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% Full title of the paper (Capitalized)
\Title{Multi-periodicity of high-frequency type III bursts as a signature of the fragmented magnetic reconnection}

% MDPI internal command: Title for citation in the left column
\TitleCitation{Multi-periodicity of reverse type III bursts}

% Author Orchid ID: enter ID or remove command
\newcommand{\orcidauthorA}{0000-0002-3963-8701} % Add \orcidB{} behind the author's name
\newcommand{\orcidauthorB}{0000-0003-3128-8396} % Add \orcidA{} behind the author's name

% Authors, for the paper (add full first names)
\Author{Marian Karlick\'{y} $^{1}$\orcidA{} and J\'{a}n Ryb\'{a}k $^{2}$\orcidB{}}

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\AuthorNames{Marian Karlick\'{y} and J\'{a}n Ryb\'{a}k}

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\address{%
$^{1}$ \quad Astronomical Institute of the Czech Academy of Sciences, Fri\v{c}ova 298, CZ -- 251 65 Ond\v rejov, Czech Republic; karlicky@asu.cas.cz\\
$^{2}$ \quad Astronomical Institute, Slovak Academy of Sciences, SK -- 05960
                  Tatransk\'{a} Lomnica, Slovakia; rybak@astro.sk}

% Contact information of the corresponding author
\corres{Correspondence: karlicky@asu.cas.cz}

% Current address and/or shared authorship
%\firstnote{Current address: Affiliation 3.} 
%\secondnote{These authors contributed equally to this work.}
% The commands \thirdnote{} till \eighthnote{} are available for further notes

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% Abstract (Do not insert blank lines, i.e. \\) 
\abstract{Using the radio spectra of the 2 April 2022 eruptive flare we analyze
   a group of high-frequency type III bursts by our new wavelet method.
   In this analysis, we found a multi-periodicity of these bursts that is interpreted
   by the electron beams accelerated in the fragmented magnetic reconnection
   in the rising magnetic rope. We propose that each period in these type III bursts
   is a result of the periodic interaction of sub-ropes formed in the rising magnetic rope.
   In each interaction, the period depends on the diameter of interacting sub-ropes and local Alfv\'{e}n
   velocity. This interpretation is supported \textbf{by detection of the 
   specific EUV structure which was, according to our knowledge, observed for
   the first time.} All proposed processes occur in the rising magnetic rope. Thus,
   this flare deviates from the standard flare model, where the main magnetic
   reconnection is located below the rising magnetic rope.}

% Keywords
\keyword{Sun; flares; radio radiaton} 

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%\textbf{What are the main findings?}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

%\endnote{This is an endnote.} % To use endnotes, please un-comment \printendnotes below (before References). Only journal Laws uses \footnote.

\section{Introduction}

Solar flares are the most powerful events in the solar system. The
strongest flares are associated with coronal mass ejections and acceleration of
particles into interplanetary space. From the physical point of view, solar flares
are explosive phenomena in the solar atmosphere, in which the energy
accumulated in the magnetic field and electric currents is rapidly transformed
into plasma heating, plasma flows, accelerated particles and emission in a
broad range of electromagnetic waves: from radio, through optical, UV, X-rays
to gamma-rays. For more details, see the reviews 
\cite{2002A&ARv..10..313P,2002SSRv..101....1A,2008A&ARv..16..155K,2009AdSpR..43..739S,2011SSRv..159...19F,2016SSRv..200...75N}.
The most powerful flares are the eruptive flares that are described by the so-called
standard CSHKP flare model
\citep{1964NASSP..50..451C,1966Natur.211..695S,1974SoPh...34..323H,1976SoPh...50...85K}
or by its generalized 3-dimensional version
\citep{2012A&A...543A.110A,2014ApJ...788...60J,2014ApJ...784..144D}. In this
model, the magnetic rope, carrying the cold and dense filament, is ejected due
to toroidal or kink instabilities. Then below the rising magnetic rope, the
narrow current sheet is formed where the main energetic process (magnetic
reconnection) occurs. Except for some papers \cite[e.g.,][]{2020ApJS..250...31K}
the energetic flare processes in the rising magnetic rope are not considered.

