text:
\section{Introduction} \label{sec:intro}
\input{sec_intro}

\section{Binary Microlensing Parameters} \label{sec:param}
\input{sec_param}

\section{Inference of time-dependent parameters} \label{sec:unet}
\input{sec_unet}

\section{Inference of time-independent parameters} \label{sec:ncde}
\input{sec_ncde}

\section{Joint Pipeline \& Real Event Application} \label{sec:application}
\input{sec_application}

\section{Discussion} \label{sec:disc}
\input{sec_disc}

\begin{acknowledgments}
We thank Zerui Liu and Minghao Liu for discussions.
This work is supported by the National Science Foundation of China (grant Nos.\ 12173021 and 12133005) and the science research grant from the China Manned Space Project with No.\ CMS-CSST-2021-B12.
We also acknowledge the Tsinghua Astrophysics High-Performance Computing platform for providing computational and data storage resources.
\end{acknowledgments}

\vspace{5mm}

\software{NumPy \citep{harris2020array}, SciPy \citep{2020SciPy-NMeth}, Pandas \citep{mckinney-proc-scipy-2010-pandas}, Matplotlib \citep{Hunter2007matplotlib}, PyTorch \citep{pytorch}, MulensModel \citep{mulensmodel}, corner \citep{corner}, Jupyter \citep{jupyter}, torchcde \citep{kidger_thesis}.}

\subsubsection*{#1}}

\pagestyle{headings}
\markright{Reference sheet: \texttt{natbib}}

\usepackage{shortvrb}
\MakeShortVerb{\|}

\begin{document}
\thispagestyle{plain}

\newcommand{\textsc{Bib}\TeX}{\textsc{Bib}\TeX}
\newcommand{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}
\begin{center}{\bfseries\Large
    Reference sheet for \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ usage}\\
    \large(Describing version \fileversion\ from \filedate)
\end{center}

\begin{quote}\slshape
For a more detailed description of the \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package, \LaTeX\ the
source file \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.dtx}.
\end{quote}
\head{Overview}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is a reimplementation of the \LaTeX\ |\cite| command,
to work with both author--year and numerical citations. It is compatible with
the standard bibliographic style files, such as \texttt{plain.bst}, as well as
with those for \texttt{harvard}, \texttt{apalike}, \texttt{chicago},
\texttt{astron}, \texttt{authordate}, and of course \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.

\head{Loading}
Load with |\usepackage[|\emph{options}|]{|\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}|}|. See list of
\emph{options} at the end.

\head{Replacement bibliography styles}
I provide three new \texttt{.bst} files to replace the standard \LaTeX\
numerical ones:
\begin{quote}\ttfamily
 plainnat.bst \qquad abbrvnat.bst \qquad unsrtnat.bst
\end{quote}
\head{Basic commands}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package has two basic citation commands, |\citet| and
|\citep| for \emph{textual} and \emph{parenthetical} citations, respectively.
There also exist the starred versions |\citet*| and |\citep*| that print
the full author list, and not just the abbreviated one.
All of these may take one or two optional arguments to add some text before
and after the citation.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
  |\citet{jon90}| & Jones et al. (1990)\\
  |\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\[0.5ex]
  |\citep{jon90}| & (Jones et al., 1990)\\
  |\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\
  |\citep[see][]{jon90}| & (see Jones et al., 1990)\\
  |\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\[0.5ex]
  |\citet*{jon90}| & Jones, Baker, and Williams (1990)\\
  |\citep*{jon90}| & (Jones, Baker, and Williams, 1990)
\end{tabular}
\end{quote}
\head{Multiple citations}
Multiple citations may be made by including more than one
citation key in the |\cite| command argument.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
  |\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\
  |\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\
  |\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\
  |\citep{jon90a,jon90b}| & (Jones et al., 1990a,b)
\end{tabular}
\end{quote}

\head{Numerical mode}
These examples are for author--year citation mode. In numerical mode, the
results are different.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
  |\citet{jon90}| & Jones et al. [21]\\
  |\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\[0.5ex]
  |\citep{jon90}| & [21]\\
  |\citep[chap.~2]{jon90}| & [21, chap.~2]\\
  |\citep[see][]{jon90}| & [see 21]\\
  |\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\[0.5ex]
  |\citep{jon90a,jon90b}| & [21, 32]
\end{tabular}
\end{quote}
\head{Suppressed parentheses}
As an alternative form of citation, |\citealt| is the same as |\citet| but
\emph{without parentheses}. Similarly, |\citealp| is |\citep| without
parentheses. Multiple references, notes, and the starred variants
also exist.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
  |\citealt{jon90}| & Jones et al.\ 1990\\
  |\citealt*{jon90}| & Jones, Baker, and Williams 1990\\
  |\citealp{jon90}| & Jones et al., 1990\\
  |\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\
  |\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\
  |\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\
  |\citetext{priv.\ comm.}| & (priv.\ comm.)
\end{tabular}
\end{quote}
The |\citetext| command
allows arbitrary text to be placed in the current citation parentheses.
This may be used in combination with |\citealp|.
\head{Partial citations}
In author--year schemes, it is sometimes desirable to be able to refer to
the authors without the year, or vice versa. This is provided with the
extra commands
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
  |\citeauthor{jon90}| & Jones et al.\\
  |\citeauthor*{jon90}| & Jones, Baker, and Williams\\
  |\citeyear{jon90}|   & 1990\\
  |\citeyearpar{jon90}| & (1990)
\end{tabular}
\end{quote}
\head{Forcing upper cased names}
If the first author's name contains a \textsl{von} part, such as ``della
Robbia'', then |\citet{dRob98}| produces ``della Robbia (1998)'', even at the
beginning of a sentence. One can force the first letter to be in upper case
with the command |\Citet| instead. Other upper case commands also exist.
\begin{quote}
\begin{tabular}{rl@{\quad$\Rightarrow$\quad}l}
  when & |\citet{dRob98}| & della Robbia (1998) \\
  then & |\Citet{dRob98}| & Della Robbia (1998) \\
     &   |\Citep{dRob98}| & (Della Robbia, 1998) \\
     &   |\Citealt{dRob98}| & Della Robbia 1998 \\
     &   |\Citealp{dRob98}| & Della Robbia, 1998 \\
     &   |\Citeauthor{dRob98}| & Della Robbia
\end{tabular}
\end{quote}
These commands also exist in starred versions for full author names.

\head{Citation aliasing}
Sometimes one wants to refer to a reference with a special designation,
rather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be
defined and used, textual and/or parenthetical with:
\begin{quote}
\begin{tabular}{lcl}
  |\defcitealias{jon90}{Paper~I}|\\
  |\citetalias{jon90}| & $\Rightarrow$ & Paper~I\\
  |\citepalias{jon90}| & $\Rightarrow$ & (Paper~I)
\end{tabular}
\end{quote}
These citation commands function much like |\citet| and |\citep|: they may
take multiple keys in the argument, may contain notes, and are marked as
hyperlinks.
\head{Selecting citation style and punctuation}
Use the command |\bibpunct| with one optional and 6 mandatory arguments:
\begin{enumerate}
\item the opening bracket symbol, default = (
\item the closing bracket symbol, default = )
\item the punctuation between multiple citations, default = ;
\item the letter `n' for numerical style, or `s' for numerical superscript
      style, any other letter for
      author--year, default = author--year;
\item the punctuation that comes between the author names and the year
\item the punctuation that comes between years or numbers when common author
      lists are suppressed (default = ,);
\end{enumerate}

The optional argument is the character preceding a post-note, default is a
comma plus space. In redefining this character, one must include a space if
one is wanted.

Example~1, |\bibpunct{[}{]}{,}{a}{}{;}| changes the output of
\begin{quote}
  |\citep{jon90,jon91,jam92}|
\end{quote}
into [Jones et al. 1990; 1991, James et al. 1992].

Example~2, |\bibpunct[; ]{(}{)}{,}{a}{}{;}| changes the output of
\begin{quote}
  |\citep[and references therein]{jon90}|
\end{quote}
into (Jones et al. 1990; and references therein).

\head{Other formatting options}
Redefine |\bibsection| to the desired sectioning command for introducing
the list of references. This is normally |\section*| or |\chapter*|.

Define |\bibpreamble| to be any text that is to be printed after the heading but
before the actual list of references.

Define |\bibfont| to be a font declaration, e.g.\ |\small| to apply to
the list of references.

Define |\citenumfont| to be a font declaration or command like |\itshape|
or |\textit|.

Redefine |\bibnumfmt| as a command with an argument to format the numbers in
the list of references. The default definition is |[#1]|.

The indentation after the first line of each reference is given by
|\bibhang|; change this with the |\setlength| command.

The vertical spacing between references is set by |\bibsep|; change this with
the |\setlength| command.

\head{Automatic indexing of citations}
If one wishes to have the citations entered in the \texttt{.idx} indexing
file, it is only necessary to issue |\citeindextrue| at any point in the
document. All following |\cite| commands, of all variations, then insert
the corresponding entry to that file. With |\citeindexfalse|, these
entries will no longer be made.

\head{Use with \texttt{chapterbib} package}

The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is compatible with the \texttt{chapterbib} package
which makes it possible to have several bibliographies in one document.

The package makes use of the |\include| command, and each |\include|d file
has its own bibliography.

The order in which the \texttt{chapterbib} and \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ packages are loaded
is unimportant.

The \texttt{chapterbib} package provides an option \texttt{sectionbib}
that puts the bibliography in a |\section*| instead of |\chapter*|,
something that makes sense if there is a bibliography in each chapter.
This option will not work when \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ is also loaded; instead, add
the option to \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.

Every |\include|d file must contain its own
|\bibliography| command where the bibliography is to appear. The database
files listed as arguments to this command can be different in each file,
of course. However, what is not so obvious, is that each file must also
contain a |\bibliographystyle| command, \emph{preferably with the same
style  argument}.
\head{Sorting and compressing citations}
Do not use the \texttt{cite} package with \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}; rather use one of the
options \texttt{sort} or \texttt{sort\&compress}.

These also work with author--year citations, making multiple citations appear
in their order in the reference list.

\head{Long author list on first citation}
Use option \texttt{longnamesfirst} to have first citation automatically give
the full list of authors.

Suppress this for certain citations with |\shortcites{|\emph{key-list}|}|,
given before the first citation.

\head{Local configuration}
Any local recoding or definitions can be put in \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.cfg} which
is read in after the main package file.

