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

The LIGO~\cite{TheLIGOScientific:2014jea} and Virgo~\cite{TheVirgo:2014hva}
detectors have observed a total of $90$ gravitational wave (GW) signals to
date~\cite{LIGOScientific:2018mvr, LIGOScientific:2021usb, LIGOScientific:2021djp}, including the landmark observations of the
first binary black hole (BH)~\cite{Abbott:2016blz}, binary neutron star
(NS)~\cite{TheLIGOScientific:2017qsa}, and BH-NS
binaries~\cite{LIGOScientific:2021qlt}. Among these observations,
GW190814~\cite{Abbott:2020khf} is unique due to its uncertain nature: a merger
of a $\sim23 \, M_{\odot}$ BH and a $\sim2.6 \, M_{\odot}$ companion that is
either the heaviest NS or the lightest BH ever discovered~\cite{Abbott:2020khf}
in a compact binary system.\footnote{A similar event, GW200210\_092254, a
merger of a $24.1 \, M_{\odot}$ BH and a $2.81 \, M_{\odot}$ compact object was
identified in Ref.~\cite{LIGOScientific:2021djp}.  However, this event is a
marginal GW candidate, with a probability of astrophysical origin
$p_{\text{astro}} \sim 0.54$~\cite{LIGOScientific:2021djp}. Therefore, we limit
our analysis to GW190814.} In addition to the intrigue about its astrophysical
origin~\cite{Godzieba:2020tjn, Dexheimer:2020rlp, Clesse:2020ghq, Tews:2020ylw,
Tsokaros:2020hli, Fattoyev:2020cws, Zhang:2020zsc, Tan:2020ics, Nathanail:2021tay}, this event
also poses new challenges for waveform models due to the highly unequal masses
of the binary components.

Numerical relativity (NR) is the only available method for solving Einstein's
equations near the merger of two compact objects, and has played a central role
in GW astronomy~\cite{Pretorius:2005gq, Campanelli:2005dd, Baker:2005vv,
Boyle:2019kee}. Unfortunately, NR simulations are prohibitively expensive
for direct GW data analysis applications, as each simulation can take up to a
few months on a supercomputer. The need for a faster alternative to NR has led
to the development of several semi-analytical waveform
models~\cite{Ossokine:2020kjp, Khan:2019kot, Matas:2020wab, Thompson:2020nei,
    Cotesta:2018fcv, Estelles:2021gvs, Estelles:2020twz, Pratten:2020ceb,
Garcia-Quiros:2020qpx, Akcay:2020qrj, Nagar:2020pcj} that rely on some
physically motivated assumptions for the underlying phenomenology, and
calibrate the remaining free parameters to NR simulations.  As a result, these
models are fast enough for GW data analysis, but are typically not as accurate
as the NR simulations~\cite{Varma:2021csh, Varma:2018mmi, Varma:2019csw}.

On the other hand, NR surrogate models~\cite{Field:2013cfa, Blackman:2017pcm,
Varma:2018mmi, Varma:2019csw} take a data-driven approach by training the model
directly on NR simulations, without the need for added assumptions. These
models have been shown to reproduce NR simulations without a significant loss
of accuracy while also being fast enough for GW data
analysis~\cite{Varma:2018mmi, Varma:2019csw}. The main limitation for surrogate
models, however, is that their applicability is restricted to the regions where
sufficient NR simulations are available. In particular, NR simulations become
expensive as one approaches large mass ratios $q=m_1/m_2$ and/or large spin
magnitudes $\chi_{1,2}$~\cite{Boyle:2019kee, Lousto:2020tnb}, where $m_{1}$
($m_2$) represents the mass of the heavier (lighter) BH, so that $q\geq1$, and
$\bm{\chi}_{1,2}$ represent the corresponding dimensionless spins, with magnitudes
$\chi_{1,2}\leq 1$.  Therefore, previous NR surrogate models have only been
trained on simulations with $q\leq8$ and $\chi_{1,2} \leq
0.8$~\cite{Varma:2018mmi}. These models are not suitable for high-mass ratio
systems like GW190814 ($q \sim 8.96^{+0.75}_{-0.62}$ at 90\%
credibility~\cite{Abbott:2020khf}).

Similarly, the calibration NR data for the semi-analytical
models~\cite{Ossokine:2020kjp, Khan:2019kot, Matas:2020wab, Thompson:2020nei}
used in the GW190814 discovery paper~\cite{Abbott:2020khf} are also very sparse
at mass ratios $q \gtrsim 8$. Fortunately, most of the events observed by
LIGO-Virgo fall at more moderate mass ratios $q\lesssim
5$~\cite{LIGOScientific:2021djp}, with a preference for $q\sim 1$
~\cite{LIGOScientific:2021psn}, where current semi-analytical models are well
calibrated. In contrast, the large mass ratio of GW190814 poses new challenges
for waveform modeling, and it is important to understand the impact of modeling
error on the source parameter estimation of this event.

For example, at large $q$, subdominant modes of radiation beyond the quadrupole
mode can play an important role. The complex waveform $\mathpzc{h} = h_{+} - i
h_{\times}$ can be decomposed into a sum of spin-weighted spherical harmonic
modes $\mathpzc{h}_{\ell m}$:
\begin{align}
\mathpzc{h}(t,\iota,\varphi_0) = \sum^\infty_{l=2} \sum^{\ell}_{m=-\ell} \mathpzc{h}_{\ell m}(t)
                        \ _{-2}Y_{\ell m}(\iota,\varphi_0),
\label{eq:hlm}
\end{align}
where $h_{+}$ ($h_{\times}$) represents the plus (cross) GW polarization,
$_{-2}Y_{\ell m}$ are the spin $=-2$ weighted spherical harmonics, and
$(\iota,\varphi_0)$ represent the direction to the observer in the source
frame.\footnote{The source frame is defined as follows: the $z$-axis points
along the orbital angular momentum $\bm{L}$ of the binary, the $x$-axis points
along the line of separation from the lighter BH to the heavier BH, and the
$y$-axis completes the triad. Therefore, $\iota$ denotes the inclination angle
between $\bm{L}$ and line-of-sight to the observer.} The $\ell=|m|= 2$ terms
typically dominate the sum in Eq.~(\ref{eq:hlm}), and are referred to as the
quadrupole modes. However, as one approaches large $q$ the subdominant modes
(also referred to as nonquadrupole or higher modes) become increasingly
important for estimating the binary source properties~\cite{Varma:2016dnf,
Varma:2014jxa, Capano:2013raa, Shaik:2019dym, Islam:2021zee}. Therefore, it is
important for waveform models to accurately capture the effect of the
subdominant modes on the observed signal. Along with developing a new surrogate
model, one of the goals of this work is to assess whether current
semi-analytical models, and in particular their subdominant modes, are accurate
enough for events like GW190814.

\subsection{The \texttt{NRHybSur2dq15}\xspace model}

In this work, we build a GW190814-targeted surrogate model that is based on NR
simulations with mass ratios up to $q=15$. Due to the computational cost of NR
simulations with large mass ratios and/or spins~\cite{Boyle:2019kee}, we
restrict the model to spins (anti-) aligned along the direction of the orbital
angular momentum $\bm{L}$, with $\chi_{1z} \in [-0.5, 0.5]$,
$\chi_{1x}=\chi_{1y}=0$, and $\bm{\chi}_{2}=0$.
We ignore the spin of the secondary BH for simplicity, as its effect is
expected to be suppressed for large $q$ systems like GW190814, at least at
current signal to noise ratio (SNR). For example, Ref.~\cite{Abbott:2020khf}
found that the secondary spin of GW190814 was unconstrained. This assumption
may need to be relaxed for louder signals that are expected in the future with
detector improvements.

Above, the $z$-direction is taken to be along $\bm{L}$, whose direction is
constant for aligned-spin systems. In addition to the dominant ($(\ell, m) =
(2,2)$) mode, the model accurately captures effects of the following
subdominant modes: (2,1), (3,3), (4,4) and (5,5). Note that the $m<0$ modes
carry the same information as $m>0$ modes for aligned-spin binaries, and do not
need to be modeled separately.

To train the model, we perform 20 new NR simulations in the range $8<q\leq15$,
using the Spectral Einstein Code (SpEC)~\cite{SpECwebsite, Boyle:2019kee}
developed by the SXS~\cite{SXSWebsite} collaboration.  Due to computational
limitations, these simulations only include about 30 orbits before the merger;
therefore, they do not cover the full LIGO-Virgo frequency band for stellar
mass binaries. More precisely, for total masses $M=m_1+m_2 \lesssim
70.0\,M_{\odot}$, the initial frequency of the $(2,2)$ mode of these waveforms
falls within the LIGO-Virgo band, taken to begin at $f_{\mathrm{low}}=20$ Hz. We extend the
validity of the model to lower masses by smoothly
transitioning~\cite{Varma:2018mmi} to the effective-one-body (EOB) model
\texttt{SEOBNRv4HM}\xspace~\cite{Cotesta:2018fcv} for the early inspiral. These NR-EOB
\emph{hybrid} waveforms are augmented with $31$ waveforms in the $q\leq8$
region, generated using the \texttt{NRHybSur3dq8}\xspace~\cite{Varma:2018mmi} surrogate model,
which is already hybridized.  The new model, \texttt{NRHybSur2dq15}\xspace is trained on these $51$
hybrid waveforms, and all modes of this model are valid for full LIGO-Virgo band
(with $f_{\mathrm{low}}=20$ Hz) for $M \gtrsim 9.5 \, M_{\odot}$.

For simplicity, \texttt{NRHybSur2dq15}\xspace ignores two physical features that can be relevant
for GW190814: precession and tidal deformability of the secondary object.
Precession occurs when the component objects have spins that are tilted with
respect to $\bm{L}$. In such binaries, the spins interact with $\bm{L}$ (as
well as with each other), causing the orbital plane to
precess~\cite{Apostolatos:1994pre}. The effective precession parameter
$\chi_{\text{p}}$~\cite{Schmidt:2014iyl} for GW190814 was constrained to
$\chi_{\text{p}} \lesssim 0.07$ at 90\% credibility by
Ref.~\cite{Abbott:2020khf}. However, including precession in the waveform model
was found to improve the component mass constraints~\cite{Abbott:2020khf}.
Therefore, while neglecting precession is a reasonable assumption, this can
limit the applicability of our results. Precessing NR surrogates can require
$\gtrsim 1000$ NR simulations~\cite{Blackman:2017dfb, Blackman:2017pcm,
Varma:2019csw}, which is not currently feasible for large mass
ratios~\cite{Boyle:2019kee}. Nevertheless, we can still compare the performance
of \texttt{NRHybSur2dq15}\xspace against other nonprecessing models.

Next, the tidal deformations of NSs within a compact binary can alter the
orbital dynamics, imprinting a signature on the GW
signal~\cite{Flanagan:2007ix}. Assuming the secondary object of GW190814 is a
NS, this effect, parameterized by the effective tidal
deformability~\cite{Flanagan:2007ix} scales as $\tilde{\Lambda} \propto 1/q^4$
(see e.g.~Eq.~(1) of Ref.~\cite{TheLIGOScientific:2017qsa}), and can be safely
ignored for GW190814~\cite{Abbott:2020khf}. For large $q$ binaries like
GW190814, the NS simply plunges into the BH before tidal deformation or
disruption can occur~\cite{Foucart:2013psa}. As a result, GW190814 shows no
evidence of measurable tidal effects in the signal, and no electromagnetic
counterpart to the GWs has been identified~\cite{Abbott:2020khf}. This
justifies our choice to ignore the effects of tidal deformation in \texttt{NRHybSur2dq15}\xspace.

To summarize, \texttt{NRHybSur2dq15}\xspace is valid for mass ratios $q\leq15$, spins $\chi_{1z}
\in [-0.5, 0.5]$ and $\chi_{1x}=\chi_{1y}=\bm{\chi}_{2}=0$, total masses $M \gtrsim
9.5 \, M_{\odot}$ (for $f_{\mathrm{low}}=20$ Hz), and zero tidal deformability. The name
of the model is derived from the fact that it is based on NR hybrid waveforms,
spans the 2-dimensional parameter space of $(q, \chi_{1z})$, and extends to
$q=15$.