In radio, the eruptive flares are associated with several types of bursts (type
II, III, IV, V, J, U) and their fine structures \citep{2004psci.book.....A}.
Among them, there are the high-frequency type III bursts with the normal and reverse
\textbf{frequency} drifts that are interpreted as those \textbf{generated by electron
  beams with the plasma emission mechanism.} While in the normal drifting type III bursts
the electron beams are propagating upwards in the solar atmosphere, in the reverse
drifting bursts the beams bombard the
dense layers of the solar atmosphere and generate the hard X-ray
emission.

In this paper, we analyze an interesting group of the type III bursts
observed during the 2 April 2022 flare. Using our new method we detected
multi-periodicity in these bursts. We interpret this multi-periodicity by
fragmented reconnection. We support our interpretation with a unique
EUV structure. Contrary to the standard flare model, the proposed processes
happen inside the rising magnetic rope.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Data}

\begin{figure}
    \begin{center}
%       \includegraphics[scale=0.28,angle=90]{fermi_1320_1332_025_101keV.eps}
        \includegraphics[width=8cm]{20220402_fig_1_top.eps}
        \includegraphics[width=8cm]{20220402_fig_1_bottom.eps}
    \end{center}
    \caption{The X-ray and radio fluxes of the 2 April 2022 flare in the time interval
      13:20-\textbf{13:32}\,UT. Upper panel: FERMI X-ray fluxes in two energy channels
      (24.9-49.4 and 49.9-101.4\,keV).
      Bottom panel: the radio fluxes at 4\,GHz (black line), 3\,GHz (red), 2\,GHz (blue)
      and 1\,GHz (green).
      The 3\,GHz -, 2\,GHz -, 1\,GHz - fluxes are step by step shifted for 100 a.u. upwards.}
    \label{fig1}
\end{figure}

\begin{figure}
    \begin{center}
%       \includegraphics[width=8cm]{20220402_13-20-00-13-32-00_100-10_CALLISTO_GREENLAND_lut_5_version_3.eps}
       \includegraphics[width=8cm]{20220402_13-20-00-13-31-59_1000-150_ORFEES_lut_5_version_2.eps}
       \includegraphics[width=8cm]{GlobalRT5.eps}
       \includegraphics[width=8cm]{GlobalRT4.eps}
    \end{center}
    \caption{Global 150\,--\,5000\,MHz radio spectrum in the time interval 13:20\,--\,13:32\,UT
      observed during the 2 April 2022 flare: the 150\,--\,1000\,MHz ORFEES spectrum (top panel)
      and the 800\,--\,2000\,MHz and 2000\,--\,5000\,MHz Ond\v{r}ejov radio spectra (bottom panels).}
    \label{fig2}
\end{figure}

\begin{figure}
	\begin{center}
 		\includegraphics[width=8cm]{a3min.eps}
		\includegraphics[width=8cm]{b3min.eps}
	\end{center}
	\caption{\textbf{Detailed 800\,--\,5000\,MHz radio spectrum in the time interval
            13:22:50\,--\,13:25:50\,UT. See, the bifurcation region at 1200\,--\,1400\,MHz
            and 13:24:50\,--\,13:25:30\,UT, where type III bursts start to drift to higher
            or lower frequencies.}}
	\label{fig3}
\end{figure}