\head{Options that can be added to \texttt{\char`\\ usepackage}}
\begin{description}
\item[\ttfamily round] (default) for round parentheses;
\item[\ttfamily square] for square brackets;
\item[\ttfamily curly] for curly braces;
\item[\ttfamily angle] for angle brackets;
\item[\ttfamily colon] (default) to separate multiple citations with
     colons;
\item[\ttfamily comma] to use commas as separaters;
\item[\ttfamily authoryear] (default) for author--year citations;
\item[\ttfamily numbers] for numerical citations;
\item[\ttfamily super] for superscripted numerical citations, as in
     \textsl{Nature};
\item[\ttfamily sort] orders multiple citations into the sequence in
     which they appear in the list of references;
\item[\ttfamily sort\&compress] as \texttt{sort} but in addition multiple
     numerical citations are compressed if possible (as 3--6, 15);
\item[\ttfamily longnamesfirst] makes the first citation of any reference
     the equivalent of the starred variant (full author list) and subsequent
     citations normal (abbreviated list);
\item[\ttfamily sectionbib] redefines |\thebibliography| to issue
     |\section*| instead of |\chapter*|; valid only for classes with a
     |\chapter| command; to be used with the \texttt{chapterbib} package;
\item[\ttfamily nonamebreak] keeps all the authors' names in a citation on
     one line; causes overfull hboxes but helps with some \texttt{hyperref}
     problems.
\end{description}
\end{document}

\subsection{Joint pipeline} \label{sec:joint}

Combining the methods described in Sections~\ref{sec:unet} and \ref{sec:ncde}, the joint pipeline of MAGIC works as follows. First, we feed the light curve into \texttt{locator}, which outputs the predicted $t_0$ and $t_{\rm E}$. If $t_{\rm E}$ is small, we can further refine it using \texttt{locator} with a larger $k$. Then we use the predicted $t_0$ and $t_E$ to shift and rescale the light curve. The resulting light curve is processed by \texttt{estimator}, which in return gives the positions and shapes of $n_G=12$ Gaussians in the parameter space. If needed, these Gaussians can be further optimized to obtain more accurate prediction or predictions (see Section~\ref{sec:ncde_method}).

The performance of the combined pipeline is largely determined by the propagation of inaccurate parameters from \texttt{locator} to \texttt{estimator}. We therefore test the joint pipeline on 16,384 light curves from the test set of \texttt{locator}. After applying \texttt{locator}, we transform the light curves using the predicted values of $t_0$ and $t_{\rm E}$. The transformed light curves are then sent to \texttt{estimator} to infer the time-independent microlensing parameters.
The set of parameters closest to the position of the input values in the parameter space is adopted as the final prediction.

The comparisons between the input and the predicted values of the time-independent parameters are shown in Figure~\ref{fig:joint_close}. The typical, fractional deviation in $q$ and $s$ are $25\%$ and $9\%$, respectively. These deviations remain small, even though they are worse than the \texttt{estimator}-only case by a factor of $\sim2$.
Overall, the joint pipeline achieves fairly high prediction accuracy in the time-independent parameters even with the use of the inaccurate time-dependent parameters.
We expect the accuracy to be further improved once the predictions are optimized via the \citet{Nelder:1965} algorithm, as seen in the \texttt{estimator}-only case.

A substantial fraction of binary events can be well described by a \citet{Paczynski:1986} model plus relatively short binary perturbation. For such events, one may mask out the anomalous feature and fit a single-lens model to obtain precise and accurate peak time $t_0$ and event timescale $t_{\rm E}$. Combining these values with \texttt{estimator}, we can achieve nearly as accurate time-independent parameters as we do in the \texttt{estimator}-alone case.

It is worth noting that the light curves generated for \texttt{locator} are similar to but not the same as those used for training \texttt{estimator}.
First, \texttt{locator} uses binary light curves with $\chi^2_{\rm pspl}/{\rm d.o.f.}>1.25$, whereas \texttt{estimator} uses binaries with $\chi^2_{\rm pspl}/{\rm d.o.f.}>2$. Both correspond to a $\Delta \chi^2>500$ between single and binary models, under the reasonable assumption that the binary signals are usually local features.
\footnote{Although we have focused on such binaries with strong signals, we expect our method to work on weak binaries as well.}
Furthermore, the time window is fixed and the timescale parameters varies in the data set for \texttt{locator}. This means the number of data points within $-2 \leq \tau \leq 2$ varies too. Nevertheless, we have shown in Section~\ref{sec:ncde_perform} that the performance of \texttt{estimator} is only mildly affected with the various length of input time series.

In addition to its high accuracy, our method demonstrates high efficiency in the parameter estimation. Specifically, \texttt{locator} and \texttt{estimator} can process $\sim1000$ and $\sim400$ light curves per second on a single GPU, respectively. As shown in Figure~\ref{fig:opt_close}, the output of \texttt{estimator} is already accurate enough for purposes like statistical studies. If needed, the optimization step can be performed. Compared to the two machine learning steps, this traditional optimization step is slower, but there are only up to 12 initial positions to be optimized, which altogether may require up to a few thousand light curve evaluations. For comparison, the traditional approach usually requires optimizations at individual grid points on a fine grid in the $(\lg{q}, \lg{s}, \alpha)$ space \citep[e.g.,][]{Wang:2022}. Our approach therefore represents a speedup by factors of $\sim10^2$--$10^5$.

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/joint_close.pdf}
    \caption{Performance of the joint pipeline. Simulated light curves for \texttt{locator} are used in this test. After the application of \texttt{locator}, the light curve is sent to \texttt{estimator} for the inference of the time-independent parameters.}
    \label{fig:joint_close}
\end{figure*}

\subsection{A Real Event Application} \label{sec:0371}

We apply MAGIC to KMT-2019-BLG-0371 (hereafter KB190371 for short), which is a short-timescale ($t_{\rm E}=6.5\,$d) binary event with $q \approx 0.08$ or $0.12$ \citep{Kim:2021}. KB190371 is chosen for this demonstration based on several considerations. First, its caustic-crossing feature is prominent and U-shaped, which is typical for binary events. Second, this event suffers from the close/wide degeneracy
\footnote{We use this term to keep consistent with the original discovery paper, even though recent studies have suggested that such ``close/wide degeneracy'' should be called offset degeneracy \citep{Zhang:2022}}
, thus enabling additional tests of MAGIC. Furthermore, except for the relatively short timescale, the other microlensing parameters are fairly typical in comparison to those used in our simulation.

The available data sets from \citet{Kim:2021} are not aligned to the same magnitude system. In order to align the data sets and then apply MAGIC, we mask out the caustic-crossing feature, fit a \citet{Paczynski:1986} model to the data, and use the best-fit flux parameters to align all other data sets to the OGLE one. We note that, although here we have made use of the special property of KB190371 to align the data sets, such a data alignment task is the same as deriving the source color and can be performed without reference to specific microlensing models \citep{Yoo:2004, Gould:2010a}, as has been practiced in the analysis of many real microlensing events \citep[e.g.,][]{Gould:2010, Zhu:2017}.

To apply our joint pipeline, we first preprocess the data and make the light curve similar to those used in the training procedure. To be more specific, we only include data points falling in the time interval of about 100 days centered at the anomaly, and filter out data points with large ($>0.1\,$mag) error bars. KB190371 was observed with high cadence, so we bin every five consecutive points into one. Finally, the light curve is shifted to the same base magnitude as those used in \texttt{locator}.

Then we feed the preprocessed light curve into \texttt{locator}. Because the timescale is relatively small, we set $k=2$. The output mask is rectified into binary values $\{0, 1\}$ with a threshold of $0.5$. This step gives the prediction $t_0~{(\rm HJD')}=8592.391$ and $t_{\rm E}=6.68\,$d, both of which are close to the values given in \citet{Kim:2021}. We transform the light curve with the predicted $t_0$ and $t_{\rm E}$ and only keep the segment within $-2 \leq \tau \leq 2$.

The rest of application follows closely the procedure given in Section~\ref{sec:joint}. With the positions of the 12 multivariate Gaussians given by \texttt{estimator}, we perform \citet{Nelder:1965} optimization to further refine the microlensing parameters. Because the light curve shows prominent finite-source effect, we also include in the optimization the scaled source size $\rho$.

The results are shown in Figure \ref{fig:kmt}. The top two Gaussians with the highest weights match closely to the close ($s<1$) and wide ($s>1$) solutions, respectively. The corresponding model light curves from the optimized solutions both match the data reasonably well, with almost indistinguishable $\chi^2$ values. According to these solutions, the event can be explained by a binary with $(\lg{q}, \lg{s})=(-1.16, -0.057)$ or $(-0.95, 0.180)$. Given the imperfect alignment as well as the slightly different preprocessing procedures, we consider these values in good match with those given by the discovery paper.

\citet{Kim:2021} found the source scaled parameter $\rho$ to be $(6.46\pm0.17) \times 10^{-3}$ and $(6.99\pm0.21) \times 10^{-3}$ for the close and wide solutions, respectively. These are quite different from the value ($10^{-3}$) that we have used to train MAGIC. Nevertheless, the optimized parameters yield $\rho=6.6 \times 10^{-3}$ and $7.3 \times 10^{-3}$ for the close and wide solutions, respectively, in good agreement with the values from the discovery paper. In other words, the presence of the finite-source effect in KB190371 does not seem to confuse MAGIC, even though MAGIC has not learned to infer the source scaled parameter $\rho$. This is probably because the finite-source effect in KB190371 only affects a limited number of data points around the caustic entrance and exit. While the majority of binary microlensing events bear the same feature, some may have their binary signals substantially modified by the finite-source effect \citep[e.g.,][]{Hwang:2018}. Therefore, it may be necessary to include in MAGIC the ability to infer $\rho$. See Section~\ref{sec:disc} for further discussion.

To further demonstrate the capability of MAGIC to handle large data gaps, we exclude observations from the South Africa site of KMTNet. This creates a substantial gap during the exit of the caustic crossing, as shown in the top right panel of Figure \ref{fig:kmt_gap}. The same procedure is applied as before, and the results are shown in Figure \ref{fig:kmt_gap}. The two degenerate solutions are again identified at high accuracy, indicating the robustness of MAGIC in handling large data gaps as well as model degeneracy.

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/kmt.pdf}
    \caption{The application of MAGIC to KB190371 and the output. The light curve and the MAGIC predictions are shown in the top right panel. Two solutions, labeled ``close'' and ``wide,'' have comparable $\chi^2$ values, and the corresponding microlensing geometries (i.e., caustic curve and source trajectory) are shown below the light curve plot. The corner plot shows the positions of the optimized solutions as well as the top two Gaussians that correspond to the ``close'' and ``wide'' solutions.
    \label{fig:kmt}}
\end{figure*}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/kmt_gap.pdf}
    \caption{Similar to Figure~\ref{fig:kmt} except that observations from the South Africa site of KMTNet are excluded in this analysis. As a result, the exit portion of the caustic crossing feature is not covered with observations. Even so, MAGIC is able to identify the two degenerate solutions at high accuracy.
    \label{fig:kmt_gap}}
\end{figure*}

\subsection{Method} \label{sec:ncde_method}

We develop a separate method to infer the time-independent microlensing parameters, which we re-parameterize as
\begin{equation}
    \omega \equiv (u_0, \lg{q}, \lg{s}, \alpha/180^\circ, \lg{f_{\rm S}}) .
\end{equation}
As shown in Figure~\ref{fig:model}, this \texttt{estimator} consists of two main parts. A neural controlled differential equation (neural CDE) is used to extract features from the microlensing light curve, which is irregularly sampled and may contain large data gaps. The second part, mixture density network (MDN), infers the probability distribution of the microlensing parameters.