The rest of the paper is organized as follows. In Sec.~\ref{sec:method}, we
describe the construction of \texttt{NRHybSur2dq15}\xspace. In Sec.~\ref{sec:surrogate_errors}, we
evaluate the accuracy of the model by computing mismatches against NR-EOB
hybrid waveforms. We demonstrate that \texttt{NRHybSur2dq15}\xspace is more accurate than existing
semi-analytical models by at least an order of magnitude, with mismatches
$\lesssim 2 \times 10^{-3}$ throughout its parameter space. In
Sec.~\ref{sec:reanalyzing_GW190814}, we reanalyze GW190814 using \texttt{NRHybSur2dq15}\xspace and
find that our constraints on the binary properties are consistent with those
reported in Ref.~\cite{Abbott:2020khf}.  We end with some concluding remarks in
Sec.~\ref{sec:conc}.  Throughout this paper, we denote redshifted the detector
frame masses as $m_1$, $m_2$, and $M=m_1+m_2$. When referring to the source
frame masses, we denote them explicitly as $\mSrc{1}$, $\mSrc{2}$, and $M^{\mathrm{src}}$.
These are related by factors of $1+z$, where $z$ is the cosmological redshift;
for example, $M=(1+z) \, M^{\mathrm{src}}$.

\section{Methods}
\label{sec:method}

In this section we describe the steps involved in building the new model
\texttt{NRHybSur2dq15}\xspace, including the generation of the required NR and hybrid waveforms,
and the surrogate model construction.

\subsection{Training set generation}
\label{sec:training_set}

In order to build the surrogate model, we need a \emph{training set} of hybrid
waveforms and their associated binary parameters. The parameter space of
interest for us is the 2D region $q \in [1, 15]$ and $\chi_{1z} \in [-0.5,
0.5]$, with fixed $\chi_{1x}=\chi_{1y}=0$, and $\bm{\chi}_{2}=0$.  The total mass
scales out for binary BHs and does not need to be modeled separately. The NR
simulations necessary for generating hybrid waveforms are expensive, especially
as one approaches large $q$~\cite{Boyle:2019kee}.  Therefore, one would ideally
like to use the fewest possible hybrid waveforms to build a surrogate model of
given a target accuracy. However, we do not know a priori how big the training
set should be or how these points should be distributed in the parameter space.
In order to determine a suitable training set, we first build a surrogate model
for post-Newtonian (PN) waveforms.

\subsubsection{PN surrogate and new NR simulations}
\label{sec:pn_sur}

We use the GWFrames package~\cite{GWFrames} to generate PN waveforms. For the
orbital phase, we use the TaylorT4~\cite{Boyle:2007ft} approximant, and include
nonspinning terms up to 4 PN order~\cite{Blanchet:2004ek, Blanchet:2004ek,
Jaranowski:2013lca, Bini:2013zaa, Bini:2013rfa} and spin terms up to 2.5 PN
order~\cite{Kidder:1995zr, Will:1996zj, Bohe:2012mr}. For the amplitudes, we
include terms up to 3.5 PN order~\cite{Blanchet:2008je, Faye:2012we,
Faye:2014fra}. For the PN surrogate, we restrict the length of the waveforms to
be $5000 \, M$, terminating at the orbital frequency of the Schwarzschild
innermost-stable-circular-orbit (ISCO): $\omega_{\mathrm{orb}} = 6^{-3/2} \,
\mathrm{rad}/M$. In addition, we only use the $(2,2)$ mode for simplicity.
Despite the restrictions in length, mode-content, and the missing
merger-ringdown section in the PN waveforms, we find that this approach
provides a good initial training set for constructing hybrid NR-EOB
surrogates~\cite{Varma:2018mmi}. Above, the orbital frequency is defined as:
\begin{equation}
    \omega_{\mathrm{orb}} = \frac{d\phi_{\mathrm{orb}}}{dt},
\label{eq:omega_orb}
\end{equation}
where $\phi_{\mathrm{orb}}$ is the orbital phase obtained from the (2,2) mode
(see Eq.~(\ref{eq:orb_phase})).

\begin{figure}[tbh]
\centering
\includegraphics[width=0.45\textwidth]{pn_surrogate.pdf}
\caption{
Largest mismatch of the PN surrogate (over the entire validation set) as a
function of number of greedy parameters used for training. The PN surrogate is
seen to converge to the validation waveforms as the size of the training set
increases.
}
\label{fig:pn_greedy}
\end{figure}

We initialize the training set for the PN surrogate with just the corner cases
of the parameter space. For our 2D model, these consist of the four points:
$(q, \, \chi_{1z}) = (1, \, \pm 0.5)$ and $(15, \, \pm 0.5)$.  We augment the
training set in an iterative \emph{greedy} manner: At each iteration, we build
a PN surrogate with the current training set, following the same methods as we
use for the hybrid surrogate (see Sec.~\ref{sec:surrogate_building}). Then, we
test this surrogate against a larger ($\sim 10$ times) \emph{validation set},
generated by randomly sampling the parameter space at each
iteration.\footnote{
The boundary parameters are expected to be more important than those in the
bulk; therefore, for 30\% of the points in the validation set, we sample only
from the boundary, which corresponds to the edges of a square in the 2D case.}
We select the parameter in the validation set that has
the largest error (computed using Eq.~(\ref{eq:mismatch_td})) and add it to the
training set for the next iteration. We repeat this procedure until the largest
validation error falls below a certain threshold.

In order to estimate the error between two complex waveforms $\mathpzc{h}_1$ and $\mathpzc{h}_2$,
we use the time-domain inner product,
\begin{align}
    & \braket{\mathpzc{h}_1,\mathpzc{h}_2} = \bigg| \int_{t_{\mathrm{min}}}^{t_{\mathrm{max}}} \mathpzc{h}_1(t) \mathpzc{h}_2^*(t) dt \bigg| ,
\end{align}
to compute the mismatch,
\begin{align}
    & \mathcal{MM} = 1 - \frac{\braket{\mathpzc{h}_1,\mathpzc{h}_2}}{\sqrt{\braket{\mathpzc{h}_1,\mathpzc{h}_1}\braket{\mathpzc{h}_2,\mathpzc{h}_2}}}
\label{eq:mismatch_td}
\end{align}
When computing mismatches for the PN surrogate, we assume a flat noise curve,
and do not optimize over time and phase shifts.

Figure~\ref{fig:pn_greedy} shows the maximum validation error at each iteration
against the size of the training set. We stop this procedure when the training
set size reaches 47, as the mismatch settles below $10^{-6}$ at this point.
Among these, 31 cases lie in the region $q\leq8$, while 16 lie in the region
$8<q\leq15$. Rather than perform new NR simulations for the $q\leq8$ cases, we
generate waveforms using the existing \texttt{NRHybSur3dq8}\xspace model~\cite{Varma:2018mmi}.
This model was trained on NR-EOB/PN hybrid waveforms with mass ratios $q\leq8$
and spins $\chi_{1z,2z}\in [-0.8, 0.8]$, and was shown to reproduce the hybrid
waveforms without a significant loss of accuracy~\cite{Varma:2018mmi}.

For the cases with $q>8$, we perform new NR simulations using
SpEC~\cite{SpECwebsite, Boyle:2019kee}. These NR waveforms include $\sim 5000M$
of evolution before the merger and are hybridized using
\texttt{SEOBNRv4HM}\xspace~\cite{Cotesta:2018fcv} waveforms to include the early inspiral (see
Sec.~\ref{sec:hybridization}). However, of the 16 cases with $q>8$, only 15
simulations were successfully completed.\footnote{The reason for failure is
large constraint violation as the binary approaches merger. We believe a better
domain decomposition may be needed for this simulation, which we plan to
explore in the future.} This leaves us with a total of 46 training waveforms
(15 NR-EOB hybrid waveforms and 31 \texttt{NRHybSur3dq8}\xspace waveforms).

From an initial attempt to build a hybrid surrogate with these 46 waveforms, we
found that the model performs poorly for low masses $\lesssim 50 M_{\odot}$,
with mismatches reaching $\sim 10^{-2}$, but performs very well for higher
masses, with mismatches $\sim 10^{-3}$. In other words, the late inspiral and
merger-ringdown stages were accurately captured, but the early inspiral was
not. This suggested that more hybrid waveforms were required. To estimate where
in parameter space to place new hybrid waveforms, we first constructed a trial
NR-only surrogate using the above training set of 46 waveforms, but restricted
to the last $5000M$ before merger; we will refer to this model as
\texttt{NRSur2dq15}\xspace. Next, we hybridized waveforms (see
Sec.~\ref{sec:hybridization}) obtained from \texttt{NRSur2dq15}\xspace to generate new
training points in the $q>8$ region. This bootstrap method allowed us to create
as many hybrid waveforms as necessary in the $q>8$ region without performing
new NR simulations. After some trial and error, we found that placing five new
hybrid waveforms at $q=14$ (uniformly distributed in $\chi_{1z}\in[-0.5, 0.5]$)
resolved the problem at low masses.

With this insight, we finally performed five new SpEC NR simulations at these
points and added the hybrid waveforms based on these to our training set for
the final model, which now includes 20 NR-EOB hybrid waveforms and 31 \texttt{NRHybSur3dq8}\xspace
waveforms, for a total of 51 waveforms. Figure~\ref{fig:parameter_distribution}
shows the distribution of these parameters, including the failed simulation and
the new $q=14$ simulations.

\begin{figure}[tbh]
\centering
\includegraphics[width=8.4cm]{parameter_distribution.pdf}
\caption{
Training set parameters used in building the surrogate model \texttt{NRHybSur2dq15}\xspace. The red
markers correspond to cases with $q\leq8$, for which \texttt{NRHybSur3dq8}\xspace is used to
generate training waveforms. The black markers represent the new NR waveforms
performed for this work, while the empty marker shows the failed NR simulation.
The distribution of the 47 parameters from Fig.~\ref{fig:pn_greedy} can be seen
by ignoring the black markers highlighted in cyan; these represent the five
additional NR simulations that were necessary to improve the model.
}
\label{fig:parameter_distribution}
\end{figure}

The new NR simulations are performed using SpEC~\cite{SpECwebsite,
Boyle:2019kee}; they have been assigned identifiers SXS:BBH:2463-SXS:BBH:2482,
and made publicly available through the SXS catalog~\cite{SXSCatalog}.  The
constraint equations are solved employing the extended conformal thin sandwich
formalism~\cite{York:1998hy, Pfeiffer:2002iy} with superposed harmonic Kerr
free data~\cite{Varma:2018sqd}. The evolution equations are solved employing
the generalized harmonic formulation~\cite{Lindblom:2005qh,Rinne:2008vn}.  The
start time of these simulations is approximately $5000M$ before the peak of the
waveform amplitude (defined in Eq.~(\ref{eq:amp_tot})), where $M = m_1 + m_2$
is the total Christodoulou mass measured after the initial burst of junk
radiation~\cite{Boyle:2019kee}.  The initial orbital parameters  are chosen
through an iterative procedure~\cite{Buonanno:2010yk} such that the orbits are
quasicircular; the largest eccentricity for these simulations is
$6.4\times10^{-4}$, while the median value is $2.9\times10^{-4}$. The waveforms
are extracted at several extraction surfaces at varying finite radii form the
origin and then extrapolated to future null infinity~\cite{Boyle:2009vi}.
Finally, the extrapolated waveforms are corrected to account for the initial
drift of the center of mass~\cite{Woodford:2019tlo}.

\subsubsection{Hybridization}
\label{sec:hybridization}

Given the new NR waveforms, we now hybridize them by smoothly attaching an
EOB waveform for the early inspiral. For the previous NR hybrid surrogate model
\texttt{NRHybSur3dq8}\xspace~\cite{Varma:2018mmi}, a combination of PN and EOB was used for the
early inspiral: the amplitudes for all modes were obtained from PN, while the
phase evolution for all modes was derived from the (2,2) mode of the \texttt{SEOBNRv4}\xspace
EOB model~\cite{Bohe:2016gbl} (see Sec.~IV.B of Ref.~\cite{Varma:2018mmi}).
This was motivated by the fact that the PN mode amplitudes were found to be
accurate enough for hybridizing $q\leq8$ NR simulations, while the PN mode
phases were not (see Fig.~\ref{fig:pn_eob_amplitudes} of
Ref.~\cite{Varma:2018mmi}).

\begin{figure}[thb]
\centering
\includegraphics[width=8.4cm]{plot_hyb.pdf}
\caption{
Mode amplitudes for NR, PN, and \texttt{SEOBNRv4HM}\xspace as a function of the characteristic
speed $v=\omega_{\mathrm{orb}}^{1/3}$, for binary parameters $(q, \chi_{1z},
\chi_{2z}) = (15, 0.5, 0.0)$. The vertical dashed lines represent the
Schwarzschild ISCO point $v=1/\sqrt{6}$. While PN deviates significantly from
NR, \texttt{SEOBNRv4HM}\xspace shows excellent agreement. We show all available modes of \texttt{SEOBNRv4HM}\xspace.
}
\label{fig:pn_eob_amplitudes}
\end{figure}

We find that the same strategy does not work for the large $q$ cases
considered in this work. Figure~\ref{fig:pn_eob_amplitudes} shows a comparison
between the mode amplitudes of NR, PN and the \texttt{SEOBNRv4HM}\xspace EOB
model~\cite{Cotesta:2018fcv}, for a $q=15$ system. We show all modes [(2,2),
(2,1), (3,3), (4,4), and (5,5)] included by \texttt{SEOBNRv4HM}\xspace, which is an extension of
the \texttt{SEOBNRv4}\xspace model. The PN waveforms are described in Sec.~\ref{sec:pn_sur}; we
include amplitudes terms up to 3.5 PN order~\cite{Blanchet:2008je, Faye:2012we,
Faye:2014fra}. In Fig.~\ref{fig:pn_eob_amplitudes}, the PN amplitudes
(especially for the subdominant modes) deviate significantly from NR , while
\texttt{SEOBNRv4HM}\xspace shows excellent agreement. This is not surprising, as \texttt{SEOBNRv4HM}\xspace is
calibrated to NR waveforms, as well as some BH perturbation theory waveforms at
extreme mass ratios~\cite{Cotesta:2018fcv}. We conclude that current PN
waveforms are not suitable for hybridizing NR waveforms at large mass ratios
like $q\sim 15$.  Therefore, in this work, we only use \texttt{SEOBNRv4HM}\xspace for hybridizing
NR waveforms. Unfortunately, this means that our new model \texttt{NRHybSur2dq15}\xspace is
restricted to the same set of modes as \texttt{SEOBNRv4HM}\xspace.

