text:
\section{APPENDIX: TECHNICAL DETAILS}
    \input{supplement}
}{
}

\end{document}

\section{TECHNICAL DETAILS}
\fi

\heading{Time-domain likelihood}
The signal model used in this analysis has a sharp transition in the
time domain.  We do not wish to incorporate any data from the detectors before
our model begins so that we avoid bias; equivalently, we want to assume that we
are infinitely uncertain about the gravitational wave signal before the start
point of our signal model.  This requires special treatment compared to the
standard LIGO data analysis
\cite{TheLIGOScientific:2016wfe,TheLIGOScientific:2016src}. Other approaches to
quasi-normal mode extraction treat the data similarly \cite{Carullo:2019flw}
except for the assumption of periodicity, enforced by tapering, which is absent
in our treatment (see below).

As in previous analyses of GW150914, we assume that the detector measures a
discrete data stream $\mathbf{d}$ that contains our signal $\mathbf{s}$ contaminated by additive, Gaussian
noise $\mathbf{n}$.  At time $t_i$,
\begin{equation}
  d_i = h( t_i ) + n_i\, ,
\end{equation}
with the noise time series $\mathbf{n}{}$ having a multivariate normal distribution,
\begin{equation}
  \label{eq:noise-multivariate-gaussian}
  \mathbf{n}{} \sim \mathcal{N} (\boldsymbol{\mu}{}, \boldsymbol{\Sigma}{} )\, ,
\end{equation}
with mean $\boldsymbol{\mu}{}$ and covariance matrix $\boldsymbol{\Sigma}{}$.  The actual noise in
the detector has very large low-frequency components, while our signal lives
primarily at high ($\gtrsim 100 \, \mathrm{Hz}$) frequencies \cite{Berti2009,BertiWebsite}.
To reduce the low-frequency components of the noise,
 we first apply a fourth-order, high-pass Butterworth filter with a roll-on frequency of 20 Hz.
 After the filter, the data are very close to zero mean, so we assume $\boldsymbol{\mu}{} = \mathbf{0}$.

The distribution of the noise implies that the log-likelihood function (i.e., the
distribution of data $\mathbf{d}$ conditioned on a signal $\mathbf{h}$) is
\begin{align}
  \label{eq:mg-log-likelihood}
  \log p\left( \mathbf{d} \mid \mathbf{h} \right) = &-\frac{1}{2} \left( \mathbf{d} - \mathbf{h} \right)^T \boldsymbol{\Sigma}{}^{-1} \left( \mathbf{d} - \mathbf{h} \right) \nonumber \\
  &- \frac{1}{2} \log \det \boldsymbol{\Sigma} - \frac{K}{2} \log 2 \pi\, ,
\end{align}
where $K$ is the total number of samples.

We assume that the noise in the detector is stationary, so that the covariance
matrix takes a special (Toeplitz) form where the $ij$ component depends only on
the time separation between samples $i$ and $j$:
\begin{equation}
  \Sigma_{ij} = \left\langle n_i n_j \right\rangle = \rho\left( \left| i - j \right| \right),
\end{equation}
where $\rho$ is the autocovariance function,
\begin{equation}
  \rho\left( k \right) = \left \langle n_i n_{i+k} \right\rangle.
\end{equation}
(The expectation above runs over all times, $i$.)  The assumptions of stationary
Gaussian noise have been checked for GW150914 specifically
\cite{TheLIGOScientific:2016zmo,TheLIGOScientific:2016uux,TheLIGOScientific:2016src,TheLIGOScientific:2016wfe}
and the LIGO events in general \cite{TheLIGOScientific:2016pea,gwtc1:2018}.

The standard LIGO data analysis imposes an additional assumption that
\begin{equation}
  \rho\left( k \right) = \rho\left( K - k \right)
\end{equation}
for $0 \leq k < K$.  This
``circularity'' assumption is appropriate for data that are periodic with
period $K$; periodicity is typically enforced by tapering the data segment at
the beginning and end \cite{Veitch:2015,TheLIGOScientific:2016wfe}.  The
benefit of this assumption is that a circular Toeplitz matrix is diagonal in the
Fourier basis (i.e.\ stationary periodic noise has statistically-independent
Fourier components), and therefore the matrix-inversion step in the
log-likelihood reduces to a sum over independent frequency components.  Such a
likelihood can also be computed directly in the time domain
\cite{Carullo:2019flw}.

A taper is not appropriate for our data analysis since we wish to ignore data
from times before the peak of the waveform, where our signal model begins. We do
not have data before the peak in which to implement a taper; and tapering past
the peak would significantly reduce our signal.  Happily, fast and stable
algorithms exist for solving linear equations with a Toeplitz structure
\cite{Levinson:1947,Durbin:1960,Jones:2001scipy}, so a direct implementation of
our likelihood in Eq.\ \eqref{eq:mg-log-likelihood} is not too costly.

We estimate the autocovariance function by the empirical autocovariance of
$64\, \mathrm{s}$ of off-source data, after high-pass filtering as above.
This is analogous to the Welch method for estimating power spectral densities
in the frequency domain \cite{Welch:1967use} used by the standard LIGO analyses
\cite{Veitch:2015,TheLIGOScientific:2016wfe}.  Our analysis is based on data at
a sample rate of $2048 \, \mathrm{Hz}$, beginning at the peak signal amplitude
at 1126259462.423{} GPS and running for $0.5 \, \mathrm{s}$.
The autocovariance estimate is truncated to that same duration.