To improve the training speed and reduce the memory requirement, we first apply the log-signature transform \citep{morrill2021neural}. This transform outputs a shorter, steadier, and higher dimensional sequence that summarizes the sub-step information. For the simulated light curves with 500 time stamps (see Section~\ref{sec:ncde_data}), we set the log-signature depth $n_D=3$ and the shortened length $l=100$. The output thus has $v=5$ dimensions.

Neural CDE approximates the underlying process of the time series as a differential equation \citep{kidger2020neural,kidger_thesis}
\begin{equation} \label{eqn:ncde}
    z_{t} = \xi_{\theta} (X_{t_0}) +\int_{t_{0}}^{t} f_{\theta}\left(z_{t'}\right) \mathrm{d} X_{t'} .
\end{equation}
Here the latent state $z_t$ is defined on the time interval of the input signal, $(t_0, t_n)$, with $w=32$ dimensions.
The continuous signal $X_t$, defined on the same time interval, is obtained from applying natural cubic splines to the input, discrete signal. The initial value and signal-derivative of the latent state, $\xi_{\theta}$ and $f_{\theta}$, are two different neural networks, where the subscript $\theta$ denotes the dependence on the learnable machine learning parameters. By Equation~\ref{eqn:ncde}, neural CDE gradually extracts features from the signal $X_t$ according to a learned policy $f_\theta$, records its knowledge in the latent state $z_t$, and output a terminal value $z_{t_n}$ that summarizes the useful information of the input signal. We refer interested readers to \citet{kidger2020neural} and \citet{kidger_thesis} for further details of neural CDE.

The summary features from neural CDE is then fed into the MDN \citep{bishop2006pattern} to estimate the probability distribution of the microlensing parameters
\begin{equation} \label{eqn:mdn}
    p(\omega|z_{t_n}) = \sum_{i=1}^{n_{\rm G}} \pi_i(z_{t_n}) \phi_{\bar{\omega}_i(z_{t_n}), \Sigma_i(z_{t_n})}(\omega).
\end{equation}
Here $\phi_{\bar{\omega}, \Sigma}$ denotes the density of a multivariate Gaussian with mean $\bar{\omega}$ and covariance matrix $\Sigma$. The normalized weight $\pi_i$, mean $\bar{\omega}_i$, and covariance matrix $\Sigma_i$ of each of the $n_{\rm G}$ multivariate Gaussians are given by the neural network of MDN.
Compared to the masked autoregressive flow method of \citet{zhang2021real}, MDN provides a straightforward and efficient alternative with explicit density estimation, which is useful for the identification of degenerate solutions and the further optimization (see below).
We adopt $n_{\rm G}=12$ Gaussians with diagonal covariance matrices.

In the joint framework, \texttt{estimator}, the initial value network $\xi_\theta$, the evolution network $f_\theta$, and MDN are constructed as residual networks (ResNet, \citealt{he2016deep}) with three residual blocks and each block consisting of two fully connected layers of width 1024. The evolution network has an additional activation of hyperbolic tangent to rectify the dynamic. To train the model, we minimize the negative log-likelihood (NLL) loss function
\begin{equation}
    \mathcal{L}_{\text{estimator}} = \mathop{\mathbb{E}}\limits
    _{{\text{light curve }}} [-\ln p(\omega_{\rm true} | z_{t_n})],
\end{equation}
To further optimize the microlensing parameters, we start from mean positions of the multivariate Gaussians and minimize the light curve $\chi^2$ values via the \citet{Nelder:1965} simplex algorithm. In this step, the source flux parameter $f_{\rm S}$ is derived analytically for a given set of $(q, s, u_0, \alpha)$ by maximizing the model likelihood \citep[e.g.,][]{mulensmodel}.

\begin{figure
    \centering
    \includegraphics[width=\linewidth]{figures/cdf_chi2.pdf}
    \caption{Cumulative distributions of the $\chi^2$ values of the output microlensing models of \texttt{estimator}. In blue and red curves models with the minimum $\chi^2$ value and the closest distance (in parameter space) to the input model are used, respectively. The vertical dashed line indicates $\chi^2=500$, which is approximately the medians for both curves.
    \label{fig:cdf_chi2}}
\end{figure}

\begin{figure
    \centering
    \includegraphics[width=\linewidth]{figures/cdf_pi.pdf}
    \caption{Cumulative distributions of the normalized weights in MDN. Here $\pi_0$, $\pi_1$, and $\pi_2$ are the top three highest weights in our MDN of 12 multivariate Gaussians. The blue curve indicates that there is one dominating Gaussian in about half of the cases, and the green curve shows that the combination of the top three Gaussians dominates the density distribution in almost all cases.}
    \label{fig:cdf_pi}
\end{figure}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/mdn_rmse.pdf}
    \caption{Comparisons between the input parameter values and the direct output of \texttt{estimator}. Out of the 12 Gaussian means of MDN, we identify the one that is closest to the ground truth in the parameter space and take it as the prediction. The median absolute deviation (MAD) is used to evaluate the agreement, and the root-mean-squared error (RMSE) is also provided for a reference.
    \label{fig:mdn_rmse}}
\end{figure*}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/opt_close.pdf}
    \caption{All except the lower right panel show comparisons between the input and the optimized parameter values. From the means of the 12 multivariate Gaussians, we apply \citet{Nelder:1965} simplex method and obtain optimized model parameters. The set of parameters closest to the ground truth in the parameter space is used in these comparisons. The lower right panel shows the prediction accuracy of microlensing parameters evaluated on events with various numbers of data points and no gap. Here MAD$_{500,\rm gap}$ is the value of MAD evaluated on events with 500 points and data gap, as incidated in the top left corner of the other panels.
    \label{fig:opt_close}}
\end{figure*}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/example_gap.pdf}
    \caption{An example event in which the binary caustic-crossing feature is partially missing due to the injected data gap, as shown by the light curve in the top right corner. Red color is for the input model, whereas blue color is for the prediction given by \texttt{estimator}. The corner plot shows the probability distributions of the four MDN Gaussians with the highest weights, with the colors indicating the relative weights. Contours denote the 1-$\sigma$ boundaries of the multivariate Gaussians with diagonal covariance matrices. As we have chosen axis limits to optimize the overall appearance, not all four contours are visible in every panel. Values and positions of the ground truth and the prediction of \texttt{estimator} on the corner plot are also indicated. Even without the final optimization, the Gaussian distribution with the highest weight gives a fairly good approximation of the input parameters.}
    \label{fig:example_gap}
\end{figure*}

\subsection{Data generation} \label{sec:ncde_data}

For the purpose of learning the time-independent parameters, we generate light curves as functions of the parameter $\tau$. This is equivalent as setting $t_0=0$ and $t_{\rm E}=1\,$d. The other parameters are sampled in the same way as in Secion~\ref{sec:unet_data}.

Each light curve contains 500 data points randomly drawn within the interval $-2 \leq \tau \leq 2$. For a typical timescale $t_{\rm E}=20\,$d, this corresponds to an averaged cadence of $\sim$4\,hr. To mimic the missing data points due to bad weather and/or instrumental failures, we throw away data points within one randomly chosen light curve segment out of the 25 equal-duration segments that constitute the whole light curve. This leaves a data gap of $3.2\,$d for the aforementioned timescale. Light curves and noises are generated in the same way as in Section~\ref{sec:unet_data}. Finally, we only keep light curves with significant binary features, defined as $\chi_{\rm pspl}^2$/d.o.f.\ $>2$.

Six data sets are generated, each having $10^5$ light curves. We use five of them for training \texttt{estimator} and the last one for validation and test.
To further understand the performance of \texttt{estimator} on light curves with different cadences, we have also generated data sets with various numbers of data points within the interval $-2 \leq \tau \leq 2$. For these additional data sets, we have assumed irregular sampling but no gap.

\subsection{Performance of \texttt{estimator}} \label{sec:ncde_perform}

We train \texttt{estimator} with the ADAM optimizer \citep{kingma2014adam}. The learning rate, initially set to $10^{-4}$, drops by ten percent after each epoch and reaches a minimum of $10^{-6}$. Training samples are labeled by their input parameters $\omega$ and divided into mini-batches of size 128. The training took $\sim1\,$day for 50 epochs on one NVIDIA Tesla V100 GPU, and the average loss drops to $-8.4$ on the validation set of size 1024.

We apply the optimized \texttt{estimator} to 16,384 light curves from the test set and obtain the probability distribution of the microlensing parameters given by the summation of 12 multivariate Gaussians. We then apply the \citet{Nelder:1965} simplex algorithm to get the refined microlensing parameters.
We show in Figure~\ref{fig:cdf_chi2} the cumulative distributions of the model $\chi^2$ values of two selected sets of parameters, one that has the smallest $\chi^2$ value and the other that is closest to the input position in the parameter space. The median $\chi^2$ values are $\sim500$ for both solutions, comparable to the number of data points in the simulated light curve and thus approximately the degrees of freedom. This suggests that the found solutions can indeed describe the data very well. Even though in $\sim$20\%--30\% of cases the model $\chi^2$ values are substantially large compared to the number of data points, the returned parameters are still good representatives of the input parameters, as will be discussed later.

We have included a mixture of $n_{\rm G}=12$ Gaussians in the density estimate. Is it flexible enough? To answer this question, we first look at the distribution of the Gaussian weights given by \texttt{estimator}. As shown in Figure~\ref{fig:cdf_pi}, the probability distribution is dominated by one Gaussian in $\sim$50\% of the tested events and by the top three Gaussians with the highest weights in nearly all tested events. As for the optimized solutions, 74\% and 85\% of the minimum $\chi^2$ and minimum RMSE solutions are given by the top four Gaussians, respectively. Only in $\lesssim1\%$ of the tested events does the Gaussian with the lowest weight lead to either solution. Therefore, we conclude that 12 Gaussians can satisfactorily describe the probability distribution of microlensing model parameters.