\begin{figure}
    \begin{center}
    \includegraphics[width=8cm]{radio_base_WT_no_overplot.eps}
       \includegraphics[width=7.3cm, bb= -40 0 450 130]{period_histogram_all_hashed_freq_2000-5000_MHz_time_132200-132700_6_95.eps}
       \includegraphics[width=8cm]{radio_7.60-11.30_s.eps}
       \includegraphics[width=8cm]{radio_5.20-7.50_s.eps}
       \includegraphics[width=8cm]{radio_3.40-5.09_s.eps}
       \includegraphics[width=8cm]{radio_1.90-3.30_s.eps}
    \end{center}
    \caption{The type III bursts observed on 2 April 2022
in the 2000-5000\,MHz range in the 13:22:00--13:27:00\,UT time interval: first panel
(from top to bottom): the radio spectrum observed by the Ond\v{r}ejov radiospectrograph;
second panel: histogram of the significant periods detected by the wavelet transform
(cross-hatched bands, separated by the vertical lines, mark intervals of periods in bottom phase maps);
and bottom four panels: the phase
maps (pink areas with the black lines showing the zero phase of oscillations) overplotted on
the radio spectrum for periods detected in the ranges of 7.6--11.3, 5.2--7.5, 3.4--5.1 and 1.9--3.3 seconds. The significance level of these phase maps is 95$\%$.}
    \label{fig4}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{radio_base_WT_no_overplot.eps}
		\includegraphics[width=7.3cm, bb= -40 0 450 130]{period_histogram_all_hashed_freq_2000-5000_MHz_time_132200-132700_6_99.eps}
		\includegraphics[width=8cm]{radio_7.60-12.20_99.eps}
		\includegraphics[width=8cm]{radio_5.20-7.50_99.eps}
		\includegraphics[width=8cm]{radio_3.40-5.09_99.eps}
		\includegraphics[width=8cm]{radio_1.90-3.30_99.eps}
	\end{center}
	\caption{The type III bursts observed on 2 April 2022
	  in the 2000--5000\,MHz range in the 13:22:00--13:27:00 UT time interval:
          the first panel (from top to bottom):
          the radio spectrum observed by the Ond\v{r}ejov radiospectrograph;
	  second panel: histogram of the significant periods detected by the wavelet transform
          (cross-hatched bands, separated by vertical lines,
          mark intervals of periods in bottom phase maps);
          and bottom four panels:
          the phase maps (pink areas with the black lines showing the zero phase of oscillations)
          overplotted on the radio spectrum for periods detected in the ranges
          of 7.6--12.2, 5.2--7.5, 3.4--5.1 and 1.9--3.3 seconds.
	  The significance level of these phase maps is 99$\%$.
}
	\label{fig5}
\end{figure}

\begin{figure}
    \begin{center}
       \includegraphics[width=6cm]{fig5_a.eps}
       \includegraphics[width=6cm]{fig5_b.eps}
    \end{center}
    \caption{The wavelet power spectrum for the radio flux detected at 3\,GHz in the
      13:22\,--\,13:27\,UT time interval with the overplotted significance levels (white lines)
      set to 95$\%$ (left panel) and 99$\%$ (right panel).}
    \label{fig6}
\end{figure}


We used data from three different radiospectrographs: the ORFEES radiospectrograph
working in the 150--1000 MHz range with resolutions of 1.0\,s and 0.98\,MHz
(publicly available data, Observation Radio Frequence pour l'Etude des
Eruptions Solaires radiospectrograph, Nancay, France), and the Ond\v{r}ejov
radiospectrographs working in the ranges 800\,--\,2000\,MHz and 2000\,--\,5000\,MHz with
resolutions of 0.01\,s and 4.7\,MHz, and 0.01\,s and 11.7\,MHz
\citep{2008SoPh..253...95J}, respectively. For comparison with radio, we 
used X-ray and EUV observations from the FERMI \citep{2009ApJ...702..791M} and
SDO/AIA \citep{2012SoPh..275...17L} instruments.

\subsection{Data description}
On April 2, 2022, the M3.9 flare occurred in NOAA AR 2975. According to the GOES
observation, its start was at 12:56 UT, maximum at 13:55\,UT and end at 14:44\,UT.
A global overview of this flare in hard X-ray and radio observations is shown in
Fig.~\ref{fig1} and~\ref{fig2}. \textbf{To see more details, in Fig.~\ref{fig3}
the 800-5000 MHz spectrum in the 13:22:50\,--\,13:25:50\,UT time interval is shown.
Here, we can see the positively drifting type III bursts (mostly at higher frequencies)
as well as the negatively drifting type III bursts (mostly at lower frequencies).
The most interesting part of the spectrum is at about 1400\,MHz and about 13:25\,UT,
where type III bursts start to drift to higher and lower frequencies. At this
bifurcation region with a frequency of about 1400\,MHz electron beams,
generating type III bursts, are accelerated and propagate to lower heights
with higher density (positive frequency drift) and to higher heights with
a lower density (negative frequency drift). Considering the plasma emission
mechanism of the type III burst generation for the emission on the fundamental
frequency, the plasma density in the bifurcation (acceleration region)
can be estimated as 2.4$\times$10$^{10}$ cm$^{-3}$. On the other hand, for the
emission on the harmonic frequency, this density is 6.0$\times$10$^{9}$ cm$^{-3}$.}
All these bursts were followed in the 1000\,--\,4000\,MHz range by a broadband continuum
that was associated with the hard X-ray flux enhancement
(see Fig.~\ref{fig1}). \textbf{In the 800\,--\,5000\,MHz range at times after 13:32\,UT
till the flare end only broadband continuum with decreasing radio flux was observed.}  
On lower radio frequencies (ORFEES spectrum) the type III bursts and
pulsations were recorded in the 300\,--\,800\,MHz range at 13:21\,--\,13:29\,UT.

