\lg$ internal energy $\mathrm{[erg\,g^{-1}]}$ and $\lg$
density $\mathrm{[g\,cm^{-3}]}$.}
\label{FigGam}%
\end{figure*}
In this section the one-zone model of \citet{DudaHart2nd},
originally used to study the Cephe{\"{\i}}d pulsation mechanism, will
be briefly reviewed. The resulting stability criteria will be
rewritten in terms of local state variables, local timescales and
constitutive relations.
\citet{DudaHart2nd} investigates the stability of thin layers in
self-gravitating,
spherical gas clouds with the following properties:
\begin{itemize}
\item hydrostatic equilibrium,
\item thermal equilibrium,
\item energy transport by grey radiation diffusion.
\end{itemize}
For the one-zone-model Baker obtains necessary conditions
for dynamical, secular and vibrational (or pulsational)
stability (Eqs.\ (34a,\,b,\,c) in Baker \citeyear{DudaHart2nd}). Using Baker's
notation:
\noindent
and with the definitions of the \emph{local cooling time\/}
(see Fig.~\ref{FigGam})
\begin{equation}
\tau_{\mathrm{co}} = \frac{E_{\mathrm{th}}}{L_{r0}} \,,
\end{equation}
and the \emph{local free-fall time}
\begin{equation}
\tau_{\mathrm{ff}} =
\sqrt{ \frac{3 \pi}{32 G} \frac{4\pi r_0^3}{3 M_{\mathrm{r}}}
}\,,
\end{equation}
Baker's $K$ and $\sigma_0$ have the following form:
\begin{eqnarray}
\sigma_0 & = & \frac{\pi}{\sqrt{8}}
\frac{1}{ \tau_{\mathrm{ff}}} \\
K & = & \frac{\sqrt{32}}{\pi} \frac{1}{\delta}
\frac{ \tau_{\mathrm{ff}} }
{ \tau_{\mathrm{co}} }\,;
\end{eqnarray}
where $E_{\mathrm{th}} \approx m (P_0/{\rho_0})$ has been used and
\begin{equation}
\begin{array}{l}
\delta = - \left(
\frac{ \partial \ln \rho }{ \partial \ln T }
\right)_P \\
e=mc^2
\end{array}
\end{equation}
is a thermodynamical quantity which is of order $1$ and equal to $1$
for nonreacting mixtures of classical perfect gases. The physical
meaning of $\sigma_0$ and $K$ is clearly visible in the equations
above. $\sigma_0$ represents a frequency of the order one per
free-fall time. $K$ is proportional to the ratio of the free-fall
time and the cooling time. Substituting into Baker's criteria, using
thermodynamic identities and definitions of thermodynamic quantities,
\begin{displaymath}
\Gamma_1 = \left( \frac{ \partial \ln P}{ \partial\ln \rho}
\right)_{S} \, , \;
\chi^{}_\rho = \left( \frac{ \partial \ln P}{ \partial\ln \rho}
\right)_{T} \, , \;
\kappa^{}_{P} = \left( \frac{ \partial \ln \kappa}{ \partial\ln P}
\right)_{T}
\end{displaymath}
\begin{displaymath}
\nabla_{\mathrm{ad}} = \left( \frac{ \partial \ln T}
{ \partial\ln P} \right)_{S} \, , \;
\chi^{}_T = \left( \frac{ \partial \ln P}
{ \partial\ln T} \right)_{\rho} \, , \;
\kappa^{}_{T} = \left( \frac{ \partial \ln \kappa}
{ \partial\ln T} \right)_{T}
\end{displaymath}
one obtains, after some pages of algebra, the conditions for
\emph{stability\/} given
below:
\begin{eqnarray}
\frac{\pi^2}{8} \frac{1}{\tau_{\mathrm{ff}}^2}
( 3 \Gamma_1 - 4 )
& > & 0 \label{ZSDynSta} \\
\frac{\pi^2}{\tau_{\mathrm{co}}
\tau_{\mathrm{ff}}^2}
\Gamma_1 \nabla_{\mathrm{ad}}
\left[ \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T }
( \kappa^{}_T - 4 )
+ \kappa^{}_P + 1
\right]
& > & 0 \label{ZSSecSta} \\
\frac{\pi^2}{4} \frac{3}{\tau_{ \mathrm{co} }
\tau_{ \mathrm{ff} }^2
}
\Gamma_1^2 \, \nabla_{\mathrm{ad}} \left[
4 \nabla_{\mathrm{ad}}
- ( \nabla_{\mathrm{ad}} \kappa^{}_T
+ \kappa^{}_P
)
- \frac{4}{3 \Gamma_1}
\right]
& > & 0 \label{ZSVibSta}
\end{eqnarray}
For a physical discussion of the stability criteria see \citet{DudaHart2nd} or \citet{anonymous}.