We compare the \texttt{estimator} predictions before and after the simplex optimization with the input parameters in Figures~\ref{fig:mdn_rmse} and \ref{fig:opt_close}.
For each tested event, we choose as the model prediction the set of parameters that is closest to the input position in the parameter space, in order to minimize the impact of degenerate solutions.
We use the median absolute deviation (MAD) to evaluate the agreement between input and prediction. Compared to RMSE, MAD is more robust against outliers.
As shown in Figure~\ref{fig:mdn_rmse}, the direct output of \texttt{estimator} already contains a set of parameters that is fairly close to the input truth.
Among all parameters, binary mass ratio $q$ and separation $s$ are of special interest to the physical interpretation of the event. The direct output of \texttt{estimator} yields a typical deviation of $0.057$ ($0.0155$) for $\lg{q}$ ($\lg{s}$), corresponding to a fractional uncertainty of $13\%$ ($3.6\%$) on $q$ ($s$).
The accuracy of the prediction is further improved after the optimization, as shown in Figure~\ref{fig:opt_close}. Specifically, the optimized prediction can achieve median deviations of $\sim0.03$ and $\sim0.006$ for $\lg{q}$ and $\lg{s}$, corresponding to fractional errors of $\sim7\%$ and $\sim1.4\%$ in $q$ and $s$, respectively.
Overall, the agreement between the input and predicted values is very well, demonstrating the high accuracy of our method on light curves with realistic sampling conditions.

We have also applied the optimized \texttt{estimator} to light curves in the additional data sets, which have different averaged cadence but no data gap.
\footnote{That is, we directly apply \texttt{estimator} without retraining.}
The results are shown in the lower right panel of Figure~\ref{fig:opt_close} for the optimized prediction. There are a few interesting trends to report. First, the performance of \texttt{estimator} is better, although fairly marginally, on events with 500 points and no data gap. Second, the prediction accuracy decreases with decreasing number of data points as expected, but the degradation is typically small. Specifically, with 200 data points the prediction accuracy is only worse by $\sim10^{0.13}-1 \approx 35\%$ compared to the default case with 500 data points. The only exception is seen in the case with 100 data points, for which the performance is reduced by a factor of $\gtrsim 2$. This is probably because the averaged cadence becomes too low to constrain binaries with mass ratios down to $10^{-3}$.
\footnote{The averaged cadence in terms of $\tau$ is $4/100$, and thus the smallest binary mass ratio that it can detect is $\gtrsim (4/100)^2 = 1.6 \times 10^{-3}$.}
Nevertheless, the overall uncertainty in the case of 100 points remains fairly small and acceptable. These results may provide useful guidance to the future application of our method to real light curves with variant cadences and/or the detection of low mass-ratio planets.

We show in Figure~\ref{fig:example_gap} an example event whose anomalous feature is only partially covered by data points. The probability distribution from \texttt{estimator}, represented by the multivariate Gaussians, is already at the neighborhood of the ground truth position. The optimization with \citet{Nelder:1965} algorithm leads to even better match in microlensing parameters as well as model the light curve. Events like this one, which have irregular samplings and large data gaps, pose challenges to machine learning methods that base on the standard RNN approach.

Our method also identify multiple solutions in events that suffer model degeneracy. We defer to Section~\ref{sec:0371} for further discussions on this issue.

\subsection{Method}

Together with $u_0$, parameters $t_0$ and $t_{\rm E}$ are the so-called Paczynski parameters \citep{Paczynski:1986}. Unlike the parameters that define the binary properties, these Paczynski parameters are constrained by the global properties of the light curve. In many low-$q$ events, especially planetary events, one can accurately constrain the Paczynski parameters by masking out the relatively brief anomalous features \citep[e.g.,][]{Gould:1992}. Of all three Paczynski parameters, the time-dependent ones, namely $t_0$ and $t_{\rm E}$, enter the microlensing phenomenon via the parameter $\tau$ (see Equation~\ref{eqn:tau}). For these reasons, we develop a stand-alone machine learning method to infer the time-dependent parameters, which is dubbed \texttt{locator}.

The basis of \texttt{locator} is the U-Net architecture. U-Net is a fully convolutional neural network that is originally designed to perform semantic segmentation of images \citep{ronneberger2015unet}. U-Net takes the original image as the input, and outputs a mask of the same shape. The mask can be interpreted as a pixel-wise classification result, representing, for example, the probability of each pixel belonging to some specific segment. U-Net has been ubiquitously used in biomedical image processing \citep{Siddique2021unet_review} since it was proposed in 2015. It has also seen a growing number of applications in astronomy \citep[e.g.,][]{VanOort:2019, Makinen:2021}.

We design \texttt{locator} to identify the light curve segment that falls within a predefined time window $\Delta T \equiv [t_0-k t_{\rm E}, t_0+k t_{\rm E}]$. Here $k$ is a parameter that can be tuned to optimize the output performance, and its choice will be discussed in Section~\ref{sec:unet_perform}. For each light curve, which is irregularly sampled (see Section~\ref{sec:unet_data}), we first interpolate it with natural cubic splines and extract the values at $n_t=4000$ equally stepped time stamps $\{t_{j}\}_{j=1}^{n_t}$.
\footnote{This crude approach of directly interpolating the irregular time series is justified given that the temporal parameters depend mostly on the macroscopic behaviour of the light curve, rather than the fine structure within a small time interval. This assumption is no longer valid for the parameters discussed in Section \ref{sec:ncde}.}
These time stamps are assigned labels depending on whether or not they are inside $\Delta T$,
\begin{equation}
    r_j = \left\{ \begin{array}{ll}
        1, & t_j \in \Delta T \\
        0, & {\rm otherwise} \\
    \end{array} \right. .
\end{equation}
At run-time, \texttt{locator} takes the time series and attempts to produce a mask $\{p_j\}_{j=1}^{n_t}$ that best approximates the ground truth labels.

We train \texttt{locator} on simulated light curves that match the key properties of real microlensing light curves (see Section~\ref{sec:unet_data}). The loss function that \texttt{locator} minimizes is given by
\begin{equation} \label{eqn:unet_loss}
    \mathcal{L}_{\text{locator}}
    = \mathcal{L}_{\text{BCE}} + \mathcal{L}_{\text{Dice}} + \mathcal{L}_{\text{reg}}.
\end{equation}
Here $\mathcal{L}_{\rm BCE}$ is the binary cross entropy (BCE) loss function
\begin{equation}
    \mathcal{L}_{\text{BCE}} \equiv \mathop{\mathbb{E}}
    [r \ln p + (1 - r)\ln (1 - p)].
\end{equation}
The expectation runs over all time stamps of all light curves in the training set. The second term in Equation~(\ref{eqn:unet_loss}) is the Dice loss function \citep{sudre2017dice}, which is introduced to assuage the class imbalance problem in events with small $t_{\rm E}$:
\begin{equation}
    \mathcal{L}_{\text{Dice}} \equiv \mathop{\mathbb{E}}
    \left[ 1 - \frac{\sum_{j} r_j p_j}{\sum_{j} (r_j + p_j)}
    \right] .
\end{equation}
The third term in Equation~(\ref{eqn:unet_loss}) is the regularization term to prevent the mask from wiggling up and down for more than twice
\begin{equation}
\mathcal{L}_{\text{reg}} \equiv \mathbb{E}
    \left(\sum_{j=1}^{n_t-1}|p_{j+1}-p_{j}| - 2\right)^2.
\end{equation}
The expectations in the last two terms are performed on all light curves in the training set.

The output mask from \texttt{locator} contains information about the event peak time, $t_0$, and the timescale, $t_{\rm E}$. Specifically, the centroid of the mask corresponds to $t_0$ and the total enclosed area is directly related to $t_{\rm E}$. In the continuum form, they are given by
\begin{equation} \label{eqn:t0te}
    t_0 = \frac{\int p(t) t dt}{\int p(t) dt}, \quad
    t_{\rm E} = \frac{1}{2k} \int p(t) dt .
\end{equation}
These integrals are evaluated via the trapezoidal rule in the discrete time series.

\subsection{Data generation} \label{sec:unet_data}

To simulate light curves for the training of \texttt{locator}, we randomly sample 2000 time stamps out of a total duration of $270\,$days. This corresponds to an averaged cadence of $\sim3\,$hr. For each light curve, we randomly draw $t_{\rm E}$ from a truncated log-normal distribution with mean of $10^{1.15}\,$days, standard deviation of $10^{0.45}\,$days, and boundaries at 5 and $100\,$ days. This log-normal distribution matches approximately the observed distribution of $t_{\rm E}$ \citep[e.g.,][]{Mroz:2017}, and the truncations are introduced to eliminate relatively short-timescale and long-timescale events, both of which may be subject to additional issues (e.g., parallax effect for long events) and are not the focus of the present work. As for $t_0$, even though it can be any value during (and possibly outside) the sampled time interval, it is reasonable to believe that a very rough by-eye estimate can limit $t_0$ to within $\sim \pm t_{\rm E}$ of the truth. We therefore randomly draw $t_0$ out of a uniform distribution between $-t_{\rm E}$ and $t_{\rm E}$. To summarize,
\begin{eqnarray*}
    &t_{\rm E}/{\rm day} &\sim \text{TruncLogNorm}(5, 100, \mu=10^{1.15}, \sigma=10^{0.45}),\\
    &t_0 &\sim \text{Uniform}(-t_E, t_E).
\end{eqnarray*}

Although the goal of \texttt{locator} is to infer $t_0$ and $t_{\rm E}$ and not the other parameters, the training data set should be representative of the target---the binary light curves. We therefore sample the remaining parameters in the following way
\begin{eqnarray*}
&u_0 &\sim \text{Uniform}(0,1), \\
&q &\sim \text{LogUniform}(10^{-3}, 10^0), \\
&s &\sim \text{LogUniform}(0.3, 3), \\
&\alpha &\sim \text{Uniform}(0, 360), \\
&f_{\rm S} &\sim \text{LogUniform}(10^{-1}, 10^0),
\end{eqnarray*}
and $\rho$ is fixed at $10^{-3}$, given that the finite-source effect is not expected to be detected for the adopted (effective) cadence. Note that the mass ratio $q$ is limited to $>10^{-3}$ simply because we want to focus on relatively massive binaries. With increased sampling cadence \texttt{locator} can be well extended down to much lower $q$ values.

With the sampled binary parameters and time stamps, we calculate the microlensing light curve using the \texttt{VBBL} algorithm \citep{Bozza:2010, Bozza:2018}. To introduce noises to the simulated light curves, we adopt a simplified noise model. Gaussian noises with signal-to-noise ratios (S/N) of 33 in flux are assumed for all data points, and the flux values are converted to magnitudes for a baseline object with $m_{\rm base}=18$.

