text:
\section{Introduction}
\label{introSec}
The Milky Way is home to billions of stars \citep{McMillan:2016uq}, many of which
are members of \emph{stellar clusters} - gravitationally bound collections of stars.
Stellar clusters are formed from low temperature and high density clouds of gas and dust
called \emph{molecular clouds}, though there
is uncertainty as to how the stars in a cluster form \citep{Beccari2017}.
Each theory of star formation yields a different prediction for the distribution
of the masses of stars that initially formed in a cluster. Hence, it is of fundamental
interest to estimate this distribution, referred to as the \emph{stellar initial mass function (IMF)},
and assess the validity of these competing theories.
\new{The IMF can be thought of as a continuous density describing the distribution of star masses that initially form in a stellar cluster.}
In fact, research advances in many areas of stellar, galactic, and extragalactic astronomy are at
least somewhat reliant upon accurate understanding of the IMF \citep{bastian2010}.
For example, the IMF is a key
component of galaxy and stellar evolution and planet formation \citep{bally2005, bastian2010, Shetty2014},
along with chemical enrichment and abundance of core-collapse supernovae \citep{weisz13}.

There is also ongoing discussion surrounding the {\it universality} of the IMF, i.e.,
if a single IMF describes the generative distribution of stellar masses for all
star clusters \citep{bastian2010}. The consensus of the astronomical community is that the IMF is not universal, however, most of the observations had been consistent with universality \citep{kroupa2001, bastian2010, Ashworth2017}.
With further research and growing sample sizes, however, there is increased theoretical
(e.g., \citealt{bonnell2006, Dib2010})
and observational
(e.g., \citealt{Treu2010, Dokkum:2010fk, Spiniello2014,Geha2013, Dib2017})
support for an IMF that can vary cluster to cluster.

\cite{salpeter55} studied the evolutionary properties of certain
populations of stars, and in the process defined the first IMF
(which he called the ``original mass function'').
This work put forth the now-classic model for the IMF, a power law
with a finite upper bound equal to the physical maximum mass of a star that
could form in a cluster \citep{salpeter55}.
More recent studies continue to use this power law form for the IMF, especially for stars of mass greater than half that
of our sun (e.g., \citealt{Massey2003, bastian2010, DaRioEtAl2012, Lim2013, weisz13, Weisz:2015kx, Jose2017}).
Similar models have been proposed and used in the astronomical literature for inference of the
stellar IMF; these will be discussed in the next section.
The estimation of the parameters of these proposed models typically relies on the assumption that
the observed stars in a stellar cluster form independently; more specifically, the assumption that the masses of the individual stars form independently.
The proposed model in this work loosens the assumption of independence in order to explore one of several possible physical formation
mechanisms of cluster formation\new{ and avoids specification of a parametric model form by using on a new simulation model}.

Despite this seemingly simple form of the power law model, the statistical challenges of estimating the
IMF using \emph{observed} stars from a cluster are significant. Many of the limitations are related
both to
observational issues and to the adequate modeling of the evolution of a star cluster after the
initial formation. For example, since stars of greater mass die more rapidly, the upper tail of the IMF is
not observed in a cluster of sufficient age.
Also, the death of massive stars can trigger additional star formation,
contaminating the lower end of the IMF with new stars \citep{Woosley2015}.
There are also issues related to missing lower-mass stars due to the sensitivity of the instruments.
The observational astronomers will often estimate a \emph{completeness function} of an observed
cluster, which is the probability of observing a star of a particular mass.  The completeness
function is discussed in more detail below.

The observational limitations and the challenge of modeling cluster evolution make approximate Bayesian
computation (ABC) appealing for estimation of the IMF, as ABC allows for relatively easy incorporation of
such effects. The difficulty of addressing these limitations is
implied by the fact that observational effects are often ignored or accounted for in
an ad hoc or unspecified manner (e.g., \citealt{DaRioEtAl2012, Ashworth2017, Jose2017, Kalari2018}),
though \cite{weisz13} discuss how some observational limitations can be incorporated into their
proposed Bayesian model.
A primary appeal of ABC for this application is the ability to incorporate more complex models for cluster
formation.
Standard IMF models do not specify the process by which
a large mass of gas (the molecular cloud) transforms into a gravitationally bound collection of stars.
ABC is based on a simple rejection-sampling approach, in which draws of model parameters from a prior
distribution are fed through a simulation model to generate a sample of data.
If the generated sample is ``close'' (based on an appropriately chosen metric) to the
observed data, the prior draw that produced that generated sample is retained. The collection of
accepted parameter values comprise draws from an approximation to the posterior.
The simulations
(the \emph{forward model}) can include any of the complex processes that make it challenging to derive a
likelihood function for the observable data.