We follow the same hybridization procedure as Sec.~V of
Ref.~\cite{Varma:2018mmi} to smoothly attach \texttt{SEOBNRv4HM}\xspace inspirals to the 20 new
$q>8$ NR simulations obtained in Sec.~\ref{sec:pn_sur}. For the remaining 31
training cases with $q\leq8$, we generate waveforms using the \texttt{NRHybSur3dq8}\xspace model,
as it is already hybridized. This completes the construction of our training
set waveforms.

\subsection{Frame alignment}

We follow Ref.~\cite{Varma:2018mmi} and apply the following post processing to
the training set waveforms. This ensures that all waveforms are in the same
frame, and therefore that the data used in the surrogate fits (see
Sec.~\ref{sec:surrogate_building}) vary smoothly across parameter space.

\subsubsection{Time alignment}
\label{sec:time_shifts}
We apply a time shift to each training waveform such that peak of the total
amplitude
\begin{equation}
    A_{\mathrm{tot}} = \sqrt{\sum_{l,m} \lvert\mathpzc{h}_{\ell m}\rvert^2},
\label{eq:amp_tot}
\end{equation}
occurs at $t=0$. The original peak time is determined by a quadratic fit using
5 time samples adjacent to the discrete maximum of
$A_{\mathrm{tot}}$~\cite{Blackman:2017pcm}.

\subsubsection{Down-sampling and common time array}
\label{sec:common_time_array}

The length of each hybrid waveform is set by choosing a starting orbital
frequency $\omega_{\mathrm{orb}}$ for the \texttt{SEOBNRv4HM}\xspace inspiral; we use
$\omega_{\mathrm{orb}} = 1 \times 10^{-3} \, \mathrm{rad}/M$ for all waveforms.
However, for the same starting frequency, the waveform length in time is
different for different mass ratios and spins. On the other hand, the
surrogate modeling procedure requires that all training waveforms have a common
time array~\cite{Field:2013cfa}. Therefore, we truncate all waveforms such that
they start at the same initial time ($\sim 2.4 \times 10^7 \,M$ before the
peak), which is determined by the shortest hybrid waveform in the training set.
Post truncation, the largest starting orbital frequency is
$\omega_{\mathrm{orb}} = 1.1 \times 10^{-3} \, \mathrm{rad}/M$, which sets the
low-frequency limit of validity of the surrogate. For LIGO and Virgo, assuming a starting
GW frequency of $20 \mathrm{Hz}$, the $(2,2)$ mode of the surrogate model is
valid for total masses $M \geq 3.7 \, M_{\odot}$. The highest spin-weighted
spherical harmonic mode included in the model is $(5,5)$, for which the
corresponding frequency is 5/2 times that of the $(2,2)$ mode. Therefore, all
modes of the surrogate are valid for $M\gtrsim 9.5 \, M_{\odot}$.

Because the hybrid waveforms are very long, it is not practical to sample the
entire waveform with a small uniform time step like $0.1\,M$, as is typically
done for NR-only surrogates~\cite{Varma:2019csw}. Fortunately, the early
low-frequency portion of the waveform does not require as dense a time sampling
as the later high-frequency portion. We therefore down-sample the time arrays
of the truncated hybrid waveforms to a common set of time samples. We choose
the time samples such that there are $5$ points per orbit for the
above-mentioned shortest hybrid waveform in the training set. However, for $t
\geq -1000\, M_{\odot}$ we switch to uniformly spaced time samples with a time
step of $0.1\,M$. This ensures that we have a sufficiently dense sampling rate
for the late inspiral and the merger-ringdown where the frequency reaches its
peak. We retain times up to $120 \, M$ after the peak, which is sufficient to
capture the entire ringdown.

Given the common down-sampled time array, we use cubic splines to interpolate
all waveforms in the training set to these times. However, we first transform
the waveforms into the co-orbital frame, defined as:
\begin{gather}
    \mathpzc{h}_{\ell m}^C = \mathpzc{h}_{\ell m} \, e^{im\phi_{\mathrm{orb}}} \label{eq:coorbital}, \\
    \mathpzc{h}_{22} = A_{22} \, e^{-i\phi_{22}},
    \label{eq:h22}\\
    \phi_{\mathrm{orb}} = \frac{\phi_{22}}{2},
    \label{eq:orb_phase}
\end{gather}
where $\mathpzc{h}_{\ell m}$ is the inertial frame waveform, $\phi_{\mathrm{orb}}$ is the
orbital phase, and $A_{22}$ and $\phi_{22}$ are the amplitude and phase of the
$(2,2)$ mode. The co-orbital frame can be seen as roughly co-rotating with the
binary, obtained by applying a time-dependent rotation about the $z-$axis, by
an amount given by the instantaneous orbital phase. Therefore, the waveform is
a slowly varying function of time in this frame, which increases the
interpolation accuracy. For the (2, 2) mode we save the downsampled amplitude
$A_{22}$ and phase $\phi_{22}$, while for all other modes we save the real and
imaginary parts of $\mathpzc{h}_{\ell m}^C$.

\subsubsection{Phase alignment}
\label{sec:phase_alignment}
Finally, we rotate the waveforms about the $z$-axis such that the orbital phase
$\phi_{\mathrm{orb}}$ is zero at $t=-1000\,M$. Note that this by itself would
fix the physical rotation up to a shift of $\pi$. When generating the EOB
inspiral waveform for hybridization, the frame is aligned such that heavier BH
is on the positive $x$-axis at the initial time, which fixes the $\pi$
ambiguity~\cite{Varma:2018mmi}. After the phase alignment, the heavier BH is
on the positive $x$-axis at $t=-1000\,M$ for all waveforms. However, keep in
mind that this frame is defined using the waveform at future null infinity, and
these BH positions do not necessarily correspond to the (gauge-dependent)
coordinate BH positions in the NR simulations.

\subsection{Data decomposition}
\label{sec:decomposing_the_data}

It is much easier to build a model for slowly varying functions of time.
Therefore, we decompose the inertial frame strain $\mathpzc{h}_{\ell m}$, which is oscillatory,
into simpler ``waveform data pieces'' and build a separate surrogate for each
data piece. When evaluating the full surrogate model, we first evaluate the
surrogate for each data piece and then combine the data pieces to get the
inertial frame strain. The $(2,2)$ mode is decomposed into its amplitude
$A_{22}$ and phase $\phi_{22}$ (which is further decomposed below). For the
other modes, we model the real and imaginary parts of the co-orbital frame
strain $\mathpzc{h}_{\ell m}^C$ (see Eq.~(\ref{eq:coorbital})).

Following Ref.~\cite{Varma:2019csw}, we further decompose $\phi_{22}$ by
subtracting the leading-order prediction from the TaylorT3 PN
approximant~\cite{Damour:2000zb}, given by:
\begin{equation}\label{eq:T3}
    \phi_{22}^{\mathrm{T3}} = \phi_{\mathrm{ref}}^{\mathrm{T3}} - \frac{2}{\eta \theta^5},
\end{equation}
where $\phi_{\mathrm{ref}}^{\mathrm{T3}}$ is an arbitrary integration constant,
$\theta = [ \eta (t_{\mathrm{ref}}-t)/(5M)]^{-1/8} $, $t_\mathrm{ref}$ is an
arbitrary time offset, and $\eta=q/(1+q)^2$ is the symmetric mass ratio.
Because $\phi_{\mathrm{ref}}^{\mathrm{T3}}$ diverges at $t_\mathrm{ref}$, we
choose $t_\mathrm{ref}=1000\,M$, long after the peak ($t=0$) of the waveform,
ensuring that we are always far away from this divergence. We choose
$\phi_{\mathrm{ref}}^{\mathrm{T3}}$ such that $\phi_{22}^{\mathrm{T3}} = 0$ at
$t=-1000\,M$, which is the same time at which we align the hybrid phase in
Sec.~\ref{sec:phase_alignment}.

By modeling the difference $\phi_{22}^\mathrm{res} = \phi_{22} -
\phi_{22}^{\mathrm{T3}}$ instead of $\phi_{22}$, we automatically capture
almost all of the phase evolution in the early inspiral of the long hybrid
waveforms.  Therefore, we simplify the problem of modeling the phase to the
same as modeling the phase of NR-only waveforms. This improves the overall
accuracy of the surrogate model for low masses, for which the inspiral
dominates. We stress that the exact form of $\phi_{22}^{\mathrm{T3}}$ (or its
physical meaning) is not important because we add the exact same
$\phi_{22}^{\mathrm{T3}}$ to our model of $\phi_{22}^\mathrm{res}$ when
evaluating the surrogate. In fact, even though TaylorT3 is known to be less
accurate than other approximants~\cite{Buonanno:2009zt, Varma:2013kna}, its
speed (being a simple, analytic, closed-form, function of time) makes it ideal
for our purpose.

To summarize, we decompose the hybrid waveforms into the following waveform
data pieces, each of which is a smooth, slowly varying function of time:
$(A_{22}, \phi_{22}^\mathrm{res})$ for the $(2,2)$ mode, and the real and
imaginary parts of $\mathpzc{h}_{\ell m}^C$ for the (2,1), (3,3), (4,4) and (5,5) modes.

\begin{figure*}[tbh]
\centering
\includegraphics[width=0.49\textwidth]{q15_ligo_mismatches.pdf}
\includegraphics[width=0.49\textwidth]{q8_ligo_mismatches.pdf}
\caption{
\emph{Left:} Mismatches as a function of the total mass M for \texttt{NRHybSur2dq15}\xspace,
\texttt{SEOBNRv4HM}\xspace and \texttt{IMRPhenomTHM}\xspace against NR-EOB hybrid waveforms with $q>8$. For \texttt{NRHybSur2dq15}\xspace,
we show leave-one-out errors. Mismatches are computed using the Advanced-LIGO
noise curve, at several points in the sky of the source frame using all
available modes: (2,2), (2,1), (3,3), (4,4), and (5,5).  The solid (dashed)
lines show the 95th percentile (median) mismatch values over points on the sky
as well as different hybrid waveforms.  \emph{Right:} Same, but now the
mismatches are computed against the \texttt{NRHybSur3dq8}\xspace model in the $q\leq8$ region.
}
\label{fig:ligo_mis}
\end{figure*}

\subsection{Surrogate construction and evaluation}
\label{sec:surrogate_building}
Given the waveform data pieces, we build a surrogate model for each data piece
using the same procedure as Sec.V.C of Ref.~\cite{Varma:2018mmi}, which we
summarize below.

For each waveform data piece, we first construct a linear basis using the
greedy basis method~\cite{Field:2011mf}, with tolerances of $10^{-2}$ radians
for the $\phi_{22}^{\mathrm{res}}$ data piece and $5 \times 10^{-5}$ for all
other data pieces. Next, we construct an empirical time
interpolant~\cite{Barrault:2004, Maday:2009, Hesthaven:2014} with the same
number of empirical time nodes as basis functions for that data piece.
Finally, for each empirical time node, we construct a parametric fit for the
waveform data piece, following the Gaussian process regression (GPR) fitting
method, as described in Refs.~\cite{Varma:2018aht, Taylor:2020bmj}. The fits
are parameterized by $(\log(q), \hat{\chi})$, where
\begin{gather}
\hat{\chi}  = \frac{\chi_{\mathrm{eff}}-38\eta(\chi_{1z}+\chi_{2z})/113}{1-76\eta/113},
\end{gather}
is the spin parameter entering the GW phase at leading
order~\cite{Ajith:2011ec}, and $\chi_{\mathrm{eff}} = \frac{q \chi_{1z} + \chi_{2z}}{1+q}$
is the effective spin. Note that in the above expressions $\chi_{2z}=0$ for the
current surrogate, but we adopt this parameterization to be consistent with
Ref.~\cite{Varma:2018mmi}. In practice, parameterizing the fits by $(\log(q),
\chi_{1z})$ also leads to a surrogate of similar accuracy.  On the other hand,
the $\log(q)$ parameterization leads to a significant improvement in model
accuracy, in agreement with Refs.~\cite{Varma:2018mmi, Rifat:2019ltp}.