Simulated light curves without significant binary features, defined as $\chi^2_{\rm pspl}$/d.o.f.\ $<1.25$,
\footnote{Here ``pspl'' represents for point-source point-lens.}
are rejected. In the end, two data sets each containing $10^5$ light curves are generated, one for training and the other for validation and test.

\begin{figure
    \centering
    \includegraphics[width=\linewidth]{figures/p_dist.pdf}
    \caption{An illustration of the output mask of \texttt{locator}. The two-dimensional histogram shows the distribution of $p$ values inside the chosen $\tau$ window for 16,384 light curves with $s<1$. The red dashed lines indicate $\tau= \pm k = \pm 1/3$, which are the boundaries of the input mask in this realization. }
    \label{fig:p_dist}
\end{figure}

\begin{figure*
    \centering
    \includegraphics[width=0.8\textwidth]{figures/t0te.pdf}
    \caption{Comparisons between input and predicted values of $t_{\rm E}$ (left) and $t_0/t_{\rm E}$ (right). Values from events with $s>1$ and $s<1$ are separated, with the former shown in red and the latter in blue. The rooted mean square errors (RMSEs) of the parameters are computed and indicated at the upper left corners. In this realization, $k=1/3$ is used.}
    \label{fig:t0te}
\end{figure*}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/lc_caustic.pdf}
    \caption{An example event in which \texttt{locator} successfully reproduces the input window and thus recovers $t_0$ and $t_{\rm E}$. The upper panel shows the microlensing geometry, with the red cross marking the center of mass (c.o.mass) of the binary lens. The lower panel shows the simulated light curve and the input and the output masks. The parameters used to reproduce this event are given on the upper right corner.
    \label{fig:lc_caustic}}
\end{figure*}

\subsection{Performance of \texttt{locator}} \label{sec:unet_perform}

For our default run, we set $k=1/3$ so that $\Delta T$ with the longest timescale ($100\,$days) can still be fully covered in the simulation interval $[-135, 135]$. We train \texttt{locator} with the ADAM optimizer \citep{kingma2014adam}. The learning rate is set to $4\times 10^{-3}$ initially and drops by ten percent after each epoch. Training samples are divided into mini-batches of size 128. After 13 epochs ($\sim$2\,hours) of training on one NVIDIA Tesla V100 GPU, the average loss drops to 0.16 on the validation set of size 1024. The optimized model is then used to predict the mask and estimate $t_0$ and $t_{\rm E}$ according to Equation~(\ref{eqn:t0te}).

We evaluate \texttt{locator} on 16,384 light curves from the test set. The output masks are illustrated in Figure~\ref{fig:p_dist}, and the comparisons between the predicted and ground truth values of $\lg t_{\rm E}$ and $t_0/t_{\rm E}$ are shown in Figure \ref{fig:t0te}.
The accuracy of the prediction is measured by the root-mean-squared error (RMSE) between the ground truth and the prediction. They are indicated in the upper left corner respectively.
We note that \texttt{locator} has very different performances on close ($s<1$) binaries and wide ($s>1$) binaries. This is especially true for the $t_0/t_{\rm E}$ parameter. We take the results of close binaries as a more reliable evaluation of the overall performance and defer to the end of this section a more detailed explanation of this phenomenon.

In the default run, \texttt{locator} achieves an accuracy in $\lg t_{\rm E}$ with RMSE$=0.049$, which translates to a fractional uncertainty of $11.3\%$ on $t_{\rm E}$. The accuracy on the ratio $t_0/t_{\rm E}$ is comparable at RMSE($s<1$)=0.062, which corresponds to an uncertainty of $1.9\,$days on $t_0$ for a typical event with $t_{\rm E}=30\,$days. An example event is shown in Figure~\ref{fig:lc_caustic}, in which the light curve substantially deviates from the \citet{Paczynski:1986} model due to the caustic crossing feature and yet \texttt{locator} successfully recovers $t_0$ and $t_{\rm E}$ at high accuracy.

\begin{figure*
    \centering
    \includegraphics[width=0.8\linewidth]{figures/t0te_small.pdf}
    \caption{Similar to Figure~\ref{fig:t0te} except that only events with $t_{\rm E}<40\,$days and $k=2$ are used.}
    \label{fig:t0te_small}
\end{figure*}

\begin{figure
    \centering
    \includegraphics[width=\linewidth]{figures/rmse_k.pdf}
    \caption{The performance of \texttt{locator}, evaluated on RMSEs of $\lg{t_{\rm E}}$, $t_0/t_{\rm E}$ and $t_0$, as a function of $k$ values. }
    \label{fig:rmse_k}
\end{figure}

\begin{figure*
    \centering
    \includegraphics[width=\linewidth]{figures/lc_s_wide.pdf}
    \caption{An example event with $s>1$, in which \texttt{locator} ``fails'' to reproduce the input window.
    The input window (red dashed curve) is defined on the center of mass (red asterisk in the upper panel) of this wide binary system.
    The prediction of \texttt{locator} (shown in green) better matches the light curve window (blue dashed curve) defined on the center of magnification (blue asterisk in the upper panel) of the primary star (black plus on the left in the upper panel, with the one on the right being the secondary). This latter window better describes the time evolution of this wide binary event.
    \label{fig:lc_s_wide}}
\end{figure*}
As shown in the left panel of Figure~\ref{fig:t0te}, the scattering in $\lg t_{\rm E}$ increases for shorter events.
This is probably because, as timescale decreases, fewer data points are available to \texttt{locator} in the parameter inference. For example, with $k=1/3$ there are only $\sim25$ time stamps fall within $\Delta T$ for the shortest ($t_{\rm E}=5\,$days) event.
To improve the performance of \texttt{locator} on short events, we select light curves with $t_{\rm E}<40\,$days and train another model with $k=2$. After 8 epochs of training with the same learning rate schedule, the loss drops to 0.083 on the validation set. We evaluate the new model on 16,384 light curves with $t_{\rm E}<40\,$days.
\footnote{This includes light curves used in Figure \ref{fig:t0te} with $t_E<40$ and additional ones in the test set that add up to 16,384.}
The results are shown in Figure~\ref{fig:t0te_small}. The prediction accuracy on the timescale increases by a factor of $\sim2$, reaching a fractional uncertainty on $t_{\rm E}$ of $6.2\%$. The performance on the ratio $t_0/t_{\rm E}$ remains largely unchanged.

The above investigation has two implications. First, it outlines a general strategy to obtain more accurate $t_{\rm E}$. For example,
one may divide the overall range of $t_{\rm E}$ into multiple smaller intervals and train separate \texttt{locator} models.
The preliminary estimate of $t_{\rm E}$ from the default run of \texttt{locator} can then be ``polished'' by the \texttt{locator} model that is optimized for the narrower $t_{\rm E}$ interval including the preliminary estimate.
Additionally, the comparison in \texttt{locator} performance between two $k$ values also sheds light on how \texttt{locator} works. In the single-lens case, the most constraining power on $t_0$ comes from the central region with width of $\sim u_0 t_{\rm E}$, whereas the event timescale $t_{\rm E}$ is best constrained by data points extending into the light curve wings. This seems to apply to \texttt{locator} as well. As shown in Figure~\ref{fig:rmse_k}, the timescale parameter can be better constrained when a wider mask window (i.e., a larger $k$) is used, whereas the event peak time $t_0$ is largely unaffected by the choice of $k$. More works are needed to better understand the mechanism of \texttt{locator} (and U-Net in general).

Finally, we discuss the issue related to wide ($s>1$) binaries. As pointed out earlier, the match between the ground truth labels and the predictions of \texttt{locator} for wide binaries is not as good as for close binaries. This larger mismatch, however, does not mean that the inference power of \texttt{locator} is reduced for wide binaries. After visually inspecting a large number of wide binary light curves, we find that the mismatch is related to the use of the reference system. When generating the light curves, we have followed the common convention and adopted the center of mass of the binary as the origin of the coordinate system. However, for wide binaries, the external shear due to a component at wide separation is better modeled in the reference frame defined on the center of magnification. For $s>1$, the offset between the center of mass and the closer center of magnification is \citep[e.g.,][]{DiStefano:1996, Chung:2005}
\begin{equation}
    \Delta x = \frac{1}{1+q} \left( s-\frac{1}{s}  \right) \cdot \left\{ \begin{array}{ll}
        q, & {\rm if~primary} \\
        1, & {\rm if~secondary} \\
    \end{array} \right. .
\end{equation}
For very wide binaries, the event timescale given by the mass of the closest lens also takes over the timescale given by the total mass of the binary. Both can lead to large offsets between the ground truth window and the predicted window. The most extreme example that has the largest deviation in $t_0$ and $t_E$ with $q=0.99$ and $s=2.8$ is shown in Figure~\ref{fig:lc_s_wide}. Once the reference frame defined on the center of magnification is adopted, the match between the ground truth and the prediction of \texttt{locator} is substantially improved (see also \citealt{zhang2021real}).

redpajama_set_name: RedPajamaArXiv


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text:
After Meeting With Eric Schmidt, France Stands By Threat To Write Law Forcing Google To Pay To Link To News Sites
Matt McGee on October 29, 2012 at 8:01 pm
The French government is digging in its heels.
After a meeting today with Google Executive Chairman Eric Schmidt, French President Francois Hollande reiterated the government's plans to create a new law to force Google to pay when it links to newspaper websites.
The spat has been going on for a few weeks, since the French government first proposed a law that would require search engines to pay to link to content.
After the original back and forth, plans were made for today's meeting in Paris. As Reuters is reporting, President Hollande didn't back down. The president reiterated that France would move ahead with the law if Google and French press organizations can't reach a deal that would involve Google paying to link to them. He wants to see a deal in place by the end of the year.
Google quickly responded to the idea a couple weeks ago, saying such a law would threaten the company's existence and saying it will simply stop linking to news sites if the law passed. We've reached out to see if the company has anything new to say after today's meeting, but I suspect Google is also digging in its heels, too.
Matt McGee
Matt McGee joined Third Door Media as a writer/reporter/editor in September 2008. He served as Editor-In-Chief from January 2013 until his departure in July 2017. He can be found on Twitter at @MattMcGee.
Channel: IndustryGoogle: Business IssuesGoogle: NewsGoogle: Outside USLegal: Regulation