This situation is typical of inference challenges that arise in astronomy. See
\cite{schafer2012, AkeretEtAl2015, IshidaEtAl2015} for reviews.
Recent years have seen a
rapid increase in the use of ABC methods for estimation in this field, including
specific application to
Milky way properties \citep{RobinEtAl2014},
strong lensing of galaxies \citep{Killedar2018,Birrer2017},
large scale structure of the Universe \citep{Hahn2017b},
estimating the redshift distribution \citep{Herbel2017},
galaxy evolution \citep{Hahn2017a},
weak lensing \citep{Peel2017,Lin2015},
exoplanets \citep{Parker2015},
galaxy morphology \citep{CameronPettitt2012},
and
supernovae \citep{WeyantEtAl2013}.

Hence, our motivation for using the proposed stochastic model for the stellar IMF includes both scientific and practical considerations. We model the observable data in a star cluster using a formation mechanism that incorporates realistic dependency in the masses of the stars. Further, this model generalizes commonly used IMF models, i.e., it can capture, but also distinguish, popular competing IMF model shapes. Flexible models of this type have great potential to test widely-held assumptions of more restrictive parametric forms, and eliminate the need for (often arbitrary) model selection exercises. Finally, the generative approach allows for the incorporation of observation effects and uncertainties within an ABC framework.

This paper is organized as follows. In Section \ref{sec:background}, background on the IMF along with inference challenges are presented along with an introduction of ABC.
The proposed stochastic model for stellar formation is discussed in Section  \ref{PAmodelSection}. A simulation study, including an application of the proposed methodology to the estimation of the IMF of a realistic astrophysical simulation \citep{Bate2012}, can be found in Section \ref{sec:sim}. Finally,  \ref{discussionSec} provides a discussion.

\section{Background}
\label{sec:background}
\subsection{Stellar Initial Mass Function}

As noted above, \cite{salpeter55} introduced the power law model for the shape of the IMF for masses larger
than 0.5$M_{\odot}$, where $M_{\odot}$ is the mass of the Sun.
\cite{kroupa2001} extended the range of the IMF by proposing a three-part broken power law model
over the range $0.01 M_{\odot} < m < M_{\text{max}}$, where $M_{\text{max}}$ is the mass of the largest star that could form with nonzero probability.
This model postulates different forms for the IMF for stars of masses $0.01 M_{\odot} < m < 0.08 M_{\odot}$,
$0.08 M_{\odot} < m < 0.5$ and $m > 0.5M_{\odot}$.
\remove{Focusing on the upper part} \new{To illustrate the form of the IMF model of \cite{salpeter55}, consider the upper tail where $m > 0.5M_{\odot}$}, and \remove{defining} \new{define} $\theta = (\alpha, M_{\text{max}}),$ \new{then} the probability density function for \remove{mass $x$} \new{ mass $m$} in the upper tail of the stellar IMF is assumed to be given by
\begin{align}
f_M(m \mid \theta) = cm^{-\alpha}\text{,}\;\; m \in [0.5M_{\odot}, M_{\text{max}}]\text{,}
	\label{eq:imf}
\end{align}

\remove{$f_M(m \mid \theta) = cm^{-\alpha}\text{,}\;\; m \in [M_{\text{min}}, M_{\text{max}}]$}

\noindent where the constant $c$ is chosen such that $f_M$ is a valid probability density.
\new{(For the form of the IMF model of \cite{kroupa2001}, see Equation~\eqref{eq:kroupa}.)}
Alternative models have been proposed that include log-normal distributions, joint power law and log-normal parts, and truncated exponential distributions  \citep{Chabrier:2003oq, Chabrier:2003om, chabrier2005,IMF50,bastian2010, OffnerEtAl2014}.  The \cite{kroupa2001} and \cite{Chabrier:2003oq, Chabrier:2003om} models are displayed in Figure~\ref{fig:imf_models} along with observational challenges discussed \S\ref{sec:observational}.
Power law distributions and log-normal distributions are closely related and may be the result of subtle differences in the underlying formation mechanism \citep{Mitzenmacher2004}.
The IMF model we propose will include, as a special case, a family of formation mechanisms that generate
power law tails, but also allow for a wider range of tail behaviors (see \remove{Section} \new{Appendix}~\ref{sec:powerlaw}).