\subsection{Method of radio data analysis}

Before calculations, the Ond\v{r}ejov radiospectrum data were resampled to 0.1\,s
temporal resolution.
In our analysis, we searched for quasi-periodic variations of radio fluxes in
these radio spectra. For this purpose, we used our new method as described in
\cite{2017SoPh..292....1K,2017SoPh..292...94K}.
It is based on the wavelet transform (WT) \citep{1998BAMS...79...61T} providing a
clear detection of the time-frequency evolution of the strong radio wave patterns.
The Morlet mother wavelet, consisting of a complex sine wave modulated by a Gaussian, was
used to search for radio signal variability, with the non-dimensional
frequency $\omega_0$ satisfying the admissibility condition
\citep{1992AnRFM..24..395F}. The WT was calculated for the period range
starting from 4-time steps with scales sampled as a fractional power of two with
${\delta j = 0.4}$. Both the calculated significance of the derived WT
periodicities and the cone-of-influence are taken into account as described in
\cite{2017SoPh..292....1K}. For analysis, the value of the significance level was set to
95$\%$. However, for verification of results, we increased the significance level to 99$\%$.
% MK
\textbf{An example of the WT power spectrum, calculated using the above mentioned
mother wavelet with the specified values of the WT parameters, is shown in
Fig.~\ref{fig4} and Fig.~\ref{fig5} with two different significance levels of 95$\%$ and 99$\%$
overplotted on the same power spectrum.}.
% JR
\textbf{An illustrative example of the WT power spectrum, calculated using the above mentioned
mother wavelet with the specified values of the WT parameters, is shown in
Fig.~\ref{fig6} with two different significance levels of 95$\%$ and 99$\%$
overplotted on the same power spectrum.}.

Similarly as in papers by \cite{2017SoPh..292....1K,2017SoPh..292...94K,2020ApJ...889...72K},
at first, \textbf{we computed the histogram of the detected periods in the whole
time-frequency domain of the radio spectra}.
Each peak in this histogram represents a group of significant periodic signals
of a roughly similar period \textbf{without the location of the periodic signal
in the time-frequency domain of the radio spectra}.

To show the temporal and frequency location of these periods, for each peak
in the period histograms we made maps of the period phases which we
overplotted on the radio spectra. The phases are drawn only in these radiospectrogram
time-frequency locations where the periodicities were detected \textbf{at least}
with the specified significance.
The values of the phases are displayed using their angular values
(0$^{\circ}$\,--\,360$^{\circ}$). 

\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{aia304_0419arr.eps}
	\end{center}
	\caption{The AIA/SDO 304\,\AA~image was taken at 13:23:53\,UT.
          The letters \textbf{HS mark three bright features showing} the helical structure of the magnetic rope and
          the letter B brightenings near the magnetic rope footpoint.}
	\label{fig7}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{aia304_0427arrt.eps}
	\end{center}
	\caption{The AIA/SDO 304\,\AA~image taken at 13:25:29\,UT.
          The arrow shows the unique semi-circular structure.}
	\label{fig8}
\end{figure}


\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{rope.eps}
	\end{center}
	\caption{A sketch of the analyzed processes scenario.}
	\label{fig9}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}

Analyzing radio spectra we found the most interesting example of the radio flux variations for
the group of type III bursts at 13:22\,--\,13:27\,UT in the 2000\,--\,5000\,MHz range, where
most of them were of the reverse type III bursts. This part of the spectrum
together with the histogram of detected periods and four examples of phase maps (for
7.6-11.3, 5.2-7.5, 3.4-5.1 and 1.9-3.3\,s quasi-periods given by peaks in the histogram),
overplotted on the radio spectrum, are shown in Figs.~\ref{fig4}.