redpajama_set_name: RedPajamaCommonCrawl


text:
Pot meet kettle
by Goldy — Sunday, 7/8/07 , 11:28 am
America's airwaves have been damaged by a series of rule changes the past two decades that have ushered in an era of bland commercial radio and television.
— The Seattle Times, 7/8/2007
Working for 710-KIRO, the commercial news/talk station with Seattle's best local news coverage, the most live and local programming, and the most only balanced lineup of talkers in the market, I can agree with the thesis of today's Seattle Times editorial entirely guilt-free. (You know, except for the part about agreeing with a Seattle Times editorial — it just makes me feel so dirty.)
But as long as we're talking about "homogenized," "formula[ic]" and "bland" commercial media, I'm wondering if the Times' editors have bothered reading their own front page recently?
Saturday's front page consisted of two soft features and two news stories, both lifted from the Los Angeles Times. Today's front page is dominated by an infomercial for Boeing's 787 Dreamliner, a bleak assessment of our military and political failures in Iraq (courtesy of the Washington Post,) and the first installment of a serial novella.
Two days. Two front pages. No hard, local news.
By comparison, every article on the front page of today's New York Times, Washington Post, Los Angeles Times and Philadelphia Inquirer was written by staff reporters. I'm just sayin'.
To be fair, the Seattle P-I isn't much better in this regard, and the kind of homogenized, formulaic and bland fare we tend to see in Seattle's dailies is pretty typical of broadsheets nationwide. While I don't mean to diminish the Times' defensible thesis that lax ownership rules and media consolidation have damaged the broadcast industry's ability (and willingness) to serve local interests, the steady decline in both quality and readership of our nation's daily newspapers suggests that there are broader forces at work.
This isn't the first time the Times has editorialized on media ownership rules, and I urge the editorial board to flog this issue with the same sort of zeal they reserve for estate tax repeal. But I would also encourage a little introspection into how staff cuts and a slavish devotion to style-book-over-substance has led to a steady decline in the quality and utility of their own product.
No doubt, local media ownership tends to better serve local community interests. But as the Times has proven by example, it is no panacea on its own.
RightEqualsStupid spews:
This exchange between me and Proud Leftist is so good, I am going to post it in every thread on HA from now on. It destroys the Publicans' stupid talking points – period.
"I'm working really hard to understand how President Bill Clinton's actions act as justifcation for the righties. Read this fuckwads.
If your lame ass argument is that Bill Clinton was a bad President – which is what you say – and if your argument now is that the AWOL coward GW Bush is in someway comparable to Bill Clinton, what you are actually saying is that GW Bush is NO BETTER THAN CLINTON.
Does it hurt to be that stupid? I really want to know.
proud leftist says:
RES @ 18
You've hit it on the nose. The rightwing fringefucks consider Clinton to be the personification of evil–indeed, he is the devil incarnate in their twisted little minds. Nonetheless, their justification for all of the Bushites' sins is, always, "well, Clinton did it, too." Methinks they never studied logic."
Mark The Redneck Goldstein spews:
I've mostly given up on terrestial radio. Wall to wall ads and except for KVI, mostly idiotic blather. Think about it… if a fucking loser like goldy can get a radio gig, does that tell ya what a wasteland it is?
Satellite radio is the only way to go. Pay a few bucks a month, and get to listen to good programming instead of filler crap.
What the fuck is the times talking about with minority and women owned microstations? Who the fuck would listen to a radio station based on who owns it?
Just a reminder for you moonbats, especially RES and PL…
Bill Clinton was impeached and disbarred because he lied to a grand jury in a failed attempt to conceal a pattern of predatory behavior in connection with an assault case that he settled for $800k.
Can you imagine being too fucking dishonest to be a lawyer? How low can you go?
scary tyler moore spews:
don't you guys ever get tired of virtual circle jerking? is it because you JUST CAN'T GET LAID?
sheesh.
YOS LIB BRO spews:
YOU CAN WELSH ON A BET YOU LOST. PAY YOUR FUCKING GAMBLING DEBT.
LOSER!
ME spews:
THANKS BILL CLINTON FOR REMOVING THE WHITE HOUSE SILVERWARE.IM SURE THAT LAURA BUSH DIDNT LIKE THE PATTEN ANYWAY.ALSO ENJOY THE HOUSEWARMING GIFTS YOUVE RECEIVED FROM YOUR FRIENDS
Dan Rather spews:
Thomas Jefferson says:
Innuendo and supposition, thy name is Leftist.
By chance, were these "minutes of a 1991 board meeting of the National Family Planning and Reproductive Health Assn." brought forth by Dan Rather and/or Mary Maples? I'm sure they're just as credible, because no left-wing organization would *ever* stoop to trying to sabotage a Republican. Right? Never happen!
Ah, hell no… never. I am typing in the right subscript, aint I?
YOS HELL YOUR NOT IN CUBA OR CANADA WITH THE FAMILY WHERE THEY HAVE BETTER HEALTH CARE THEN THE US FUCKIN IDIOT.
Sunday, 7/8/07 at 1:21 pm
MY WINGNUT BRO YO @ 8,
I PREFER CANADA TO CUBA INDEED. TOO HOT AND STICKY DOWN THERE IN THE CARRIBEAN AND I DON'T AGREE WITH THE GOVERNMENT.
THE QUESTION DOOFUS @ 7 CAN'T ANSWER.
WHAT HAPPENED TO THE ACVR?
No better way to increase awareness of global warming than a concert by Hoobastank. hehehehehehe Liberals are so hilarious. hehehehehehe
Broadway Joe spews:
I don't even bother with the local paper here in Reno, it'd only be worth something to me if I had a bird cage or litterbox to line. I read the local edition of the News & Review, use eBay and Craigslist to replace the want-ads, and (shockingly) like MTR, have abandoned the crapfest that is terrestrial radio for satellite radio or the occasional internet simulcast. After KISW became a mostly-talk-radio station (let's call it 'dude-radio'), I flew the white flag and got XM.
One question though for MTR: XM or Sirius? Okay, moot point, now that they're merging…..
SeattleJew spews:
ANYONE HERE read THE THREAD?
Goldy is right in spades. We still get the PI mainly for the Wed food ads. There is basically no daily newspaper in Seattle now.
I find it hard to believe there is not just a need but a market.
headless lucy spews:
Speaking of media laziness…
http://hlrp.blogspot.com/2007/.....y-bad.html
Marc Rich and Scooter Libby: A Bad Comparison
Let us put to rest any comparison of the commutation of I. Lewis "Scooter" Libby's sentence and the pardon that Marc Rich received from President Bill Clinton.
They are in no way comparable.
By the degree of the individual's guilt and traitorous behavior, Libby's sentence-commutation by George W. Bush could be more aptly compared to GWB's father's pardon of Oliver North. North could have implicated the elder Bush in the illegal arms for hostages deal, which funded the subsequent murder of hundreds of thousands of Central American peasants, at the hands of the Reagan/Bush administration. Similarly, Libby could implicate the current underdstandably secretive holders of the offices of president and vice-president, of covering up the deliberate outing of a covert CIA operative.
Bill Clinton, in his autobiography, My Life, on pp. 940-941 has this to say about his own last-minute pardon of Marc Rich:
"The most controversial pardons went to Marc Rich and his partner, Pincus Green. Rich, a wealthy businessman, had left the United States for Switzerland shortly before he was indicted on tax and other charges for allegedly falsely reporting the price of certain oil transactions to minimize his tax liability. There were several such cases in the 1980s, when some oil was under price controls and some was not, inviting the dishonest to underestimate their income or to overcharge their customers. During that time, several people and companies were charged with violating the law, but the individuals were usually charged with a civil offense. It was extremely rare for tax charges to be prosecuted under the racketeering statutes, as Rich and Green were, and after they were charged, the Justice Department ordered U.S. attorneys to stop doing it. After he was indicted, Rich stayed overseas, mostly in Israel and Switzerland.
The government had allowed Rich's business to continue to operate after he agreed to pay $200 million in fines, more than four times the $48 million in taxes the government claimed he had evaded. Professor Marty Ginsberg, a tax expert and husband of Justice Ruth Bader Ginsberg, and Harvard law professor Bernard Wolfman had reviewed the transactions in question and concluded that Rich's companies were right in their tax computations, which meant that Rich himself had not owed any taxes on these transactions."
I call bullshit on Republicans for using the Marc Rich pardon as an example that is in any way comparable to the commutation of Scooter Libby richly deserved sentence. Incidentally, one of Rich's former lawyers was the ubiquitous Scooter Libby.
Yeah, but Oliver North and Scooter, eventhough they broke the law, are American heros. Marc Rich is not.
How are they "heros" (sic)? Any more than Willie Horton, who was released on the instructions of a Republican Governor, is a hero?
DOOFUS @ 15
WAS THE CHILD RAPIST A LIBERAL OR A CONSERVATIVE?
WHY CAN'T YOU ANSWER A SIMPLE QUESTION?
BILL CLINTON. I WANT TO THANK YOU AND YOUR STAFF IN THE WEST WING OF THE WHITE HOUSE FOR VANDALIZING AND DESTROYING GOVERMENT PROPERTY ON THE WAY OUT.I
ALSO APPRECIATE REMOVING ALL OF THAT EXCESS WEIGHT(CHINA,SILVERWARE,LINEN,TOWELS,ASH TRAYS,SOAP,PENS,MAGNETIC COMPASS,FLIGHT MANUALS,ETC)OUT OF AIR FORCE ONE.THE WEIGHT SAVINGS MEANS THAT IT WILL BURN LESS FUEL,THUS LESS TAX DOLLARS SPENT ON JET FUEL.THANK YOU!
MY GENIUS BRO YO @ 18
ANSWER MY QUESTION THEN. (DOOFUS IS CATATONIC OR SOMETHING).
WAS THE CROSS-DRESSER A LIBERAL OR A CONSERVATIVE?
klake spews:
By the way Roger is today what Paul Joseph Goebbels was in past the present day Minister for Public Enlightenment and Propaganda for the Socialist Democrats Party. His Nazi friends taught him the trade when he visited them in jail.
YOS I HEAR THAT THE LINES ARE SHORT GOING INTO CANADA AT THE BLAINE CROSSING NOW WOULD BE A GREAT TIME FOR YOU AND THE FAMILY TO MAKE THE MOVE GO FOR GUSTO BRO
John Barelli spews:
Wow. Roger leaves town, and all the wingnuts come out of hiding. You folks must be really scared of that poor old bunnyrabbit.
I wish that I had as much time as Roger to show you wingnuts for the absurd cowards that you are, but he's retired, and I still have to work for a living.
However, since Roger isn't here to say it, I'll do the honors.
klake is a Nazi.
MY WINGNUT BRO YO @ 21
I DO APPRECIATE CANADA AND WHAT THE PEOPLE THERE HAVE CREATED FOR THEMSELVES.
BUT I PREFER TO STAY HERE IN MY NATIVE COUNTRY THE USA TO FIX UP THE GODAWFUL MESS DUMBSHITS LIKE YOU HAVE MADE OF THE LAND OF MY BIRTH.
Thanks for again proving my point. We need more publican wingnuts like you comparing GW Bush to Bill Clinton. It completely destroys your talking pionts and you're just too stoooooopid to realize it. So please keep it up. Thanks.
K spews:
@ 18-
Keep up those tired old lies, they're all you've got.
"Speaking for the Bush administration in an April 18 letter, Phillip Larsen, the director of the White House Office of Administration, told the GAO that it could not document any damage or repairs."
http://sfgate.com/cgi-bin/arti.....169709.DTL
M. Yass spews:
Mark "the redneck" Goldstein bloviated:
Wall to wall ads is exactly right. KKKVI seems to run a lot of ads for various kinds of diet ripoffs and investment scams. They promoted the hell out of Wade Crook back in the day, before he was sentenced to several years in federal lockup.
I'm more interested in how losers like Kirby "meaner than catshit" Wilbur can get radio gigs. What are his qualifications? That he was a crooked real estate salesman at some point? Wait, I forgot, that qualifies him to be governor if Dino "cheap and mean" Rossi is to be believed.
Amen, brother! Hours of commercial-free radio of substance. CBC Radio One comes to mind immediately. It's so nice to get a perspective other than "why do President Bush's detractors hate America and want the terrorists to win?" Other good bets include BBC, PRI and Air America.
BJ @ 12 – I got xm because they used boeing satellites, didn NOT carry librual propaganda NPR, and DID carry nascar.
But really, ya can't go wrong with either XM or sirius… you don't hafta torture yourself with "robin and maynard" ever agin…
Yass – So tell me… how is air murka doing in the free market? How are their ratings?
Oh yeah… and are they still stealing money from poor chirrens in NYC to pay the bills?
Thought so…
Here's the deal… the free market solves all problems. If there's REALLY money to be made, you can bet your ass somebody will provide the product or service. What you guys REFUSE to admit is that murka is fundamentally a Conservative place, and your european socialist shit is NEVER gonna be mainstream here. NEVER. Not gonna happen. EVER. Even if The Smartest Woman In The World is POTUS or Empty Skirt Obama or The Inventor of The Internet.
Not gonna happen.
RD spews:
Here's the deal… the free market solves all problems.
Hooooo! That thar's a good 'un! Har har har har har!
murka is fundamentally a Conservative place
Hoooooooo! That'n's even better! Har har har har har har!
Thanks for the chuckle.
ass's ass .org spews:
Monday, 7/9/07 at 10:09 am
Diversity in Seattle means from left to farther left, from dumb to dumber, from the Times to the P-I. Bland homogeneity? Read our stupid evil-twin "news"papers, if you can gag them down, then go gag.
Maybe Goldy's too young to remember (it's difficult to discuss the age & inexperience of someone like Goldy who was the unfortunate victim of post-birth abortion) when all of America's Big Media, BM, were like Seattle's Big Media. They wallowed almost obliviously in fashionable leftwing fascist enthusiasms while throwing in, for comic relief, the equivalent of a goofy Blethen endorsement or a goofy Randy Hearst editorial. Media were as monochrome as Mayberry was then or Seattle is now. NBC ABC CBS told us what they thought we needed to know for our own good, dammit, and then they told us "that's the way it is."
Rush, wrong about almost everything since 1994, was right-on right that we were being twisted and manipulated by the Goldycrat media monopoly. Thanks, Rush, for trying to give us some doctrinal fairness, and thanks for diversity. AM radio rules, so of course you uncloseted tyrants want to change the rules.
You still have NBC ABC CBS. You have CNN MSNBC NPR, the Post and the P-I and the Times (Seattle) and the Times (LA) and the Times (New York). You even have Michael Savage's wife, Dan, and you have Stephanie Miller. Isn't that more than enough?
chadt spews:
Amazing, isn't it? The worse the publicans look, the more shrieking trolls we get here insisting that their fantasies are real.
It's called "denial".
Shrieking, wailing, FLAMING denial. They will prevail. The world will recognize the holiness of their position and they will triumph.
Watch for pork passing overhead.
Monday, 7/9/07 at 12:17 pm
Mark The Redneck Goldstein again bloviated:
Does it now?
Has the free market provided us with a quality operating system to run most of the world's computers? It has, has it? You're talking about the execrable, bug-laden, security-hole-infested monstrosity that is Windows?
How has the free market done with health care? Number one in the world in spending but 37th in outcomes. Every civilized country on this planet, including every country on the list above us, has a fully funded national health care system. It is precisely because the "free market" has not been given control of health care in those places that they spend less and and have better outcomes. Now, say again please, all after "free market solves everything"?
If there's REALLY money to be made, you can bet your ass somebody will provide the product or service.
And that is precisely why high culture is reserved for places like PBS and NPR. America is a fatuous, ignorant, culturally and intellectually bankrupt backwater whose citizens only care about who Paris Hilton isn't fucking this week. Anything that involves, you know, critical thinking or rational thought causes the average American's atrophied brain to freeze up.
That explains why you (I'm assuming) voted against your economic self-interest and elected Bush. Since you're not of the top 0.01 percent of wage earners, you had no business voting for him. Because of him and his buddies, you have lost ground under this administration. You are not getting ahead, but are working harder and harder just to slow the pace of the ground you're losing.
What you guys REFUSE to admit is that murka is fundamentally a Conservative place, and your european socialist shit is NEVER gonna be mainstream here.
You're absolutely right, "murka" is a fundamentally conservative (read: hide-bound, closed-minded, unable to accept new ideas) place. Hell, half of my countrymen think the Earth is 6,000 years old. See above re: ignorant. Also see above re: countries with national health care do better. But don't let me confuse you with the facts. You just go right on listening to Rush Limbaugh and his OxyContin-induced rantings.
NEVER. Not gonna happen. EVER. Even if The Smartest Woman In The World is POTUS or Empty Skirt Obama or The Inventor of The Internet.
You're absolutely right. And that's why the smart money is looking outside of the U.S. for innovation. That's why I'm moving to a new country at the first opportunity. I'd be only too happy to accept your inevitable offer to help me pack.
Incidentally, at least you're honest in admitting that you're a redneck. Your views lay bare your ignorance and lack of education or acculturation.
Mark The Redneck Goldstein scrawled with a crayon on a piece of tobacco juice-stained onion skin paper:
So you're one of those rednecks who drools all over himself while watching rolling billboards drive 500 miles in a circle? What a corporate tool you are. You must get a chubby for the virile young men who drive the cars. It's okay, we understand. Stop torturing yourself. It is not true that there exists a god that is going to send you to hell for all eternity. That's a myth.
For fuck's sake, though, learn to spell and capitalize. NASCAR is an acronym. And I think you meant "liberal" not "libural." This is what happens when you go to the public schools the GOP defunded to give the money to their rich friends, the religious schools run by pederastic sadists.
I don't hate you. I pity you. I see that you now represent the mainstream of the country of my birth. And it is for that reason that I want no further part of it and that I must leave.
sandalista spews:
Tuesday, 7/10/07 at 4:06 pm
"I want no further part of it and that I must leave …"
Not a moment too soon, don't let Ellis Island hit you in the ass on your way out, and I'll join you if thick Hillary gets elected.
(Don't forget to take a fistful of Steinbrueck t-shirts for foreign exchange …)