We observe that these criteria for dynamical, secular and
vibrational stability, respectively, can be factorized into
\begin{enumerate}
\item a factor containing local timescales only,
\item a factor containing only constitutive relations and
their derivatives.
\end{enumerate}
The first factors, depending on only timescales, are positive
by definition. The signs of the left hand sides of the
inequalities~(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta})
therefore depend exclusively on the second factors containing
the constitutive relations. Since they depend only
on state variables, the stability criteria themselves are \emph{
functions of the thermodynamic state in the local zone}. The
one-zone stability can therefore be determined
from a simple equation of state, given for example, as a function
of density and
temperature. Once the microphysics, i.e.\ the thermodynamics
and opacities (see Table~\ref{KapSou}), are specified (in practice
by specifying a chemical composition) the one-zone stability can
be inferred if the thermodynamic state is specified.
The zone -- or in
other words the layer -- will be stable or unstable in
whatever object it is imbedded as long as it satisfies the
one-zone-model assumptions. Only the specific growth rates
(depending upon the time scales) will be different for layers
in different objects.
\begin{table}
\caption[]{Opacity sources.}
\label{KapSou}
$$
\begin{array}{p{0.5\linewidth}l}
\hline
\noalign{\smallskip}
Source & T / {[\mathrm{K}]} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Yorke 1979, Yorke 1980a & \leq 1700^{\mathrm{a}} \\
Kr\"ugel 1971 & 1700 \leq T \leq 5000 \\
Cox \& Stewart 1969 & 5000 \leq \\
\noalign{\smallskip}
\hline
\end{array}
$$
\end{table}
We will now write down the sign (and therefore stability)
determining parts of the left-hand sides of the inequalities
(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) and thereby
obtain \emph{stability equations of state}.
The sign determining part of inequality~(\ref{ZSDynSta}) is
$3\Gamma_1 - 4$ and it reduces to the
criterion for dynamical stability
\begin{equation}
\Gamma_1 > \frac{4}{3}\,\cdot
\end{equation}
Stability of the thermodynamical equilibrium demands
\begin{equation}
\chi^{}_\rho > 0, \;\; c_v > 0\, ,
\end{equation}
and
\begin{equation}
\chi^{}_T > 0
\end{equation}
holds for a wide range of physical situations.
With
\begin{eqnarray}
\Gamma_3 - 1 = \frac{P}{\rho T} \frac{\chi^{}_T}{c_v}& >& 0\\
\Gamma_1 = \chi_\rho^{} + \chi_T^{} (\Gamma_3 -1)& >& 0\\
\nabla_{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} & >& 0
\end{eqnarray}
we find the sign determining terms in inequalities~(\ref{ZSSecSta})
and (\ref{ZSVibSta}) respectively and obtain the following form
of the criteria for dynamical, secular and vibrational
\emph{stability}, respectively:
\begin{eqnarray}
3 \Gamma_1 - 4 =: S_{\mathrm{dyn}} > & 0 & \label{DynSta} \\
\frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 )
+ \kappa^{}_P + 1 =: S_{\mathrm{sec}} > & 0 & \label{SecSta} \\
4 \nabla_{\mathrm{ad}} - (\nabla_{\mathrm{ad}} \kappa^{}_T
+ \kappa^{}_P)
- \frac{4}{3 \Gamma_1} =: S_{\mathrm{vib}}
> & 0\,.& \label{VibSta}
\end{eqnarray}
The constitutive relations are to be evaluated for the
unperturbed thermodynamic state (say $(\rho_0, T_0)$) of the zone.
We see that the one-zone stability of the layer depends only on
the constitutive relations $\Gamma_1$,
$\nabla_{\mathrm{ad}}$, $\chi_T^{},\,\chi_\rho^{}$,
$\kappa_P^{},\,\kappa_T^{}$.
These depend only on the unperturbed
thermodynamical state of the layer. Therefore the above relations
define the one-zone-stability equations of state
$S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
and $S_{\mathrm{vib}}$. See Fig.~\ref{FigVibStab} for a picture of
$S_{\mathrm{vib}}$. Regions of secular instability are
listed in Table~1.
\begin{figure}
\centering
\caption{Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
}
\label{FigVibStab}
\end{figure}
# Conclusions
\begin{enumerate}
\item The conditions for the stability of static, radiative
layers in gas spheres, as described by Baker's (\citeyear{DudaHart2nd})
standard one-zone model, can be expressed as stability
equations of state. These stability equations of state depend
only on the local thermodynamic state of the layer.
\item If the constitutive relations -- equations of state and
Rosseland mean opacities -- are specified, the stability
equations of state can be evaluated without specifying
properties of the layer.