The histogram of periods and phase maps in Fig.~\ref{fig4} show several quasi-periodic
processes (for 7.6-11.3, 5.2-7.5, 3.4-5.1 and 1.9-3.3\,s quasi-periods) around 13:24\,UT,
especially close to 3\,GHz.
To verify these results we increased the significance level from 95$\%$ to 99$\%$
(Fig.~\ref{fig5}). As seen by the comparison of Fig.~\ref{fig4} and Fig.~\ref{fig5}, the peaks
in histograms and forms of phase maps are nearly the same.
To see more details we made the wavelet analysis of the radio flux at 3\,GHz in the
13:22\,--\,13:27\,UT time interval with the significance levels set to 95$\%$ and 99$\%$,
see Fig.~\ref{fig6}.
As expected the area specified by the significance level with 99$\%$ is smaller
than the area for level 95$\%$.
Both these wavelet spectra show a broad interval of periods (0.2--$\sim$10\,s) around 13:24\,UT.

At the time interval of the analyzed radio spectra, i.e., at the time of the type III bursts,
AIA/SDO observations show a huge rising magnetic rope (Fig.~\ref{fig7}).
\textbf{Three bright features, distributed along the magnetic rope and designated in this figure as HS, indicate helical magnetic field structure.}
The most of flare EUV brightenings (B) are located near the magnetic
rope footpoint. \textbf{Furthermore, during the time interval with type III bursts, we observed a very unique EUV structure in the cross-section of the magnetic rope, see the structure 
shown by the arrow in the bottom panel of the Fig.~\ref{fig8}. This structure has 
a semi-circular form with four blobs located around a circle. According to our knowledge,
such a structure is observed for the first time. Observations in other AIA/SDO EUV lines confirm its reality.} 

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\section{Discussion and conclusions}

According to the previous findings \cite[e.g.,][]{2004psci.book.....A}, the type III bursts
with normal (negative) frequency drift are generated by the electron beams propagating
upwards in the solar atmosphere and the reverse (positively) drifting type III bursts by
beams propagating downwards. In our case, the type III bursts were generated by beams that were
accelerated multi-periodically with the quasi-periods in the period ranges 7.6-11.3, 5.2-7.5 and 1.1-1.7 s. 

Most of the flare AIA/SDO EUV 304\,\AA~ brightenings at the time of the analyzed type III
bursts were located near the magnetic rope
footpoint.
Because the electron beams, generating the type III bursts, propagate along the magnetic field
lines, these beams need to be accelerated and propagating inside the magnetic rope.

\textbf{The found multi-periodicity of the type III bursts we interpret as follows (see also the scenario of processes in Fig.~\ref{fig9}):
the magnetic rope, as shown by its helical structure, carries the electric current. The unique structure, presented in Fig.~\ref{fig8}, shows that the magnetic rope is not homogeneous in its cross-section. It is structured, it consists of sub-ropes. These sub-ropes can be a result of the rope formation before its rising or they are generated during the rising of the magnetic rope. Some sub-ropes can mutually interact.
When the electric current in interacting sub-ropes is oriented in the same direction then between them in their interaction region the current sheet is formed. Such a current sheet can be unstable owing to the tearing mode instability \cite{2007PhPl...14j0703L,2010PhRvL.105w5002U}. As shown by \cite{2001EP&S...53..473S,2011ApJ...733..107K} this instability can produce a fragmented current sheet, consisting of sub-ropes with different cross-section diameters. Some of these sub-ropes interact. As presented by \cite{1987ApJ...321.1031T, 1996SSRv...77....1S} the process of interaction
between two sub-rope is periodic. Its period depends on the diameter of
cross-sections of the interacting sub-ropes and local Alfv\'{e}n velocity.
Because in our case the interacting sub-ropes have different cross-sections,
the interaction between different sub-ropes has a different period.
Each sub-rope interaction periodically accelerates electron beams 
and thus the type III bursts are generated periodically. The beams, generating the reverse
type III bursts bombard the chromosphere and cause the EUV brightenings there.} 