At Busch Gardens Tampa, There are nursing areas in Sesame Street Safari of Fun's family restroom. There's an electrical outlet for pumping moms in the companion restroom/baby care center. There is also a small microwave oven for moms to heat up bottles.


text:
Explore .base
AFRICAN METHODIST EPISCOPAL STATEMENT RE: VIOLENCE IN CHARLOTTESVILLE, VIRGINIA
August 14, 2017 BTL Admin Articles
The Council of Bishops of the African Methodist Episcopal Church condemns the violence that exploded in Charlottesville, Virginia. The "Unite the Right" march was sponsored by white supremacists, white nationalists, and the Ku Klux Klan to protest the removal of a statue of General Robert E. Lee. It was responded to by counter protestors.
The result was a face-off between the two groups marked with punching, kicking, water bottle hurling, racial taunting and even the deployment of chemical sprays. A car plowed into a group of peaceful protesters killing one woman and injuring about 19 others. The driver, James Alex Fields is being held on charges including second degree-murder. A helicopter monitoring the rally later crashed killing two Virginia State Troopers.
There are those who argue that white nationalists have the right to free speech, and in a democracy, we support their right. But what was on display before the nation yesterday was more than free speech. It was an unruly event designed to intimidate and provoke violence. In a word, what happened yesterday was a hate crime and domestic terrorism. It was demonic and does not represent what the United States claims it stands for.
We are also disappointed with the response of President Donald Trump. Let us be clear, this is not a partisan issue, this is a matter of failed leadership. President Trump while condemning hatred and violence, claimed it was on "many sides", but it wasn't. This violence was initiated by the white supremacist. We have heard President Trump specifically call out Attorney General Jeff Sessions, Senator Mitch McConnell and come down hard on others, but he did not explicitly condemn white nationalist, supremacist or the Ku Klux Klan. It must be noted that hate crimes have increased since President Trump's campaign and inauguration.
The Council of Bishops calls upon President Trump at a scheduled Monday press availability, to categorically denounce white supermacist, nationalist and the Ku Klux Klan that threatened to divide our nation. The Council wants to make it clear that he does not want or seeks their support. This would help unite and assure the nation that our president, embraces all the citizens of the United States.
The African Methodist Episcopal Church reaffirms our opposition to hate crimes, racism and anti–Semitism, and calls upon the nation, particularly our national leadership to condemn racism and hatred.
Hate never wins. It never has and it never will.
Bishop McKinley Young, Senior Bishop
Bishop Clement W. Fugh, President, Council of Bishops
Bishop Frank Madison Reid III, Ecumenical & Urban Affairs
Stay up to date with the LATEST news from .Base
Tags:AMECharlottesvilleKKKVirginiaWhite Supremacy
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text:
package org.restlet.data;

import java.util.Collections;
import java.util.Comparator;
import java.util.List;
import java.util.Map;
import java.util.concurrent.ConcurrentHashMap;

import org.restlet.engine.Engine;

/**
 * Method to execute when handling a call.
 *
 * @author Jerome Louvel
 */
public final class Method implements Comparable<Method> {

    /** Map of registered methods. */
    private static final Map<String, Method> _methods = new ConcurrentHashMap<String, Method>();

    /**
     * Pseudo-method use to match all methods.
     */
    public static final Method ALL = new Method("*",
            "Pseudo-method use to match all methods.");

    private static final String BASE_HTTP = "http://www.w3.org/Protocols/rfc2616/rfc2616-sec9.html";

    private static final String BASE_WEBDAV = "http://www.webdav.org/specs/rfc2518.html";

    /**
     * Used with a proxy that can dynamically switch to being a tunnel.
     *
     * @see HTTP
     *      RFC - 9.9 CONNECT
     */
    public static final Method CONNECT = new Method("CONNECT",
            "Used with a proxy that can dynamically switch to being a tunnel",
            BASE_HTTP + "#sec9.9", false, false);

    /**
     * Creates a duplicate of the source resource, identified by the
     * Request-URI, in the destination resource, identified by the URI in the
     * Destination header.
     *
     * @see WEBDAV
     *      RFC - 8.8 COPY Method
     */
    public static final Method COPY = new Method(
            "COPY",
            "Creates a duplicate of the source resource, identified by the Request-URI, in the destination resource, identified by the URI in the Destination header",
            BASE_WEBDAV + "#METHOD_COPY", false, true);