When evaluating the surrogate waveform, we first evaluate each surrogate
waveform data piece. Next, we compute the $(2,2)$ mode phase:
\begin{equation}
    \phi_{22}^{\mathrm{S}} \equiv \phi_{22}^{\mathrm{res,S}} +
    \phi_{22}^{\mathrm{T3}},
\end{equation}
where $\phi_{22}^{\mathrm{res,S}} \approx \phi_{22}^{\mathrm{res}}$ is the
surrogate model for $\phi_{22}^{\mathrm{res}}$, and $ \phi_{22}^{\mathrm{T3}}$
is given by Eq.~(\ref{eq:T3}). If the waveform is required at a uniform
sampling rate, we interpolate each waveform data piece from the sparse time
samples to the required time samples using a cubic-spline interpolation scheme.
Finally, we use Eqs.~(\ref{eq:coorbital}),~(\ref{eq:h22}),
and~(\ref{eq:orb_phase}) to reconstruct the inertial frame strain.

\section{Surrogate errors}
\label{sec:surrogate_errors}

In this section, we evaluate the accuracy of \texttt{NRHybSur2dq15}\xspace by comparing against
NR-EOB hybrid waveforms. Similarly, we compute errors for two semi-analytic
waveform models, the phenomenological model \texttt{IMRPhenomTHM}\xspace~\cite{Estelles:2020twz} and
the EOB model \texttt{SEOBNRv4HM}\xspace~\cite{Cotesta:2018fcv}. Both of these models are
calibrated against nonprecessing NR simulations and include the same set of
modes as \texttt{NRHybSur2dq15}\xspace and the hybrid waveforms: (2,2), (2,1), (3,3), (4,4) and
(5,5).  Other semi-analytic nonprecessing models that include subdominant modes
exist in literature, including Refs~\cite{Nagar:2020pcj,
Garcia-Quiros:2020qpx}, but we do not consider these models for simplicity (as
they have accuracies comparable~\cite{Nagar:2020pcj, Garcia-Quiros:2020qpx,
Nagar:2022icd} to \texttt{IMRPhenomTHM}\xspace and \texttt{SEOBNRv4HM}\xspace).

In order to estimate the difference between two waveforms, $\mathpzc{h}_1$ and $\mathpzc{h}_2$,
we compute the mismatch (Eq.~\ref{eq:mismatch_td}) using the
noise-weighted inner product in frequency-domain, defined as
\begin{equation}
\braket{\mathpzc{h}_1,\mathpzc{h}_2} = 4 \Re \left[ \int_{f_{\mathrm{min}}}^{f_{\mathrm{max}}} \frac{\tilde{\mathpzc{h}}_1(f)\tilde{\mathpzc{h}}^*_2(f)}{S_n(f)} df \right],
\label{eq:mismatch_fd}
\end{equation}
where $\tilde{\mathpzc{h}}(f)$ indicates the Fourier transform of the complex strain
$\mathpzc{h}(t)$, $^*$ indicates a complex conjugation, $\Re$ indicates the real part,
and $S_n(f)$ is the one-sided power spectral density of a GW detector. We use
the Advanced-LIGO design sensitivity Zero-Detuned-HighP noise
curve~\cite{aLIGODesignNoiseCurve}, with $f_{\mathrm{min}} = 20$ Hz and
$f_{\mathrm{max}} = 2000$ Hz. We compute the mismatches following the
procedure described in Sec.VII of Ref.~\cite{Varma:2018mmi}: the mismatches are
optimized over shifts in time, polarization angle, and initial orbital phase.
Both plus and cross polarizations are treated on an equal footing by using a
two-detector setup where one detector sees only the plus and the other only the
cross polarization.  We use all the available modes of a given waveform model,
and compute the mismatches at 37 points uniformly distributed on the sky in the
source frame.

Figure~\ref{fig:ligo_mis} shows mismatches computed using the Advanced-LIGO
noise curve for \texttt{NRHybSur2dq15}\xspace, \texttt{SEOBNRv4HM}\xspace and \texttt{IMRPhenomTHM}\xspace against hybrid waveforms. As
these depend on the total mass, we show mismatches for various masses, starting
near the lower limit of the range of validity of the surrogate $M\gtrsim 9.5 \,
M_{\odot}$. At each mass, we show the median and 95th percentile mismatches,
over many hybrid waveforms and points in the source frame sky.

The left panel of Fig.~\ref{fig:ligo_mis} shows mismatches against the 20 $q>8$
NR-EOB hybrid waveforms in Fig.~\ref{fig:parameter_distribution}. As these
hybrid waveforms were also used in the training of \texttt{NRHybSur2dq15}\xspace, we conduct a
\emph{leave-one-out} analysis: we generate 20 trial surrogates, leaving out one
of the $q>8$ hybrid waveforms from the training set in each trial, but
including the rest of the training cases (both $q>8$ and $q\leq8$) in
Fig.~\ref{fig:parameter_distribution}. For each trial surrogate, we compute
errors against the $q>8$ hybrid waveform that was left out. In this manner, we
only compare \texttt{NRHybSur2dq15}\xspace against waveforms not used in the model training.
Therefore, these errors are indicative of the true modeling error.

For the $q>8$ region, 95th percentile mismatches for \texttt{NRHybSur2dq15}\xspace fall below $\sim
2\times10^{-3}$ over the entire mass range in Fig.~\ref{fig:ligo_mis}. The
errors for \texttt{IMRPhenomTHM}\xspace and \texttt{SEOBNRv4HM}\xspace are generally larger by at least an order of
magnitude.  However, for \texttt{SEOBNRv4HM}\xspace, the errors at low masses overlap with the
surrogate errors. This is most likely because \texttt{SEOBNRv4HM}\xspace was used to generate the
early inspiral waveform for the NR-EOB hybrid waveforms. At low masses, where
the early inspiral dominates the overall error budget, these errors are
therefore not representative of the true error in \texttt{SEOBNRv4HM}\xspace.

The right panel of Fig.~\ref{fig:ligo_mis} shows mismatches in the $q<8$
region. In this region, rather than conduct leave-one-out tests, we simply
generate 100 new hybrid waveforms using the \texttt{NRHybSur3dq8}\xspace model for testing.  These
test cases are uniformly distributed in the region $q\in[1,8]$ and
$\chi_{1z}\in[-0.5, 0.5]$, with $\chi_{2z}=0$. Once again \texttt{NRHybSur2dq15}\xspace has
mismatches that are at least an order of magnitude smaller than that of \texttt{SEOBNRv4HM}\xspace
and \texttt{IMRPhenomTHM}\xspace. In this case, \texttt{SEOBNRv4HM}\xspace errors are broadly uniform across all masses.
This is most likely explained by the fact that the early inspiral of \texttt{NRHybSur3dq8}\xspace
was based on PN as well as EOB waveforms; more precisely, PN was directly used
to generate the mode amplitudes while the (2,2) mode of \texttt{SEOBNRv4HM}\xspace (the
\texttt{SEOBNRv4}\xspace~\cite{Bohe:2016gbl} model) was used to correct the PN mode phases.

\begin{figure*}[h]
\centering
\includegraphics[width=0.45\textwidth]{error_histogram.pdf}
\includegraphics[width=0.45\textwidth]{error_histogram_q8.pdf}
\caption{
\emph{Left:} Normalized error, $\mathcal{E}$ (Eq.~(\ref{eq:L2_error})),
computed for \texttt{NRHybSur2dq15}\xspace, \texttt{SEOBNRv4HM}\xspace, and \texttt{IMRPhenomTHM}\xspace against NR-EOB hybrid waveforms
with q>8, but restricting the start time of the waveforms to $-4500M$ before
the peak amplitude. In the first row, $\mathcal{E}$ is computed using all
available modes, and in the subsequent rows, single-mode errors are computed by
restricting Eq.~(\ref{eq:L2_error}) to individual modes. \emph{Right:} Same,
but now the error is computed against the \texttt{NRHybSur3dq8}\xspace model in the $q\leq8$
region.
}
\label{fig:error_histgoram}
\end{figure*}

\begin{figure*}[tbh]
\centering
\includegraphics[width=\textwidth]{WorstCasePlot_h_l2m2.pdf}
\caption{
The (2,2) modes of the three waveform models compared against NR, for the cases
that lead to the largest (2,2) mode error in the left panel of
Fig.~\ref{fig:error_histgoram}. The top (middle) [bottom] panel shows the case
for which \texttt{NRHybSur2dq15}\xspace (\texttt{SEOBNRv4HM}\xspace) [\texttt{IMRPhenomTHM}\xspace] has the largest (2,2) mode error.
}
\label{fig:WorstCasePlot_22}
\end{figure*}

While Fig.~\ref{fig:ligo_mis} shows model errors when including all available
modes, it can be useful to also understand the errors in the individual modes.
We quantify this using the normalized $L_2-$norm between two waveforms $\mathpzc{h}$ and
$\mathpzc{h}'$:
\begin{align}
    \mathcal{E}(\mathpzc{h}, \mathpzc{h}') = \frac{1}{2}\frac{\sum_{l,m}\int^{t_2}_{t_1}\vert \mathpzc{h}_{\ell m}(t) - \mathpzc{h}_{\ell m}^{'}(t) \vert^2 dt}{\sum_{l,m}\int^{t_2}_{t_1}\vert \mathpzc{h}_{\ell m}(t)\vert^2 dt}.
\label{eq:L2_error}
\end{align}
This error measure was introduced in Ref.~\cite{Blackman:2017dfb} and is
related to weighted average of the mismatch over the sky in the source frame.
When computing $\mathcal{E}$, we only consider the late inspiral and
merger-ringdown region by choosing $t_1=-4500M$ and $t_2=115M$. As the NR
waveforms used in generating the hybrid waveforms had typical start times $\sim
-5000M$ (see Sec.~\ref{sec:training_set}), this ensures that $\mathcal{E}$ is
independent of which model was used in the hybridization procedure.
Furthermore, rather than optimizing over time or phase shifts, we simply align
the frames of the two waveforms such that the peak amplitude
(Eq.~(\ref{eq:amp_tot})) occurs at $t=0$, and the orbital phase
(Eq.~(\ref{eq:orb_phase})) is zero at $t = -4500M$.  This makes $\mathcal{E}$
much cheaper to evaluate than the mismatches in Eq.~(\ref{eq:mismatch_fd}).  In
addition to computing normalized errors using all available modes, we also
consider single-mode errors by restricting the sums in Eq.~(\ref{eq:L2_error})
to individual modes.

Figure~\ref{fig:error_histgoram} shows normalized errors for \texttt{NRHybSur2dq15}\xspace, \texttt{SEOBNRv4HM}\xspace
and \texttt{IMRPhenomTHM}\xspace against hybrid waveforms. The left panel of
Fig.~\ref{fig:error_histgoram} follows the left panel of
Fig.~\ref{fig:ligo_mis}, and shows errors for the three waveform models (using
a leave-one-out analysis for \texttt{NRHybSur2dq15}\xspace) against the 20 $q>8$ NR-EOB hybrid
waveforms. The right panel of Fig.~\ref{fig:error_histgoram} follows the right
panel of Fig.~\ref{fig:ligo_mis}, and shows errors against the same 100
uniformly distributed \texttt{NRHybSur3dq8}\xspace waveforms in the region $q\in[1,8]$ and
$\chi_{1z}\in[-0.5, 0.5]$, with $\chi_{2z}=0$. For both $q>8$ and $q\leq8$, we
once again find that \texttt{NRHybSur2dq15}\xspace is more accurate than the other models by at
least an order of magnitude, both for the full waveform and for the individual
modes.

Considering the individual mode errors in Fig.~\ref{fig:error_histgoram}, we
note that the fractional errors in the nonquadrupole modes of \texttt{SEOBNRv4HM}\xspace and
\texttt{IMRPhenomTHM}\xspace reach large values. In particular, the errors in the (5,5) mode for
\texttt{SEOBNRv4HM}\xspace for $q>8$ can reach values $\mathcal{E} \sim 1$. While the nonquadrupole
modes are still subdominant for $q\gtrsim10$ binaries like GW190814 (which is
why the full waveform errors do not reach such large values in
Fig.~\ref{fig:error_histgoram}), it may be important for models like \texttt{IMRPhenomTHM}\xspace
and \texttt{SEOBNRv4HM}\xspace to improve accuracy in these modes for future observations. Finally,
to illustrate the (in)accuracy of the individual modes,
Figs.~\ref{fig:WorstCasePlot_22}, \ref{fig:WorstCasePlot_21_33} and
\ref{fig:WorstCasePlot_44_55} show the cases leading to the largest individual
mode errors in the left panel of Fig.~\ref{fig:error_histgoram}.