The type III bursts analyzed here indicate that they were generated by the \textbf{fragmented} magnetic
reconnection inside the rising and unstable magnetic rope. It means that this flare deviates
from the standard CSHKP flare model, where the main magnetic reconnection is located below
the rising magnetic rope and reconnection processes inside the magnetic rope are not considered. \textbf{On the other hand, it does not mean that energy release processes according to the standard flare model do not work in later phases of this flare. In future works, it would be interesting to compare the released energy in the rising magnetic rope with that released in the whole flare.}

\textbf{Remark: From a global point of view the magnetic rope can be
considered as half of the toroidal plasma in laboratory systems. Although phenomena in both cases
cannot be directly compared, owing to different conditions, the processes like magnetic islands, local 2D magnetic reconnection and particle acceleration found in the unstable toroidal 
plasma support our interpretation, see Figs~9, 11, 18-22 and chapter 4.5 in \cite{2022RvMPP...6...18P}.}


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\authorcontributions{
%For research articles with several authors, a short paragraph specifying their individual contributions must be provided. The following statements should be used ``Conceptualization, X.X. and Y.Y.; methodology, X.X.; software, X.X.; validation, X.X., Y.Y. and Z.Z.; formal analysis, X.X.; investigation, X.X.; resources, X.X.; data curation, X.X.; writing---original draft preparation, X.X.; writing---review and editing, X.X.; visualization, X.X.; supervision, X.X.; project administration, X.X.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript.'', please turn to the  \href{http://img.mdpi.org/data/contributor-role-instruction.pdf}{CRediT taxonomy} for the term explanation. Authorship must be limited to those who have contributed substantially to the work~reported.
Conceptualization, M.K.; methodology, M.K., J.R.; software, J.R.; validation, M.K., J.R.; investigation, M.K.; resources, M.K., J.R.; data curation, M.K., J.R.; writing -- original draft preparation, M.K.; writing -- review and editing, M.K., J.R.; visualization, M.K., J.R. Both authors have read and agreed to the published version of the manuscript.
}

\funding{
M. K. acknowledges support from the project RVO-67985815 and GA \v{C}R grants
21-16508J and 22-34841S.
J.R. acknowledges support by the Science Grant Agency project VEGA 2/0048/20 (Slovakia).}

%\institutionalreview{In this section, you should add the Institutional Review Board Statement and approval number, if relevant to your study. You might choose to exclude this statement if the study did not require ethical approval. Please note that the Editorial Office might ask you for further information. Please add “The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of NAME OF INSTITUTE (protocol code XXX and date of approval).” for studies involving humans. OR “The animal study protocol was approved by the Institutional Review Board (or Ethics Committee) of NAME OF INSTITUTE (protocol code XXX and date of approval).” for studies involving animals. OR “Ethical review and approval were waived for this study due to REASON (please provide a detailed justification).” OR “Not applicable” for studies not involving humans or animals.}

\informedconsent{
%Any research article describing a study involving humans should contain this statement. Please add ``Informed consent was obtained from all subjects involved in the study.'' OR ``Patient consent was waived due to REASON (please provide a detailed justification).'' OR ``Not applicable'' for studies not involving humans. You might also choose to exclude this statement if the study did not involve humans.
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\dataavailability{The data presented in this study are available on request from the
corresponding author.} 

\acknowledgments{We thank the RSDB service at LESIA/USN (Observatoire de Paris) for making the
NRH/ORFEES/NDA data available. Courtesy of NASA/SDO and the AIA science teams. We also thank J. Dud\'{\i}k for the preparation
of two AIA/SDO images.}

\conflictsofinterest{
%Declare conflicts of interest or state ``The authors declare no conflict of interest.'' Authors must identify and declare any personal circumstances or interest that may be perceived as inappropriately influencing the representation or interpretation of reported research results. Any role of the funders in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results must be declared in this section. If there is no role, please state ``The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the~results''.
The authors declare no conflict of interest.
} 

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