    /**
     * Requests that the origin server deletes the resource identified by the
     * request URI.
     *
     * @see HTTP
     *      RFC - 9.7 DELETE
     */
    public static final Method DELETE = new Method(
            "DELETE",
            "Requests that the origin server deletes the resource identified by the request URI",
            BASE_HTTP + "#sec9.7", false, true);

    /**
     * Retrieves whatever information (in the form of an entity) that is
     * identified by the request URI.
     *
     * @see HTTP
     *      RFC - 9.3 GET
     */
    public static final Method GET = new Method(
            "GET",
            "Retrieves whatever information (in the form of an entity) that is identified by the request URI",
            BASE_HTTP + "#sec9.3", true, true);

    /**
     * Identical to GET except that the server must not return a message body in
     * the response but only the message header.
     *
     * @see HTTP
     *      RFC - 9.4 HEAD
     */
    public static final Method HEAD = new Method(
            "HEAD",
            "Identical to GET except that the server must not return a message body in the response",
            BASE_HTTP + "#sec9.4", true, true);

    /**
     * Used to take out a lock of any access type on the resource identified by
     * the request URI.
     *
     * @see WEBDAV
     *      RFC - 8.10 LOCK Method
     */
    public static final Method LOCK = new Method("LOCK",
            "Used to take out a lock of any access type (WebDAV)", BASE_WEBDAV
                    + "#METHOD_LOCK", true, false);

    /**
     * MKCOL creates a new collection resource at the location specified by the
     * Request URI.
     *
     * @see WEBDAV
     *      RFC - 8.3 MKCOL Method
     */
    public static final Method MKCOL = new Method("MKCOL",
            "Used to create a new collection (WebDAV)", BASE_WEBDAV
                    + "#METHOD_MKCOL", false, true);

    /**
     * Logical equivalent of a copy, followed by consistency maintenance
     * processing, followed by a delete of the source where all three actions
     * are performed atomically.
     *
     * @see WEBDAV
     *      RFC - 8.3 MKCOL Method
     */
    public static final Method MOVE = new Method(
            "MOVE",
            "Logical equivalent of a copy, followed by consistency maintenance processing, followed by a delete of the source (WebDAV)",
            BASE_WEBDAV + "#METHOD_MOVE", false, false);

    /**
     * Requests for information about the communication options available on the
     * request/response chain identified by the URI.
     *
     * @see HTTP
     *      RFC - 9.2 OPTIONS
     */
    public static final Method OPTIONS = new Method(
            "OPTIONS",
            "Requests for information about the communication options available on the request/response chain identified by the URI",
            BASE_HTTP + "#sec9.2", true, true);

    /**
     * Requests that the origin server accepts the entity enclosed in the
     * request as a new subordinate of the resource identified by the request
     * URI.
     *
     * @see HTTP
     *      RFC - 9.5 POST
     */
    public static final Method POST = new Method(
            "POST",
            "Requests that the origin server accepts the entity enclosed in the request as a new subordinate of the resource identified by the request URI",
            BASE_HTTP + "#sec9.5", false, false);

    /**
     * Retrieves properties defined on the resource identified by the request
     * URI.
     *
     * @see WEBDAV
     *      RFC - 8.1 PROPFIND
     */
    public static final Method PROPFIND = new Method(
            "PROPFIND",
            "Retrieves properties defined on the resource identified by the request URI",
            BASE_WEBDAV + "#METHOD_PROPFIND", true, true);

    /**
     * Processes instructions specified in the request body to set and/or remove
     * properties defined on the resource identified by the request URI.
     *
     * @see WEBDAV
     *      RFC - 8.2 PROPPATCH
     */
    public static final Method PROPPATCH = new Method(
            "PROPPATCH",
            "Processes instructions specified in the request body to set and/or remove properties defined on the resource identified by the request URI",
            BASE_WEBDAV + "#METHOD_PROPPATCH", false, true);

    /**
     * Requests that the enclosed entity be stored under the supplied request
     * URI.
     *
     * @see <a
     *      href="http://www.w3.org/Protocols/rfc2616/rfc2616-sec9.html#sec9.6"
     *      HTTP RFC - 9.6 PUT
     */
    public static final Method PUT = new Method(
            "PUT",
            "Requests that the enclosed entity be stored under the supplied request URI",
            BASE_HTTP + "#sec9.6", false, true);

    /**
     * Used to invoke a remote, application-layer loop-back of the request
     * message.
     *
     * @see HTTP
     *      RFC - 9.8 TRACE
     */
    public static final Method TRACE = new Method(
            "TRACE",
            "Used to invoke a remote, application-layer loop-back of the request message",
            BASE_HTTP + "#sec9.8", true, true);

    /**
     * Removes the lock identified by the lock token from the request URI, and
     * all other resources included in the lock.
     *
     * @see WEBDAV
     *      RFC - 8.11 UNLOCK Method
     */
    public static final Method UNLOCK = new Method(
            "UNLOCK",
            "Removes the lock identified by the lock token from the request URI, and all other resources included in the lock",
            BASE_WEBDAV + "#METHOD_UNLOCK", true, false);

    /**
     * Adds a new Method to the list of registered methods.
     *
     * @param method
     *            The method to register.
     */
    public static void register(Method method) {
        String name = (method == null) ? null : method.getName().toLowerCase();
        if ((name != null) && !name.equals("")) {
            _methods.put(name, method);
        }
    }

    /**
     * Sorts the given list of methods by name.
     *
     * @param methods
     *            The methods to sort.
     */
    public static void sort(List<Method> methods) {
        Collections.sort(methods, new Comparator<Method>() {
            public int compare(Method m1, Method m2) {
                return m1.getName().compareTo(m2.getName());
            }
        });
    }

    /**
     * Returns the method associated to a given method name. If an existing
     * constant exists then it is returned, otherwise a new instance is created.
     *
     * @param name
     *            The method name.
     * @return The associated method.
     */
    public static Method valueOf(final String name) {
        Method result = null;

        if ((name != null) && !name.equals("")) {
            result = Method._methods.get(name.toLowerCase());
            if (result == null) {
                result = new Method(name);
            }
        }

        return result;
    }

    /** The description. */
    private final String description;

    /**
     * Indicates if the side-effects of several requests is the same as a single
     * request.
     */
    private volatile boolean idempotent;

    /** The name. */
    private volatile String name;

    /** Indicates if the method replies with a response. */
    private final boolean replying;

    /**
     * Indicates if it should have the significance of taking an action other
     * than retrieval.
     */
    private final boolean safe;

    /** The URI of the specification describing the method. */
    private volatile String uri;

    {
        // Let the engine register all methods (the default ones and the ones to
        // be discovered) as soon as the Method class is loaded or at least
        // used.
        Engine.getInstance();
    }

    /**
     * Constructor for unsafe and non idempotent methods.
     *
     * @param name
     *            The technical name of the method.
     * @see org.restlet.data.Method#valueOf(String)
     */
    public Method(final String name) {
        this(name, null);
    }

    /**
     * Constructor for unsafe and non idempotent methods.
     *
     * @param name
     *            The technical name of the method.
     * @param description
     *            The description.
     * @see org.restlet.data.Method#valueOf(String)
     */
    public Method(String name, String description) {
        this(name, description, null, false, false);
    }

    /**
     * Constructor for unsafe and non idempotent methods.
     *
     * @param name
     *            The technical name.
     * @param description
     *            The description.
     * @param uri
     *            The URI of the specification describing the method.
     * @see org.restlet.data.Method#valueOf(String)
     */
    public Method(String name, String description, String uri) {
        this(name, description, uri, false, false);
    }

    /**
     * Constructor for methods that reply to requests with responses.
     *
     * @param name
     *            The technical name.
     * @param description
     *            The description.
     * @param uri
     *            The URI of the specification describing the method.
     * @param safe
     *            Indicates if the method is safe.
     * @param idempotent
     *            Indicates if the method is idempotent.
     * @see org.restlet.data.Method#valueOf(String)
     */
    public Method(String name, String description, String uri, boolean safe,
            boolean idempotent) {
        this(name, description, uri, safe, idempotent, true);
    }

    /**
     * Constructor.
     *
     * @param name
     *            The technical name.
     * @param description
     *            The description.
     * @param uri
     *            The URI of the specification describing the method.
     * @param safe
     *            Indicates if the method is safe.
     * @param idempotent
     *            Indicates if the method is idempotent.
     * @param replying
     *            Indicates if the method replies with a response.
     * @see org.restlet.data.Method#valueOf(String)
     */
    public Method(String name, String description, String uri, boolean safe,
            boolean idempotent, boolean replying) {
        this.name = name;
        this.description = description;
        this.uri = uri;
        this.safe = safe;
        this.idempotent = idempotent;
        this.replying = replying;
    }

    /**
     * Compares this method to another. Based on the method name.
     *
     * @param o
     *            The other method.
     */
    public int compareTo(Method o) {
        if (o != null) {
            return this.getName().compareTo(o.getName());
        }
        return 1;
    }

    /** {@inheritDoc} */
    @Override
    public boolean equals(final Object object) {
        return (object instanceof Method)
                && ((Method) object).getName().equals(getName());
    }

    /**
     * Returns the description.
     *
     * @return The description.
     */
    public String getDescription() {
        return this.description;
    }

    /**
     * Returns the name.
     *
     * @return The name.
     */
    public String getName() {
        return name;
    }

    /**
     * Returns the URI of the specification describing the method.
     *
     * @return The URI of the specification describing the method.
     */
    public String getUri() {
        return this.uri;
    }

    /** {@inheritDoc} */
    @Override
    public int hashCode() {
        return (getName() == null) ? 0 : getName().hashCode();
    }

    /**
     * Indicates if the side-effects of several requests is the same as a single
     * request.
     *
     * @return True if the method is idempotent.
     */
    public boolean isIdempotent() {
        return idempotent;
    }

    /**
     * Indicates if the method replies with a response.
     *
     * @return True if the method replies with a response.
     */
    public boolean isReplying() {
        return replying;
    }

    /**
     * Indicates if it should have the significance of taking an action other
     * than retrieval.
     *
     * @return True if the method is safe.
     */
    public boolean isSafe() {
        return safe;
    }

    /**
     * Returns the name.
     *
     * @return The name.
     */
    @Override
    public String toString() {
        return getName();
    }
}

redpajama_set_name: RedPajamaGithub


"The schedule might change depending on the situation, but the episode will air somewhere between late May and early June," the source concluded.
As a loved member of the first season of "1N2D," Eun Ji Won's sudden return to the show has been met with an excited response. This will be his first time appearing on the show after leaving his position in 2012.
In related news, "1N2D" returned to the air on May 11 after being canceled for three weeks due to the tragic Sewol ferry incident.


Beautiful Wedding created using Modern Flowers and designed around Callas , Phalaenopsis Orchids , Roses . Collection of Bridal Bouquet, Bridesmaid and Buttonholes.
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