\subsubsection{Observational Challenges} \label{sec:observational}
Observing all stars comprising an IMF
is not feasible, as the most massive stars ($m > 10 M_{\odot}$) have lifetimes of
only a few million years.
The lifetime of a star (the time it takes for the star to burn through its
hydrogen) depends strongly on its mass:~the most massive
stars have shorter lives due to the hotter temperatures they must maintain
to avoid collapse from the strong gravitational forces. In particular,
stellar life is
approximately proportional to $m^{-\rho}$
where $\rho \approx 3$ (\citealt[p. 30]{hansen2004}, \citealt[p. 439]{Chaisson:2011}).
Hence, the mass of the largest star observed in a given
cluster depends on the cluster age.

Furthermore, the IMF is estimated using a noisy, incomplete view of that cluster.  Whether or not a star is observed is dependent on several factors including its mass, its location in the cluster, and its neighbors.  Some of these factors are described by a data set's \emph{completeness function}, which quantifies a given star's probability of being observed.  This depends on its luminosity (i.e. intrinsic brightness) since it needs to be sufficiently bright to be observable; in particular, completeness depends on stellar flux in comparison with the flux limits of the observations.
There are also issues with {\it mass segregation}:~stars with lower mass tend to be on the edge of the cluster, while the most massive stars are often
found in the center \citep{weisz13}. Due to {\it stellar crowding} in the center, stars in this region can be more difficult to observe.  Additionally, binary stars (star systems consisting of a pair of stars)
are difficult to distinguish from a single star, creating the potential for overstating the mass of an object and understating the number of stars in the cluster.

\begin{figure}[htbp]
   \centering
\includegraphics[width = .5\textwidth]{figures/CompareOthers3.pdf}
   \caption{The broken power law model of \cite{kroupa2001} (red, dashed) and the log-normal with a power law tail model of \cite{Chabrier:2003oq, Chabrier:2003om} (black, solid) are displayed along with vertical lines representing several observational challenges.  The blue vertical dotted-dashed lines indicate the range of values ($C_{\min}, C_{\max}$) on which the completeness function may be defined, and the vertical green dotted line indicates the maximum observable mass (Mobs$_{\max}$) due to the aging cutoff.  The observational challenges are discussed further in \S\ref{sec:obs_challenges}.}
   \label{fig:imf_models}
\end{figure}

There are additional uncertainties involved in translating the actual observables  (e.g. photometric magnitudes) into a mass measurement; that is, the mass values for observable stars are only estimates. The {\it Hertzsprung-Russell (H-R) Diagram} is a classic visual summary of the distribution of the luminosity and temperature of a collection of stars. A typical H-R Diagram includes a {\it main sequence} of stars that trace a line from bright and hot stars to dim and cool stars. Stellar mass also evolves along this one-dimensional feature, and since luminosity and temperature are estimable, mass can thus also be estimated. The mass of binary stars can be determined via Kepler's Laws, and hence a {\it mass-luminosity relationship} can be fit to binaries and then extended to other stars on the main sequence. Unfortunately, luminosity and temperature are nontrivial to estimate, as corrections for effects such as {\it accretion} and {\it extinction} are required, along with an accurate estimate of the distance to the stars \citep{Da-Rio:2010aa}.
The process is further complicated by the dependence of how these transformations are made on the {\it spectral type} of the star. Careful budgeting of the errors that accumulate is required in order to produce a reasonable error bars on mass estimates; \cite{Da-Rio:2010aa} utilize a Monte Carlo approach in which the errors in magnitudes are propagated forward through to uncertainties in the spectral type, the accretion and reddening corrections, and finally to an uncertainty on the mass.

\subsection{Approximate Bayesian Computation}

Standard approaches to Bayesian inference, either analytical or built on
MCMC, require the specification of a
likelihood function, $f(m \mid \boldsymbol{\theta})$, with data $m_{\text{obs}} \in \mathcal D$,
and parameter(s) $\boldsymbol{\theta} \in \Theta$. In many modern scientific inference
problems, such as for some emerging models for the stellar IMF, the likelihood is too complicated to be derived or otherwise
specified.
As noted previously, ABC provides an approximation to the posterior
without specifying a likelihood function, and instead relies on forward simulation of the data generating
process.