\subsection{Extrapolating outside the training region}
The errors computed so far were restricted to the training region of
\texttt{NRHybSur2dq15}\xspace: $q\leq 15$, $\chi_{1z} \in [-0.5, 0.5]$, and $\chi_{2z}=0$.  It is
possible to extrapolate the model to larger $q$ and $|\chi_{1z}|$, but it is
difficult to assess the model accuracy in this region due to a lack of NR
simulations. Instead, through a visual inspection of the evaluated waveforms,
we find that extrapolating beyond $q=20$ or $|\chi_{1z}|=0.7$ leads to
unphysical ``glitches'' in the time series for the mode amplitudes and the
derivatives of the mode phases. Therefore, while we allow the model to be
evaluated in the region $q\leq20$, $\chi_{1z} \in [-0.7, 0.7]$, and
$\chi_{2z}=0$, we advise caution when extrapolating the model.

\begin{figure*}[p]
\centering
\includegraphics[width=\textwidth]{WorstCasePlot_h_l2m1.pdf}

\vspace{0.6cm}

\includegraphics[width=\textwidth]{WorstCasePlot_h_l3m3.pdf}
\caption{
Same as Fig.~\ref{fig:WorstCasePlot_22}, but now showing the worst cases for
the (2,1) [top] and (3,3) [bottom] modes.
}
\label{fig:WorstCasePlot_21_33}
\end{figure*}

\begin{figure*}[p]
\centering
\includegraphics[width=\textwidth]{WorstCasePlot_h_l4m4.pdf}

\vspace{0.6cm}

\includegraphics[width=\textwidth]{WorstCasePlot_h_l5m5.pdf}
\caption{
Same as Fig.~\ref{fig:WorstCasePlot_22}, but now showing the worst cases for
the (4,4) [top] and (5,5) [bottom] modes.
}
\label{fig:WorstCasePlot_44_55}
\end{figure*}

\begin{figure*}[htb]
\centering
\includegraphics[width=0.49\textwidth]{cornerplot_mass.pdf}
\includegraphics[width=0.49\textwidth]{cornerplot_chirp.pdf}
\centering
\includegraphics[width=0.49\textwidth]{cornerplot_distance.pdf}
\caption{
Constraints on GW190814 parameters obtained using the \texttt{NRHybSur2dq15}\xspace, \texttt{IMRPhenomTHM}\xspace and
\texttt{IMRPhenomPv3PHM}\xspace models. We show posterior distributions for the source-frame
component masses $\mSrc{1}$ and $\mSrc{2}$ (top-left), the effective spin
$\chi_{\mathrm{eff}}$ and the source-frame chirp mass $\mathcal{M}^{\mathrm{src}}$ (top-right), and the
extrinsic parameters $cos(\theta_{JN})$ and luminosity distance $D_L$ (bottom).
The solid (dashed) contours represent the central 50\% (90\%) credible regions
of the joint posteriors. Marginalized 1D posteriors are shown on the plot
edges. In the top-left panel, we include lines of constant mass ratios
($q=7,9,11,13$) for comparison. The bimodality in the bottom panel is due to a
well known degeneracy between distance and inclination~\cite{Cutler:1994ys}.
\texttt{IMRPhenomTHM}\xspace and \texttt{NRHybSur2dq15}\xspace show good agreement, suggesting that \texttt{IMRPhenomTHM}\xspace is
accurate enough for GW190814-like events at current SNRs. The constraints on
the component masses and $\chi_{\mathrm{eff}}$ improve for \texttt{IMRPhenomTPHM}\xspace compared to the
nonprecessing models, suggesting that precession should be included in
\texttt{NRHybSur2dq15}\xspace.
}
\label{fig:cornerplots}
\end{figure*}

\section{Reanalyzing GW190814}
\label{sec:reanalyzing_GW190814}

\texttt{NRHybSur2dq15}\xspace is targeted towards GW events like GW190814~\cite{Abbott:2020khf},
with mass ratios $q\gtrsim9$. As \texttt{NRHybSur2dq15}\xspace is more accurate than alternative
models in this region, we now reanalyze GW190814 with \texttt{NRHybSur2dq15}\xspace. In addition,
we consider two phenomenological models, \texttt{IMRPhenomTHM}\xspace~\cite{Estelles:2020twz} and
\texttt{IMRPhenomTPHM}\xspace~\cite{Estelles:2021gvs}. Both of these models include the effects of
subdominant modes, but only \texttt{IMRPhenomTPHM}\xspace includes precession effects. Precession
effects are included in \texttt{IMRPhenomTPHM}\xspace by ``twisting'' the frame of the nonprecessing
model \texttt{IMRPhenomTHM}\xspace to mimic orbital precession~\cite{Estelles:2021gvs}. The GW190814
discovery paper~\cite{Abbott:2020khf} instead considered the
\texttt{SEOBNRv4PHM}\xspace~\cite{Ossokine:2020kjp} and \texttt{IMRPhenomPv3PHM}\xspace~\cite{Khan:2019kot} binary BH
models, both of which include the effects of subdominant modes and precession
(through a similar twisting procedure). For simplicity, we do not consider
these models here, but we have verified that our results with \texttt{IMRPhenomTPHM}\xspace are
consistent with Ref.~\cite{Abbott:2020khf}. Ref.~\cite{Abbott:2020khf} also
considered models~\cite{Matas:2020wab, Thompson:2020nei} with tidal effects,
but found no measurable tidal signatures; therefore, we only show results for
binary BH models.

Source properties can be inferred from GW data following Bayes' theorem (see
e.g.~Ref.~\cite{Thrane:2019pe} for a review). We analyze the GW190814 data made
public by the LIGO-Virgo-Kagra Collaboration~\cite{Abbott:2020khf,
GW_open_science_center}, using the \texttt{Parallel Bilby}~\cite{Smith:2019ucc}
parameter estimation package with the \texttt{dynesty}~\cite{dynesty_paper}
sampler. Following Ref.~\cite{LIGOScientific:2021djp}, we choose a prior that
is uniform in detector frame component masses, and isotropic in sky location
and binary orientation. For the distance prior, we use the
\texttt{UniformSourceFrame} prior~\cite{Romero-Shaw:2020owr} assuming a
cosmology from~\cite{Planck:2015fie} as implemented in
\texttt{Astropy}~\cite{Astropy:2013muo, Price-Whelan:2018hus}.

When using the nonprecessing models \texttt{NRHybSur2dq15}\xspace and \texttt{IMRPhenomTHM}\xspace, we use the
\texttt{AlignedSpin} prior~\cite{Romero-Shaw:2020owr, Callister:2021gxf}, with
$-0.5 \leq \chi_{1z}\leq 0.5$ and $\chi_{2z}=0$.  The \texttt{AlignedSpin}
prior follows the generic-spin assumptions of a prior that is uniform in
magnitude and isotropic in orientation for each of the two spin vectors, which
in the nonprecessing case is projected onto the orbital angular momentum. Even
though \texttt{IMRPhenomTHM}\xspace allows generic aligned-spins on both BHs, we restrict the model
to the same spin range as \texttt{NRHybSur2dq15}\xspace for easy comparison.
We have, however,
verified that using unrestricted aligned-spins for \texttt{IMRPhenomTHM}\xspace has a negligible
impact on GW190814 posteriors; this is expected as Ref.~\cite{Abbott:2020khf}
placed a constraint of $\chi_1\lesssim 0.07$ at 90\% credibility, and found
that $\chi_2$ cannot be constrained for GW190814.
When using the precessing
model \texttt{IMRPhenomTPHM}\xspace, our prior is uniform in spin magnitudes (with $0 \leq
\chi_{1},\chi_{2}\leq 1$) and isotropic in spin orientations for both BHs. The
reason for considering a precessing model with no spin restrictions is to gauge
the impact of neglecting precession in \texttt{NRHybSur2dq15}\xspace.

Figure~\ref{fig:cornerplots} shows posterior distributions for the GW190814
source parameters obtained using \texttt{NRHybSur2dq15}\xspace, \texttt{IMRPhenomTHM}\xspace and \texttt{IMRPhenomTPHM}\xspace. We show
constraints on the source-frame component masses $\mSrc{1}$ and $\mSrc{2}$, the
effective spin $\chi_{\mathrm{eff}}$, the source-frame chirp mass $\mathcal{M}^{\mathrm{src}} = M^{\mathrm{src}} \,
\eta^{3/5}$, the luminosity distance $D_L$, and cosine of the inclination angle
$\theta_{JN}$ between the total angular momentum $\bm{J}$ and the line of sight
direction $\hat{\bm{N}}$. As \texttt{NRHybSur2dq15}\xspace is significantly more accurate (see
Fig.~\ref{fig:ligo_mis}), the differences between \texttt{NRHybSur2dq15}\xspace and \texttt{IMRPhenomTHM}\xspace can be
used to gauge systematic uncertainties in \texttt{IMRPhenomTHM}\xspace. In
Fig.~\ref{fig:cornerplots} we find good agreement between \texttt{NRHybSur2dq15}\xspace and
\texttt{IMRPhenomTHM}\xspace for all parameters shown, which suggests that semi-analytical models
like \texttt{IMRPhenomTHM}\xspace are accurate enough for events like GW190814. However, this may
not be the case as detector sensitivity improves and GW190814-like signals are
observed at larger SNRs. At larger SNRs, the differences noted in
Figs.~\ref{fig:ligo_mis} and \ref{fig:error_histgoram} can become significant.

Finally, comparing the posteriors for \texttt{IMRPhenomTHM}\xspace and \texttt{IMRPhenomTPHM}\xspace in
Fig.~\ref{fig:cornerplots}, we find that including the effects of precession
leads to stronger constraints on the component masses and $\chi_{\mathrm{eff}}$, while the
chirp mass, distance and inclination constraints are not significantly
affected. This is in agreement with Ref.~\cite{Abbott:2020khf}, and implies
that precession effects should be included in \texttt{NRHybSur2dq15}\xspace. While this can be done
by a frame twisting procedure similar to \texttt{IMRPhenomTPHM}\xspace, this method does not capture
the full effects of precession like the asymmetries between pairs of $(\ell,
m)$ and $(\ell, -m)$ spin-weighted spherical harmonic
modes~\cite{Varma:2019csw, Varma:2021csh}. While precessing NR surrogate
models~\cite{Varma:2019csw} capture these effects, they require $\gtrsim 1000$
NR simulations, which are not currently possible at large mass ratios.
Therefore, we leave this exploration to future work.

\section{Conclusion}
\label{sec:conc}
We present \texttt{NRHybSur2dq15}\xspace, a surrogate waveform model targeted at large mass ratio
GW events like GW190814. The model is trained on 51 binary BH hybrid waveforms
with mass ratios $q\leq 15$ and aligned spins $ \chi_{1z} \in [-0.5, 0.5]$,
$\chi_{2z}=0$, includes the (2,2), (2,1), (3,3), (4,4), and (5,5) spin-weighted
spherical harmonic modes, and spans the entire LIGO-Virgo bandwidth (with $f_{\mathrm{low}}=20$
Hz) for total masses $M \gtrsim 9.5 \, M_{\odot}$.  Through a leave-one-out
study, we show that \texttt{NRHybSur2dq15}\xspace accurately reproduces the hybrid waveforms, with
mismatches below $\sim 2 \times 10^{-3}$ for total masses $10 \, M_{\odot} \leq
M \leq 300 \, M_{\odot}$. This is at least an order-of-magnitude improvement
over existing semi-analytical models.  The model is made publicly available
through the easy-to-use Python package \textit{gwsurrogate}~\cite{gwsurrogate}.