\heading{Other details}
\ifdefined
We handle polarizations by projecting the complex-valued strain in \eq{qnm} onto each LIGO detector by means of the corresponding antenna patterns.
\else
We handle polarizations by projecting the complex-valued strain in Eq.~(1) of the main text onto each LIGO detector by means of the corresponding antenna patterns.
\fi
To do so, we assume the source of GW150914 had right ascension $\alpha = 1.95 $ rad and declination $\delta = -1.27$ rad, with polarization angle $\psi = 0.82$ rad and inclination $\iota = \pi$ rad.
These parameters are consistent with the maximum a posteriori estimates inferred for GW150914 \cite{gw150914,gw150914_pe,GWOSC:GWTC}.
We also time-shift the LIGO Livingston data by the corresponding arrival-time delay of 7 ms \cite{gw150914,gw150914_pe}, so as to align the signal at the two detectors.
As noted in the main text, we may make these simplifications because all rindgown modes we consider are subject to the same angular dependence ($\ell=m=2$).
A version of this analysis with the more simplified approach of \cite{gw150914_tgr,bayesian_qnm} yields compatible results.

Our priors are such that quasinormal-mode amplitudes $A_n$ are allowed to vary in the range $[0,\,2.5\times10^{-19}]$, an arbitrary range found to offer full support to the posterior in all cases.
The corresponding phases $\phi_n$ are unrestricted in the full range $[0,\,2\pi]$.
For computational efficiency, we internally parameterize the amplitude and phase of each mode using the two quadratures $c_n = A_n \cos \phi_n$ and $s_n = A_n \sin \phi_n$, but set priors uniform in $A_n$ and $\phi_n$.
The remnant mass $M_f$ is allowed to vary within $[50,\,100]\,M_\odot$, while the dimensionless spin magnitude $\chi_f$ varies within $[0,\,1]$.
When considering explicit deviations from the no-hair theorem, we set uniform priors such that $-0.5 < \delta f_1 < 0.5$ and $-1 < \delta \tau_1 < 1$.
In all cases, samples are drawn from the posterior using \texttt{kombine} \cite{kombine}.

\ifdefined
\else

redpajama_set_name: RedPajamaArXiv


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text:
NewsWorldShunning virus lockdown, defiant Belarus stages Victory Day parade
Shunning virus lockdown, defiant Belarus stages Victory Day parade
People attend the Victory Day parade, which marks the anniversary of the victory over Nazi Germany in World War Two, amid the coronavirus disease (COVID-19) outbreak, in Minsk, Belarus May 9, 2020. REUTERS/Vasily Fedosenko
Thousands of soldiers marched in Belarus on Saturday to celebrate the Soviet victory in World War Two, as President Alexander Lukashenko rejected calls for lockdown measures to fight the coronavirus pandemic.
Lukashenko, who has ruled the eastern European country since 1994, has called fears over the coronavirus a "psychosis" and variously suggested drinking vodka, visiting saunas or playing ice hockey to beat the disease.
Spectators in stands in the capital Minsk, a few of whom wore masks, looked on as soldiers marched, tanks rolled past and Su-30 fighter jets flew in formation overhead.
Lukashenko's insistence on going ahead with the display contrasted with neighbour Russia, which scaled back celebrations amid a jump in coronavirus cases and postponed its usual massive military parade on Red Square.
Dressed in military uniform and surrounded by generals, Lukashenko said it was unacceptable for Belarus to even think about cancelling the parade.
"There will be people who will condemn us," Lukashenko said. He told such critics: "do not rush to draw conclusions, let alone condemn us, the heirs of the Victory, the Belarusians … We simply could not асt differently, we had no other choice."
Belarus has not imposed lockdown measures or social distancing rules, and kept its borders open while countries around the world have closed them.
"This is a demonstration of determination, will, strength, not so much for society as for the inner circle of the elite," said Andrey Egorov, senior analyst at the Center for European Transformation. "It's a demonstration that everything remains under control."
Another reason for staging the parade could be an act of one-upmanship against Russian President Vladimir Putin. Ties between the two traditional allies have been strained, especially over Moscow's decision to scale back subsidies and loans that prop up Lukashenko's rule.
"Against the background of Putin's cancelled parade, Lukashenko has the opportunity to draw attention to himself," said political analyst Alexander Klaskovsky.
"This is such a kind of revenge for the numerous humiliations. Putin hid in the bunker, and Lukashenko at that time will be standing on the podium in a beautiful uniform."
The World Organisation has called on Belarus to introduce tougher measures to fight the coronavirus and the head of its Minsk office has expressed concern about holding the parade.
There are 21,107 confirmed coronavirus cases in Belarus, with 121 deaths. But some in the country believe the official statistics underestimate the true toll.
Pictured: People attend the Victory Day parade, which marks the anniversary of the victory over Nazi Germany in World War Two, amid the coronavirus disease (COVID-19) outbreak, in Minsk, Belarus May 9, 2020. REUTERS/Vasily Fedosenko
By Josephine Koumettou
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redpajama_set_name: RedPajamaCommonCrawl


Okay so where do I start??? I guess I will just start and go from here!
One - today is Day 1 of my Project 365 - a photo a day. I have started a new blog just for that and will be placing the link in my sidebar and making an announcement once I have loaded my first photo. FUN!
Lastly, A PROJECT! I have been dying to learn how to make wire wrapped rings. I knew it was simple I just never sat down to try. So this morning I did! I honestly didn't finish the first one, I missed a vital step to keep the wraps neat and I hated it...so I didn't waste my time in trying to finish it I just tossed it aside.