The basic algorithm for sampling from the ABC posterior is attributed to
\cite{TavareEtAl1997} and \cite{PitchardEtAl1999}, used for applications to population
genetics. The algorithm has three main steps which are repeated until a sufficiently
large sample is generated:
\emph{Step 1}, Sample $\boldsymbol{\theta}^*$ from the prior; \emph{Step 2},
Generate $m_{\text{sim}}$ from the forward process assuming $\boldsymbol{\theta}^*$;
 \emph{Step 3},
Accept $\boldsymbol{\theta}^*$ if $\rho(m_{\text{obs}}, m_{\text{sim}}) \leq \epsilon$, where $\rho(\cdot, \cdot)$ is a user-specified distance function and $\epsilon$ is a tuning parameter that should be close to 0.
This last step typically consists of comparing low-dimensional summary statistics generated
for the observed and simulated datasets.
Adequate statistical and computational performance of ABC algorithms depends greatly on the
selection of such summary statistics
\citep{JoyceMarjoram2008,BlumFrancois2010, Blum2010, FearnheadPrangle2012, BlumEtAl2013}.

The basic ABC algorithm can be inefficient in cases where the parameter space is of moderate
or high dimension.
Hence, important adaptations of the basic ABC algorithm incorporate ideas of sequential sampling in
order to improve the sampling efficiency \citep{MarjoramEtAl2003,SissonEtAl2007,beaumont2009, DelMoralEtAl2011}.  A nice overview of ABC can be found in \cite{MarinEtAl2012}.
Here, we use a sequence of decreasing tolerances $\epsilon_{1:T} = (\epsilon_1, \ldots, \epsilon_T)$
with the tolerance $\epsilon_t$ shrinking until further reductions do not significantly affect the
resulting ABC posterior.
The improvement in efficiency is due to the modification that happens after the first time step:
instead of sampling from the prior distribution, the proposed $\boldsymbol{\theta}$ are drawn from the previous time step's ABC posterior.  Using this adaptive proposal distribution can help to improve the sampling efficiency.  The resulting draws, however, are not targeting the correct posterior, and so importance weights, $W_t$, are used to correct this discrepancy.

\section{Forward Model for the IMF}
\label{PAmodelSection}
Due to their simple interpretations, mathematical ease, and demonstrated consistency with observations, power law IMFs (or similar variants) have been widely adopted in the astronomy literature \citep{kroupa2012}; however, open questions remain about stellar formation processes.  The proposed forward model is a way to link a possible stellar formation process with the realized mass function (MF).

\new{
One known underlying mechanism for producing data with power-law tails is based on {\em preferential attachment} (PA) \citep{Mitzenmacher2004}.
The earliest PA model, the Yule-Simon process, was popularized by \cite{simon55}, and was originally used to model biological genera and word frequencies. Other PA models include the classic {\it Chinese restaurant process}
and its generalizations \citep{Bloem2017}.
Interest in PA models grew within the study network evolution
\citep{BarabasiAlbert1999}.
Such evolution is described by the {\it attachment function}, which
describes the probability that a node acquires an additional edge,
usually as an increasing function of its current degree.
Most of the work done on estimation of the attachment function
makes the assumption that observations are available regarding
the full or partial evolution of the network.
This includes the nonparametric methods of
\cite{Jeong2003}, \cite{Newman2001}, and \cite{Pham2015};
the maximum likelihood approaches of
\cite{Gomez2011}, \cite{Wan2017a}, \cite{Onodera2014}; and
the Bayesian approach (using MCMC) taken by \cite{Sheridan2012}.
\cite{Wan2017a} also describes
an approximation to the MLE that can be utilized when only a snapshot view of
the network is available.
\cite{Wan2017b} uses a semiparametric approach to fit to the upper tail of the
network degree distribution. The focus is on how the estimator performs
under deviations from the linear PA model and the ``superstar" linear PA
model, in which one node to which most of the other nodes attach. Estimation
is based on extreme value theory.
\cite{Kunegis2013} use a simple least squares method to estimate the exponent
in a nonlinear, but parametric, PA model.}

\new{
In what follows, the proposed data generating process will use ideas of PA
to model the the evolution of a star cluster. The ABC approach will be
well-suited to perform inference with this model, given its complexity and
the available data.
}

\subsection{Preferential attachment for the IMF} \label{sec:pa}
The formation of a star cluster is a complicated and turbulent process with different theories on the physical processes that lead to the origin of the stellar IMF \citep{chabrier2005, Bate2012, OffnerEtAl2014, Pokhrel:2018nr}.  It is generally understood that the molecular cloud fragments and then forms stellar cores with a distribution referred to as the \emph{core mass function} (CMF).  Whether evolution from the CMF to the IMF is random, deterministic, or something in between is debated \citep{OffnerEtAl2014}.  In the proposed work, we consider the case where star cores can increase in mass by accreting material from the surrounding cloud and a particular star's final mass can be affected by its neighbors through turbulence or dynamical interactions.
That is, rather than assuming that stellar masses in a cluster arise independently of each other, our PA model proposes a resource-limited mass accretion process between stellar cores whose ability to accumulate additional mass is a function of their existing masses.
This dependence feature is particularly important for statistical inference, as models that assume independent observations of stellar masses are vulnerable to incorrect and misleading inference.
Additionally, the mass of the largest star to form in a cluster is limited by the total cluster mass.