We reanalyze GW190814 using \texttt{NRHybSur2dq15}\xspace and find results consistent with the
discovery paper Ref.~\cite{Abbott:2020khf}. This suggests that current
semi-analytical models are accurate enough for events like GW190814. However,
as detector sensitivity improves, we can expect to see similar signals at a
higher SNR. We anticipate that accurate models like \texttt{NRHybSur2dq15}\xspace will be necessary
for analyzing such signals. With that goal, we identify precession as an
important feature to be added to \texttt{NRHybSur2dq15}\xspace in the future.

\begin{acknowledgments}
We thank Hector Estelles and Alessandro Nagar for comments on the manuscript.
This work was supported in part by the Sherman Fairchild Foundation and by
National Science Foundation (NSF) Grant Nos. PHY-2011961, PHY-2011968, and
OAC-1931266 at Caltech, and NSF Grant Nos. PHY-1912081 and OAC-1931280 at
Cornell.
V.V. acknowledges funding from the European Union's Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No.~896869.
V.V. was supported by a Klarman Fellowship at Cornell.
C.-J.H. acknowledges support of the NSF and the LIGO Laboratory.
NR simulations were conducted on the Frontera computing project at the Texas
Advanced Computing Center.  Additional computations were performed on the
Wheeler cluster at Caltech, which is supported by the Sherman Fairchild
Foundation and by Caltech; and the High Performance Cluster at Caltech.
This material is based upon work supported by NSF's LIGO Laboratory which is a
major facility fully funded by the NSF.
This research made use of data, software and/or web tools obtained from the
Gravitational Wave Open Science Center~\cite{GW_open_science_center}, a service
of the LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo
Collaboration.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


The Sidney Myer Fund and the Myer Foundation are private philanthropic organizations committed to support of programs for the arts and humanities, education, alleviating poverty and supporting the disadvantaged and the environment and sustainability. Australian Sidney Myer, born Jewish under Imperial Russian rule in Belarus, relocated to Melbourne to escape persecution, and there made his fortune, becoming a philanthropist deeply involved in the wellbeing of his community. A committed advocate of the arts, he founded the Sidney Myer Foundation in 1934 as a means to continue to contribute to the advancement of arts and culture in Australia. His sons in 1952 founded the Myer Fund to further contribute to collaborative international projects, science and technology and the environment. The philanthropies are managed by a single team which in 2016 donated a total of 13,574,404 AUD. Largest by category of the recipients fall under the arts and humanities, and in 2015 this category received 34% of all donations. Included below is a list of the chief grants offered by the Sidney Myer Foundation.
While artists are the driving creative force behind the art community, their pay is too often insufficient to amount to a livelihood. The Sidney Myer Foundation views this as a problem and seeks to remedy it. The Fellowship's aim is to support artists through the critical stages of their careers in order to help them make their way as professionals. Intended for artists, writers, etc., the Fellowship is not contingent on any results and is unrestricted in how the grants are spent. Prospective recipients are between seven and 15 years into their careers, with outstanding talent and exceptional courage; Australian or permanent residents of Australia; and can have a range of backgrounds including artists, art managers and thought leaders in the humanities. There are approximately eight recipients each year receiving the grant, which totals 160,000 AUD (tax-free) over a two-year period. Recipients, to date, include artist Vernon Ah Kee (2014), writer/playwright Courtney Collins (2015), and Alex Kelly (2015), who takes on various roles in community art including production and direction.
This grant program open to applications from the general public, selects projects in the fields of education, alleviating poverty and aiding the disadvantaged and the environment in addition to the arts and humanities. In the arts and humanities, grants are given to artist groups and ensembles that create outstanding work through networking with other organizations, with the ultimate aim of strengthening extant independent artistic and cultural groups in Australia, and helping to support emergent ones, focusing chiefly on small- to mid-sized organizations. Applications are accepted from specific regions which change yearly: organizations based in the states of Victoria and Tasmania were the target for 2015, and South Australia for 2016. Funding lies between 20,000 and 50,000 AUD per organization, split into payments over two years. The grant has supported such organizations as the Polyglot Theatre, which gives interactive and participation-type performances led by children, and All the Queens Men, known for their spectacular large-scale performances in conjunction with communities from diverse cultural backgrounds.
This arts award created in memory of Sidney Myers, began in the 50th year since his passing, 1984. The Award's aims are to recognize achievements in dance, drama, comedy, music, opera, circus and puppetry, and to contribute to the advancement of the performing arts in Australia, giving individual (60,000 AUD), group (90,000 AUD) and facilitator (25,000 AUD) prizes. Past recipients include the circus act Circus Oz (Group Award, 1984), singer/actress Ursla Yovich (Individual Award) and musician and educator Mandawuy Yunupingu (Facilitator's Prize, 2009).
The Award's other activities includes an annual contribution of 2 million AUD to the Performing Arts Triennale, which the Sidney Myer foundation began as the first of such events in the Asia-pacific region in conjunction with Arts Centre Melbourne in 2015-16.


I'm Natasha and I've been with Reflect since November 2013. I was studying Drama and Creative Writing at university at the time and have since then graduated. I've taken part in a number of projects with Reflect from a TIE workshop on the BFG to a Drama club performance of Mary Poppins. I also do the Reflect Choi Kwang Do admin as well as being a student. I thoroughly enjoy working at Reflect and look forward to taking part in more projects in the future.


Anyway, back to our scene: the Seattle sun peeked through the raindrops and the pop-flecked jazz filled the air, which became increasingly thick with the scent of melted butter and dark chocolate and, despite the baffling circus that is the political news cycle (you guys, WHAT IS HAPPENING), in that moment, all felt right with the world. And then we ate brownies.
As today is a holiday (albeit one of those weird ones that not everyone recognizes — sorry if you are, indeed, at the office), you might think about creating your own happy place, with music and brownies? It's a start, at least.
Preheat oven to 350 degrees F. Grease a 9-inch square baking pan with butter or cooking spray, line it with parchment (leaving some overhang on 2 sides) and grease the parchment, too.
In a medium saucepan, melt the butter and coconut oil over low heat. Once melted, add the chocolate and stir occasionally until melted and smooth. Remove the pan from the heat.
Add the sugars to the melted butter mixture and stir to combine (the mixture will be a bit grainy). Add the eggs and vanilla, stirring vigorously to combine. Fold in the dry ingredients until no streaks remain in the batter.
Pour the batter evenly into the prepared pan and smooth the top with a spatula.
Bake until the top looks crackly near the edges and a tester comes out with a few moist crumbs, about 30 minutes. Let the brownies cool in the pan for about 20 minutes, then use the parchment overhang to pull the brownies out of the pan and cool completely on a rack.
Once cool, cut into squares and serve!
Brownies keep, well-wrapped, for about a week at room temperature.
Makes about 16 brownies (depending on how you cut them).
YES… these look like the very best of the brownie world. Can't wait to make!
Will you share what pan you use? Because I like to copy you.
What?! Really? Ok, thank you! I've never heard of this! I'll go find one!
I LOVED these! Probably underbaked them a little, but they were delicious cold.


The author has traveled extensively to study esoteric philosophy, the magical arts, and alternative religion. He's the editor and a contributing author to the e-zing 'The Familiar'. This is his first book.
The book provides a program of study and activities aimed at improving the reader's mental control and magical power through layered techniques. The developmental process utilizes an allegorical beast that's part unicorn, horse, and lion that the author calls Dragonflame.
Chapters 1 through 3 provide an explanations of intent, magical rituals, and visualizations that highlight the alchemical process of transformation. These are supplemented with information about focal deities, astrological signs, the related Major Arcana cards in the tarot (based on the Golden Dawn attributions), and its Hebrew letter assignment. Each exercise takes several days and progress is supposed to be tracked in the reader's magical diary. Ritual supplies like candles, oils, and incenses are recommended.
Chapter Four progresses into talisman-making. Instructions include detailed illustrations and a ritual for making one or two talismans that will be used in further workings. Chapter Five "The Mechanics of Your Interior Spaces" gives instructions for casting a sacred circle, setting up an altar, charging candles, and consecrating tools used in ritual work. The characteristics of the Sephiroth in the Kabbalistic Tree of Life are described.
Chapter 6 is "Rituals of Transformation." Each ritual has a goal and a purpose: Come Back Done (to recover from tragedy); Moon Rite (to open the mind's eye); SOS Ritual (for help in a desperate situation); Spiral Rite (for magical progress and development); Vesta Ritual (to manifest continued success); Mars Ritual (tap into your reservoirs of power); Lakshmi-Ganesha Ritual (for monetary growth). Each ritual includes a list of supplies, procedures that include petitions and will-power building activities, a group of sacred names to chant. Rituals are performed over a prescribed number of days. Some suggestions for the timing of rituals is indicated; this generally coincides with particular moon phases and/or days of the week. Planetary hours are mentioned but not explained.
The book concludes with supplemental sections: a glossary of terms, a Willpower List of ideas, References and books for recommended reading, and an index.
This is a decent guide for someone who is getting into magic and paganism on a solitary basis and needs instructions on working complicated spells and rituals that require an intense mental focus over several days. The book emphasizes purpose and intent, necessary ingredients in building magical powers. Rituals offer extra options for "advanced practitioners."
There are pluses and minuses to the extremely diverse selection of deities covered by the rituals. Some may find the mix of practical Qabalah, Hebrew names of God, Golden Dawn rites, deities of several pantheons, and practices of other religions mixed in with Wiccan ritualistic procedures a bit jumbled. The extensive supply list includes essential oils, incenses, and plant substances that can be expensive or difficult to obtain. The fragrances corresponded to deities/purposes appear to reflect the author's eclectic preferences.
The book is carefully written and well organized, but it pushes the boundaries of outlining a complex, multi-layered magical practice in a limited amount of space. It's recommended for experienced beginner/intermediate solitary practitioners with a relaxed, eclectic view of tapping into various pantheons. It's suitable for readers who want to learn techniques for developing magical power and are ready to perform more complicated rituals and spells.


text:
using UnityEngine;
using UnityEngine.UI;
using System.Collections.Generic;
using System.Linq;
using System;

// This is the Hierarchy Panel, the panel that shows the list of actors
// that are on stage and off stage and lets the player select actors.
// (NOT TO BE CONFUSED with the Actor Picker Dialog!)
public class HierarchyPanelController : MonoBehaviour
{
  const int COLLAPSED_WIDTH = 35;
  const string ShowCopiesPrefKey = "SHOW_COPIES";
  const string SortByDistancePrefKey = "SORT_BY_DISTANCE";

  // Container of all the panels. This is what gets activated/deactivated to
  // hide/show everything when we enter/leave edit mode.
  [SerializeField] RectTransform container;
  // The panel to show when we're in expanded state.
  [SerializeField] RectTransform expandedPanel;
  // The widget that shows the list of actors on stage.
  [SerializeField] HierarchyActorList onStageList;
  // The widget that shows the list of actors off stage.
  [SerializeField] HierarchyActorList offStageList;
  // Checkbox for "show copies".
  [SerializeField] Toggle showCopiesToggle;
  [SerializeField] TMPro.TMP_InputField searchInput;
  [SerializeField] Button clearSearchButton;
  // Sort by button.
  [SerializeField] Button sortByButton;
  // Close button.
  [SerializeField] Button closeButton;
  [SerializeField] ActorListItemUI draggingGhost;

  // Are we in expanded mode right now?
  bool expanded;
  bool monitoringActors;

  // References to other modules/services:
  UserMain userMain;
  EditMain editMain;
  ToolMemory toolMemory;
  VoosEngine engine;
  HudManager hudManager;
  UndoStack undoStack;

  // If true, we need to refresh the list.
  bool listNeedsRefresh = false;

  // If true, sort by distance; else, by name.
  bool sortByDistance = false;

  List<VoosActor> selectedActors = new List<VoosActor>();
  // VoosActor selectedActor;

  float timeSinceLastRefresh;

  private System.Action<VoosActor> selectCallback;

  private VoosActor draggingActor;

  void Awake()
  {
    Util.FindIfNotSet(this, ref userMain);
    Util.FindIfNotSet(this, ref editMain);
    Util.FindIfNotSet(this, ref toolMemory);
    Util.FindIfNotSet(this, ref engine);
    Util.FindIfNotSet(this, ref hudManager);
    Util.FindIfNotSet(this, ref undoStack);
    SetExpanded(false);
    editMain.targetActorsChanged += TargetActorsUpdatedExternally;
    onStageList.AddClickListener(OnActorEntryClicked);
    onStageList.onActorDrag += OnActorDrag;
    offStageList.AddClickListener(OnActorEntryClicked);
    offStageList.onActorDrag += OnActorDrag;
    sortByDistance = PlayerPrefs.GetInt(SortByDistancePrefKey, 0) != 0;
    sortByButton.GetComponentInChildren<TMPro.TMP_Text>().text = sortByDistance ? "Sort: distance" : "Sort: name";
    sortByButton.onClick.AddListener(OnSortButtonClicked);
    showCopiesToggle.isOn = PlayerPrefs.GetInt(ShowCopiesPrefKey, 1) != 0;
    showCopiesToggle.onValueChanged.AddListener(v =>
    {
      PlayerPrefs.SetInt(ShowCopiesPrefKey, v ? 1 : 0);
      RefreshActorList();
    });
    searchInput.onValueChanged.AddListener(s =>
    {
      RefreshActorList();
      clearSearchButton.gameObject.SetActive(!s.IsNullOrEmpty());
    });
    clearSearchButton.onClick.AddListener(() =>
    {
      searchInput.text = "";
    });
    closeButton.onClick.AddListener(OnCloseButtonClicked);
  }
  public bool IsExpanded()
  {
    return expanded;
  }

  public void SetExpanded(bool expanded)
  {
    this.expanded = expanded;
    if (!expanded)
    {
      SetDraggingActor(null);
    }
    UpdateUi();
  }

  void UpdateUi()
  {
    hudManager.UpdateHorizontalRightOffset(expanded ? expandedPanel.rect.width : 0);
    if (expanded)
    {
      // Start monitoring for actor changes that are interesting to us.
      RefreshActorList();
      StartMonitoringActors();
    }
    else
    {
      // Stop monitoring for actor changes.
      StopMonitoringActors();
    }
    expandedPanel.gameObject.SetActive(expanded);
  }

  private void StartMonitoringActors()
  {
    if (monitoringActors) return;
    monitoringActors = true;
    engine.onActorCreated += OnActorCreated;
    engine.onBeforeActorDestroy += OnActorDestroyed;
    foreach (VoosActor actor in engine.EnumerateActors())
    {
      StartMonitoringActor(actor);
    }
  }

  private void StartMonitoringActor(VoosActor actor)
  {
    actor.AddDisplayNameChangedListener(OnSomeActorChanged);
    actor.offstageChanged += OnSomeActorChanged;
  }

  private void StopMonitoringActors()
  {
    if (!monitoringActors) return;
    engine.onActorCreated -= OnActorCreated;
    engine.onBeforeActorDestroy -= OnActorDestroyed;
    foreach (VoosActor actor in engine.EnumerateActors())
    {
      StopMonitoringActor(actor);
    }
    monitoringActors = false;
  }

  private void StopMonitoringActor(VoosActor actor)
  {
    actor.RemoveDisplayNameChangedListener(OnSomeActorChanged);
    actor.offstageChanged -= OnSomeActorChanged;
  }