\item For solar composition gas the $\kappa$-mechanism is
working in the regions of the ice and dust features
in the opacities, the $\mathrm{H}_2$ dissociation and the
combined H, first He ionization zone, as
indicated by vibrational instability. These regions
of instability are much larger in extent and degree of
instability than the second He ionization zone
that drives the Cephe{\"\i}d pulsations.
\end{enumerate}
# Format of the Paperlighter
Format of paperlighter is defined in this section.
## Dimensions
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
margins will depend on whether you print on US letter or A4 paper, but
all final versions must be produced for US letter size.
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.
## 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.
## Author Information for Submission
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.
## 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.
## Partitioning the Text
You should organize your paper into sections and paragraphs to help
readers place a structure on the material and understand its
contributions.
### 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.
### Paragraphs and Footnotes
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.}
\begin{figure}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{Figure/icml_numpapers.eps}}
\caption{Historical locations and number of accepted papers for International
Machine Learning Conferences (ICML 1993 -- ICML 2008) and International
Workshops on Machine Learning (ML 1988 -- ML 1992). At the time this figure was
produced, the number of accepted papers for ICML 2008 was unknown and instead
estimated.}
\label{icml-historical}
\end{center}
\vskip -0.2in
\end{figure}
## Figures
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.
## Algorithms
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.
\begin{algorithm}[tb]
\caption{Bubble Sort}
\label{alg:example}
\begin{algorithmic}
\STATE {\bfseries Input:} data $x_i$, size $m$
\REPEAT
\STATE Initialize $noChange = true$.
\FOR{$i=1$ {\bfseries to} $m-1$}
\IF{$x_i > x_{i+1}$}
\STATE Swap $x_i$ and $x_{i+1}$
\STATE $noChange = false$
\ENDIF
\ENDFOR
\UNTIL{$noChange$ is $true$}
\end{algorithmic}
\end{algorithm}
## Tables
You may also want to include tables that summarize material. Like
figures, these should be centered, legible, and numbered consecutively.
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.
\begin{table}[t]
\caption{Classification accuracies for naive Bayes and flexible
Bayes on various data sets.}
\label{sample-table}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccr}
\toprule
Data set & Naive & Flexible & Better? \\
\midrule
Breast & 95.9$\pm$ 0.2& 96.7$\pm$ 0.2& $\surd$ \\
Cleveland & 83.3$\pm$ 0.6& 80.0$\pm$ 0.6& $\times$\\
Glass2 & 61.9$\pm$ 1.4& 83.8$\pm$ 0.7& $\surd$ \\
Credit & 74.8$\pm$ 0.5& 78.3$\pm$ 0.6& \\
Horse & 73.3$\pm$ 0.9& 69.7$\pm$ 1.0& $\times$\\
Meta & 67.1$\pm$ 0.6& 76.5$\pm$ 0.5& $\surd$ \\
Pima & 75.1$\pm$ 0.6& 73.9$\pm$ 0.5& \\
Vehicle & 44.9$\pm$ 0.6& 61.5$\pm$ 0.4& $\surd$ \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
Tables contain textual material, whereas figures contain graphical
material. Specify the contents of each row and column in the table’s
topmost row. Again, you may float tables to a column’s top or bottom,
and set wide tables across both columns. Place two-column tables at the
top or bottom of the page.
## Theorems and such
The preferred way is to number definitions, propositions, lemmas, etc.
consecutively, within sections, as shown below.
\begin{definition}
\label{def:inj}
A function $f:X \to Y$ is injective if for any $x,y\in X$ different, $f(x)\ne f(y)$.
\end{definition}
Using \cref{def:inj} we immediate get the following result:
\begin{proposition}
If $f$ is injective mapping a set $X$ to another set $Y$,
the cardinality of $Y$ is at least as large as that of $X$
\end{proposition}
\begin{proof}
Left as an exercise to the reader.
\end{proof}
\cref{lem:usefullemma} stated next will prove to be useful.
\begin{lemma}
\label{lem:usefullemma}
For any $f:X \to Y$ and $g:Y\to Z$ injective functions, $f \circ g$ is injective.
\end{lemma}
\begin{theorem}
\label{thm:bigtheorem}
If $f:X\to Y$ is bijective, the cardinality of $X$ and $Y$ are the same.
\end{theorem}
An easy corollary of \cref{thm:bigtheorem} is the following:
\begin{corollary}
If $f:X\to Y$ is bijective,
the cardinality of $X$ is at least as large as that of $Y$.
\end{corollary}
\begin{assumption}
The set $X$ is finite.
\label{ass:xfinite}
\end{assumption}
\begin{remark}
According to some, it is only the finite case (cf. \cref{ass:xfinite}) that is interesting.
\end{remark}
## Citations and References
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.
# 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.
Hasan al Rasyid
3 years ago