  /*  bool ActorListsAreDifferent()
   {
     return true;
     ////int deltaA = selectedActors.Except(editMain.GetTargetActors()).Count();
     //int deltaB = editMain.GetTargetActors().Except(selectedActors).Count();
     //return deltaA + deltaB != 0;
   } */

  void TargetActorsUpdatedExternally(IEnumerable<VoosActor> actors)
  {
    selectedActors.Clear();
    selectedActors.AddRange(actors);
    listNeedsRefresh = true;
  }

  void Update()
  {
    timeSinceLastRefresh += Time.deltaTime;

    if (!expanded) return;

    if (listNeedsRefresh)
    {
      RefreshActorList();
    }

    if (sortByDistance)
    {
      // Actors move, the player moves, everything changes, so refresh the list
      // every 2 seconds or so to keep it up to date.
      listNeedsRefresh = timeSinceLastRefresh > 2;
    }

    if (draggingActor != null)
    {
      UpdateDrag();
    }
  }

  void UpdateDrag()
  {
    Vector2 localPos;
    RectTransformUtility.ScreenPointToLocalPointInRectangle(
      container, Input.mousePosition, null, out localPos);
    draggingGhost.container.anchoredPosition = localPos;

    VoosActor onstageHint;
    bool onstageUnderMouse = onStageList.IsDragTarget(out onstageHint);
    if (onstageHint == draggingActor) onstageHint = null;

    VoosActor offstageHint;
    bool offstageUnderMouse = offStageList.IsDragTarget(out offstageHint);
    if (offstageHint == draggingActor) offstageHint = null;

    if (Input.GetMouseButtonUp(0))
    {
      if (onstageUnderMouse)
      {
        if (onstageHint != null)
        {
          if (VoosActor.IsValidParent(draggingActor, onstageHint))
          {
            draggingActor.SetTransformParent(onstageHint.GetName());
            draggingActor.SetSpawnTransformParent(onstageHint.GetName());
          }
        }
        else
        {
          draggingActor.SetPreferOffstage(false);
          draggingActor.SetTransformParent(null);
          draggingActor.SetSpawnTransformParent(null);
        }
      }
      else if (offstageUnderMouse)
      {
        if (offstageHint != null)
        {
          if (VoosActor.IsValidParent(draggingActor, offstageHint))
          {
            draggingActor.SetTransformParent(offstageHint.GetName());
            draggingActor.SetSpawnTransformParent(offstageHint.GetName());
          }
        }
        else
        {
          draggingActor.SetPreferOffstage(true);
          draggingActor.SetTransformParent(null);
          draggingActor.SetSpawnTransformParent(null);
        }
      }
      SetDraggingActor(null);
    }
  }

  void OnSomeActorChanged()
  {
    // We can't just update the actor that changed, because changes to on/off stage
    // and display name require us to re-sort the lists, etc.
    listNeedsRefresh = true;
  }

  void OnActorCreated(VoosActor actor)
  {
    if (!monitoringActors)
    {
      Debug.LogError("Shouldn't get OnActorCreated when not monitoring actors...");
      return;
    }
    StartMonitoringActor(actor);
    listNeedsRefresh = true;
  }

  void OnActorDestroyed(VoosActor actor)
  {
    if (!monitoringActors)
    {
      Debug.LogError("Shouldn't get OnActorDestroyed when not monitoring actors...");
      return;
    }
    StopMonitoringActor(actor);
    listNeedsRefresh = true;
    if (actor == draggingActor)
    {
      SetDraggingActor(null);
    }
  }

  void OnSortButtonClicked()
  {
    sortByDistance = !sortByDistance;
    PlayerPrefs.SetInt(SortByDistancePrefKey, sortByDistance ? 1 : 0);
    sortByButton.GetComponentInChildren<TMPro.TMP_Text>().text = sortByDistance ? "Sort: distance" : "Sort: name";
    listNeedsRefresh = true;
  }

  void OnCloseButtonClicked()
  {
    SetExpanded(false);
  }

  void RefreshActorList()
  {
    onStageList.SetActors(engine.EnumerateActors().Where(actor => ShouldActorBeListed(false, actor)), sortByDistance);
    onStageList.HighlightActors(selectedActors);
    offStageList.SetActors(engine.EnumerateActors().Where(actor => ShouldActorBeListed(true, actor)));
    offStageList.HighlightActors(selectedActors);
    listNeedsRefresh = false;
    timeSinceLastRefresh = 0;
  }

  private bool ShouldActorBeListed(bool isOffstageList, VoosActor actor)
  {
    return (actor.GetIsOffstageEffective() == isOffstageList) &&
      !actor.GetWasClonedByScript() &&
      (showCopiesToggle.isOn || actor.GetCloneParentActor() == null) &&
      actor.GetDisplayName().ToLower().Contains(searchInput.text.ToLower());
  }

  public void SetSelectCallback(System.Action<VoosActor> callback)
  {
    this.selectCallback = callback;
  }

  void OnActorEntryClicked(HierarchyActorEntry entry, HierarchyActorEntry.ActionType actionType)
  {
    // The user has clicked on an actor in the list.
    VoosActor actor = engine.GetActor(entry.GetActorName());
    if (actor == null)
    {
      // Shouldn't happen.
      Debug.LogError("Clicked on actor that doesn't exist " + entry.GetActorName() + ". Bug?");
      return;
    }
    if (selectCallback != null)
    {
      selectCallback(actor);
      selectCallback = null;
      return;
    }
    switch (actionType)
    {
      case HierarchyActorEntry.ActionType.SELECT:
        if (Input.GetKey(KeyCode.LeftShift) || Input.GetKey(KeyCode.RightShift))
        {
          AddActorToSelection(actor);
        }
        else
        {
          SelectActor(actor);
        }
        break;
      case HierarchyActorEntry.ActionType.INFO:
        SelectActor(actor, true);
        break;
      case HierarchyActorEntry.ActionType.MOVE:
        ToggleActorOffstage(actor);
        break;
    }
  }

  private void AddActorToSelection(VoosActor actor)
  {
    listNeedsRefresh = true;
    selectedActors.Add(actor);
    editMain.TryForceAddingActor(actor);
  }

  private void SelectActor(VoosActor actor, bool forceShowInfo = false)
  {
    if (forceShowInfo)
    {
      editMain.SelectInfoTool();
    }
    listNeedsRefresh = true;
    selectedActors.Clear();
    selectedActors.Add(actor);
    editMain.TryForceSelectingActor(actor);
  }

  private void ToggleActorOffstage(VoosActor toggleActor)
  {
    bool currentOffstage = toggleActor.GetPreferOffstage();
    if (!toggleActor.IsLockedByAnother() && !toggleActor.IsParentedToAnotherActor())
    {
      undoStack.PushUndoForActor(
        toggleActor,
        $"Toggle offstage for {toggleActor.GetDisplayName()}",
        actor =>
        {
          if (!actor.IsParentedToAnotherActor())
          {
            actor.SetPreferOffstage(!currentOffstage);
          }
        },
        actor =>
        {
          if (!actor.IsParentedToAnotherActor())
          {
            actor.SetPreferOffstage(currentOffstage);
          }
        });
      RefreshActorList();
    }
  }

  private void OnActorDrag(VoosActor actor)
  {
    if (draggingActor != null) return;
    SetDraggingActor(actor);
  }

  private void SetDraggingActor(VoosActor actor)
  {
    if (actor == draggingActor) return;
    draggingActor = actor;

    onStageList.SetDraggingActor(draggingActor);
    offStageList.SetDraggingActor(draggingActor);
    draggingGhost.gameObject.SetActive(draggingActor != null);
    if (actor != null)
    {
      draggingGhost.GetComponentInChildren<TMPro.TextMeshProUGUI>().text = actor.GetDisplayName();
    }
  }

}

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text:
Q: How many triplets (a, b, p) satisfy $a^3 - 2ba^2 + 16ap^2 + 32p^2b = 0 $ where $p$ is a prime and $a,b$ $\epsilon$ $N$ How many triplets (a, b, p) satifsy $a^3 - 2ba^2 + 16ap^2 + 32p^2b = 0 $ where $p$ is a prime and $a,b$ $\epsilon$ $N$
I tried solving this question using the quadratic formula since this equation is quadratic in $p$. But I wasn't successful in solving this question.

A: From your eq we have $$a^2(2b-a)=16p^2(2b+a).$$
If $p\nmid a$ then $p^2\mid 2b-a$.
Write $2b-a=kp^2$.  So $2b+a=kp^2+2a$.  That is
$ka^2=16(kp^2+2a)$ or
$$ka^2-32a-16kp^2=0.$$
The discriminant has to be square so
$$32^2+4\cdot 16k^2p^2=n^2.$$
We must have $n=8m$ and so
$$m^2=4^2+k^2p^2.$$
The only possible value of $kp$ is $3$ so $k=1$ and $p=3$.  Then
$$a^2-32a-16\cdot 9=0$$
or $a=36$ (since $a\in \Bbb N$).  But $2b-a=kp^2=9$ we get $2b=45$ but this is impossible.
Therefore $p \mid a$.  That is $a=px$.  We get
$$x^2(2b-px)=16(2b+px).$$
Then
$$2b(x^2-16)=px(x^2+16).$$
If $p\mid 2b$, then write $2b=py$ so that
$$y(x^2-16)=x(x^2+16)=x(x^2-16)+32x.$$  That is $x^2-16\mid 32 x$.  That is $x-4\mid x^2-16 \mid 32x$.  So $x-4\mid 32\cdot 4=128$.  Therefore $x-4=2^i$ for some $i=0,1,..,7$.  But $x+4\mid x^2-16\mid 32 x$ too, so $x+4=2^j$ for some $j=0,1,...,7$.  That is $2^j-2^i=8$ so $i=3$ and $j=4$, so $x=12$.  Therefore $a=px=24$ and
$$y=\frac{12(12^2+16)}{12^2-16}=15.$$
That is $2b=15p$ so $p=2$. Then we have an answer $(a,b,p)=(24,15,2)$.
If $p\nmid 2b$, then $p$ is odd.  But $2b(x^2-16)=px(x^2+16)$, so $x$ is even.  Write $x=2z$.  We have
$$b(z^2-4)=pz(z^2+4).$$
If $z$ is odd, then $\gcd(z,z^2-4)=1$ and so $z\mid b$.  Write $b=zu$.  We get
$$u(z^2-4)=p(z^2+4).$$
Since $p\nmid 2b$ and $b=zu$, we conclude that $p$ is a factor of $z^2-4=(z-2)(z+2)$.  Therefore after dividing by $p$, either $z-2$ or $z+2$ divides $z^2+4$.  Thus, $z\pm 2\mid 2^2+4=8$.  But $z$ is odd, so $z\pm 2=1$.  Hence, $z=3$ is the only possible value.  Then
$u(3^2-4)=p(3^2+4)$ so $5u=13p$.  That is, $p=5$ and $u=13$.  Hence $b=zu=39$.  Also $x=2z=6$ and $a=px=30$.  We have one more answer $(a,b,p)=(30,39,5)$.
If $z$ is even, then $z=2v$.  So $$b(v^2-1)=2pv(v^2+1).$$  Because $\gcd(v,v^2-1)=1$, we get $v\mid b$.  Write $b=vw$.  So
$$w(v^2-1)=2p(v^2+1).$$
Thus $2(v^2+1)\mid w(v^2-1)$.  Note that $\gcd(v^2+1,v^2-1)=1,2$ depending on whether $v$ is even or odd.
If $v$ is even, then $2(v^2+1)\mid w$.  So
$$p=\frac{w}{2(v^2+1)}(v-1)(v+1).$$
But $p$ is prime, we get $\frac{w}{2(v^2+1)}=1$ and $v-1=1$.  So $v=2$, $w=10$, $b=vw=20$, $z=2v=4$, $x=2z=8$, $p=v+1=3$, and $a=px=24$.  We get another solution $(a,b,p)=(24,20,3)$.
If $v$ is odd, then $8\mid v^2-1$ but $8\nmid 2p(v^2+1)$ because $p$ is odd and $v^2+1\equiv 2\pmod{4}$.
Therefore, all $(a,b,p)$ are $(24,15,2)$, $(30,39,5)$, $(24,20,3)$.  This problem is so long.

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Readers of this blog may be aware of the recent scandal at literary agency, Donadio & Olson, which filed for bankruptcy in December 2018 as a result of an accounting scandal. The agency's bookkeeper Darin Webb had for years been embezzling money from clients of the agency with respect to royalty payments. Webb himself was sentenced to a two year jail term. Although he had embezzled over $3 million, no criminal fine was imposed because the judge held that Webb was not in a financial position to pay it. The agency was unable to compensate its client authors, many of whom had lost significant sums of money as a result of Webb's actions.
This situation has led many in the publishing world, particularly authors and agents, to consider how to avoid these kinds of risks in practice. Obviously, partners/managers in literary agencies need to take proper care to supervise the financial side of things, and authors should do the same, and not rely totally on agents to do the right thing.
Are there any particular legal steps that can be taken to avoid fraudulent accounting and similar practices? Can authors use the law to protect themselves upfront before such a scandal occurs?
The answer is largely "no" or "perhaps, but just a little bit". You can't meaningfully include an anti-embezzlement clause in your agency contract, or at least there's little point in doing so, because embezzlement is a crime and the agency can't contract out of committing crimes. In other words, you don't need a clause saying the agency won't commit crimes that affect you as an author, because it's illegal for the agency to engage in criminal activity no matter what the contract says.
You can—and should—have an attorney go through your agency contract, ideally at the point where you're first offered representation, to check on other pitfalls that might be in the contract. However, the contract itself won't likely contain any promises about not committing crimes. That goes without saying.
Your attorney will be able to advise you about clauses in the contract that explain how the agency plans to handle your money. In situations where a publisher pays your royalties direct to your agent, and your agent passes the funds to you (minus their commission), you'll want to ensure that, in your agent's hands, the money is kept in an appropriate account that's earmarked for you (or you and other agency clients). There are a number of ways of doing this, and a good contracts attorney should be able to advise you on the safest options.
These days, publishers are increasingly also offering the option of splitting royalty payments direct between agents and authors, so your share of the royalties comes direct to you from your publisher, and the publisher likewise sends the agent's commission direct to them. You could ask if your agent is prepared to include a clause in the agency agreement that, when she negotiates publishing contracts for you, she'll do her best to secure these kinds of payment arrangements with the publishers. If your agent is not prepared to do this, you can look out for it yourself in your publishing contracts. While the agent will typically negotiate the contract, you get to read it before you sign it. See if you can ask for this kind of split payments arrangement to be included before you sign. Some publishers are even prepared to send the entire royalty payment to you direct and leave it to you to forward the agent's commission to her. Many agents won't agree to these arrangements, but you could ask if yours will.
At the end of the day, one important lesson from the Donadio & Olson situation is that authors can't afford to be complacent about their royalty payments. As creative artists, most authors like to focus on the creative side of their lives and leave the business/legal/financial issues to agents and attorneys. However, an ounce of prevention is worth a pound of cure. At the very least try to read and understand your contracts, even if you occasionally have to hire an attorney to help you, and, make sure you're keeping track of your royalty payments. If your agent is giving you a hard time about seeking payment information, it might be a sign that she's not the best agent for you. After all, it's your money.
And, of course, the usual disclaimers apply to this column. This is not legal advice, but hopefully is helpful background information. Those concerned about author finances should consult a legal professional.


"If you look there in the lower right where the shadow's being cast by the gnomon, you can see that the shadow is orange. We didn't have a word for this in English, so I made it up; orangidescence.
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text:
Kailee Rhianne Marie Hundley
Kailee Rhianne Marie Hundley had recently celebrated her very first birthday. Pictures were taken of her as she ate birthday cake and opened presents.
One month later, the family who loved her so much, would have to prepare themselves to say good bye to her.
"We had big plans for her. We knew what we wanted for her.
And we can't believe this"
Tressa Hays - Kailee's Aunt
In spite of an open investigation against her by the Kansas Department Of Health And Environment, Jessica Cummings ran a home daycare and was licensed in the state of Kansas, BY the state of Kansas to take care of children in her home. On March 25, 2008, Kailee was dropped off at this home daycare. That would be the last time her family would see her alive.
Tiffanie Williams was bothered by the crying and noises that Kailee was making
while drinking a bottle while she was trying to sleep after working late and then going to visit her boyfriend. Jessica Cummings decided to take care of the problem.
Jessica Cummings strapped Kailee into her car seat, which was too small for her, tightly strapping her in she the put her in a laundry room bathroom for a nap. Two hours went by before she would check on Kailee and by then, she would already be dead. It would later be revealed that Kailee strangled on the car seat strap. Kailee's family said that if they had known naps were taken in car seats, she never would have been going to that daycare and they wouldn't have to attend a funeral for her:
"We can't undo it now and that's just really sucks."
Tressa Hays
Tressa Hays said that Kailee was big enough to use a toddler car seat which was in the car at all times. Comments by Tressa Hays confirm that Kailee enjoyed going to this daycare and that it was thought that Jessica did a good job. Tressa says she believes that the death of her niece was an accident, though she still said Jessica is to blame:
"I do believe Jessica loved her. That's why I'm trying to find in my heart to not hate her
but right now I do"
A neighbor and friend of Jessica's said that she had spent time in her daycare when she brought her daughter over for play dates. The friend said that Jessica loved children and was good at her job. The friend also said that she would trust Jessica with her daughter and that even though she was hurting for both families, she would support Jessica through this case.
The Kansas Department Of Health And Environment was investigating Jessica and said they couldn't speak about the case since it was still an open case. Jessica had been licensed since August, though that license had been suspended since the death of Kailee.
Wichita Police turned over the case to the district attorney which is procedure in cases of the death of a child. The police also wouldn't comment on the Kansas Department Of Health And Environments on going investigation and didn't immediately release the cause of Kailee's death.
Mike Hundley had sole custody of Kailee since her mother had recently been jailed on suspicion in the first degree murder of a stepdaughter, Daytona Robertson who was the daughter of her then husband, Tommy Robertson.
On Saturday, March 29, 2008, family and friends gathered at the Westlink Christian Church to pay their respects and say their good byes to Kailee at her funeral.
On April 16, 2008, Jessica Cummings was in court facing the charge of involuntary manslaughter. It was revealed around that same time that the investigation against her was due to not reporting suspected child abuse and instead just taking pictures of the abuse that took place towards a two year old girl that she took care of. The complaint was filed the day after Daytona Robertson died of child abuse. Kailee and Daytona were stepsisters who both attended Jessica's daycare. A preliminary hearing was scheduled for April 30th and Jessica was let
out of jail on a $10,000. bond.
Kailee's family was upset about the charge saying it was not severe enough for what she had done to Kailee:
"This is as bad as it gets. Losing your child, your baby. It doesn't get any worse than that.
It's just another slap in the face. Another punch in the gut"
Prosecutors said the even though they hold Jessica Cummings accountable for the death of Kailee, they feel that her death was not done intentionally. Tressa Hays disagreed with them:
"Everything about that day was preventable"
Kailee's family was going to meet with the Safe Kids Coalition to help in finding ways to improve regulations for daycare providers.
In December of 2008, Jessica Cummings was back in court where she would say that she didn't mean any harm to Kailee Jessica also stated that she knew the procedure when putting children down for naps and that it included that children under the age of 18 months MUST sleep in a crib or a play pen:
"Not a car seat, not an infant carrier, not a stroller attachment"
Shannon Wilson's closing argument
Defense attorney John Stang said that Jessica could not have known that Kailee was in any danger by being put down for a nap in the car seat since this is something parents have done often when taking their children on long car trips. John Stang had a diagram showing how close Jessica was to the area where Kailee was sleeping:
"She was, what, 8 feet away?"
John Stang again said that Jessica could not have foreseen what would happen to Kailee adding that Deborah Johnson, the coroner who did the autopsy on Kailee, said she had never seen a case like this.
"But what did Dr. Johnson also say? She said in no uncertain terms never leave a child
unattended in a car seat."
It was determined that Kailee had been improperly strapped into the car seat and had slipped down allowing the straps to strangle her.
A jury took about five hours to find her guilty of the charge of involuntary manslaughter. Jessica is facing more than 2 1/2 years in prison for her part in the death of Kailee Robertson. In order to find Jessica guilty the jury had to come to the conclusion that she had put Kailee in danger deliberately by strapping her into the car seat and leaving her alone in the bathroom:
"In an extremely difficult case, they followed the law"
Assistant District Attorney Shannon Wilson
After hearing the verdict, Jessica broke down into tears. Kailee's family was also crying.
"Hopefully, if any good comes out of this, it will get people to realize the risks to children by leaving
them unattended in a car seat"
Jessica was seven months pregnant at the time of her hearing and would leave the court house free on bond.
On October 9, 2008, Kailee's great grandmother, Opal Marie Hundley died. Kailee is now safely with a grandmother who loved her. A memorial fund has been set up for Kailee at Intrust Bank.
Jessica was sentenced to two years and eight months in prison, such a small price to pay for what she did to this little Angel. In March of 2013, the Kansas Supreme Court was considering an appeal based on questions of intent and if the jury should have been given instruction requested by the defense, on rules related to convicting someone of involuntary manslaughter if it was believed to be a "freak accident".
I find it disgusting that this woman killed a child, took that child away from her parents for the rest of their lives, yet she doesn't seem to feel she should have to do less than three years in prison.
The Supreme Court ruled that Jessica deserved a new trial based on the improper instructions to the jury about endangering a child. Jessica said that the state needed to prove that she knew putting Kailee into a car seat would be dangerous for her. I guess all of the warnings all over car seats about how you SHOULD NOT leave a child unattended in a cars eat were not enough to make her realize the danger she put children in.
A Forensic Pathologist said that Kailee was too big for this car set and her weight had caused the seat to tip forward putting a strain on her neck and regardless of how small or big the car seat was, a child should not be left unattended in a car seat, also pointing out that the seat warnings indicated the same. Jessica said that thought it was safe and she had even put her own children in that very car seat.
Jessica was paroled on April 5, 2013.
Kailee,
I love you so much. I miss your beautiful smile and your bright blue eyes. I miss your big big wet kisses and your high fives. This world does not seem fair but one day I will be in heaven with you.
Until then have fun with your big sister "Day Day" and your little brother Daylee. I miss you and love you, You are forever my baby.
Love Daddy
Christopher Keane
In Kansas, Jessica Cummings was found guilty of involuntary manslaughter in the strangulation death of a 13-month-old girl found dead partially strapped in a car seat while at day care. On March 25, 2008, the child died and was found two hours later in the bathroom partially strapped in her car seat. New stories report she was placed in a bathroom partially strapped in her infant seat to finish her bottle.
Studies show that caregiver supervision and an effective strategy that keeps the child in the caregiver's view reduces the frequency of injury. It is also known that a lapse in supervision where the child is not in view correlates with a higher incidence of child injury. This may seem to be an obvious statement. However, even if the caregiver is highly conscientious there remains an increased risk for injury during intervals when the child is not in view. So whether a child caregiver is regarded as highly conscientious or not-highly conscientious, the risk for injury to a child is still higher when the child is not in the view of the caregiver. Close supervision contributes directly to reduced events of injury.
<>A child has died tragically at a day care, from a day care provider's neglect. This is wrongful death. The Keane Law Firm represents children that have been catastrophically injured from neglect by day care providers. Please call us, the child injury law firm for a timely case evaluation at 888-592-KIDS or 415-398-2777. You may visit our website at www.keanelaw.com – we will determine the facts of the case and file a claim to hold those responsible to pay for the damages. In a child death lawsuit, we will recover for our clients.
Please visit the pages of all three siblings in this family:
Daytona Joe-Lei Robertson
Daylee Joe Robertson
Kailee Rhianne Hundley
Death occurred in the state of Kansas

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