Our proposed stochastic model for stellar formation is as follows:~we first fix a total available cluster mass $M_{\text{tot}}$.
This quantity can be physically interpreted as the total mass available for stellar formation in a molecular cloud.
At each time step $t = 1, 2, \ldots$, a random quantity of mass $m_t \sim \text{Exponential}(\lambda)$ enters the collection of stars; $M_{1,1} = m_1$ becomes the mass of the first star.
Subsequent masses entering the system form a new star \remove{with probability $\pi_t$} \new{with probability $\alpha$} or join existing star $k = 1, \ldots, n_t$ with probability $\pi_{kt}$\remove{.
These probabilities are specified as} \new{, defined as}
\begin{align}
	\pi_{kt} = \frac{M_{k,t}^{\gamma}}{\alpha+ \sum_{j = 1}^{n_t} M_{j,t}^{\gamma}} \text{.}
\label{eq:PAstars}
\end{align}

\remove{$\pi_t = \min \left (1, \alpha \right ) \;\;\; \text{and} \;\;\; \pi_{kt} \propto M_{k,t}^{\gamma}$}

\noindent The generating process is complete when the total mass of formed stars \remove{reaches} \new{first exceeds} $M_{\text{tot}}$.
The possible ranges of the three parameters are $\lambda > 0$, $\alpha \in [0,1)$, and $\gamma > 0$.

\remove{The parameter $\alpha$ defines the probability that entering mass forms a new star in a cluster.}  For the growth component, the model allows for linear ($\gamma = 1$), sublinear ($\gamma < 1$), and superlinear ($\gamma > 1$) behavior; the limiting case of $\gamma \to 0$ gives a uniform attachment model.  Finally, the parameter $\lambda$ acts as a scaling factor which controls the average `coarseness' of masses joining the forming stellar cores.

The proposed PA mass generation model offers considerable flexibility to approximate existing IMF models in the literature.  To illustrate the generality of the proposed model, IMF realizations were drawn assuming the \cite{kroupa2001}
broken power-law model as the true model, defined as
\begin{equation}
f(m) \propto \left\{
  \begin{array}{lr}
   m^{-0.3}, &  m \leq 0.08 \\
   k_1 \cdot m^{-1.3}, &  0.08 < m \leq 0.5 \\
   k_2 \cdot m^{-2.3}, &  m > 0.5,
  \end{array}
\right. \label{eq:kroupa}
\end{equation}
and the \cite{Chabrier:2003oq, Chabrier:2003om} log-normal model, defined as
\begin{equation}
f(m) \propto \left\{
  \begin{array}{lr}
    \frac{0.158}{m} \times \exp \left ( -\frac{(\log_{10}(m) - \log_{10}(0.079))^2}{2(0.69)^2}\right), &  m \leq 1\\
   k_3 \cdot m^{-2.3}, &  m > 1,
  \end{array}
\right. \label{eq:chab}
\end{equation}
where constants $k_1$, $k_2$, and $k_3$ are defined to make the densities continuous.
Our proposed PA ABC procedure was then used for inference and Figure~\ref{fig:otherModels} displays the resulting posterior predictive IMFs.  The proposed model captures the general shape of the true model.  Figures~\ref{subfig:kc_k} - \ref{subfig:kc_gamma} display ABC marginal posteriors for the broken power-law model of \cite{kroupa2001} and the log-normal model of  \cite{Chabrier:2003oq, Chabrier:2003om}.  Both the broken power-law and log-normal models have similar ABC posterior means for $\alpha$ (0.293 and 0.304, respectively).  However, the ABC posterior means for $\gamma$ are notably different.  The broken power-law model has an ABC posterior mean of 0.889 while the estimate for the log-normal model is 1.050.  Since the  \cite{kroupa2001} and \cite{Chabrier:2003oq, Chabrier:2003om} models use the same power-law slope for masses greater than 0.5 $M_{\odot}$ and 1 $M_{\odot}$, respectively, this suggests that the differences in $\gamma$ are due to differences in the shape of the lower-mass end of the IMF.  The proposed model offers an approach for discriminating these models.

\begin{figure} \centering
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/kroupa_abc_imf.pdf}
\caption{ABC posterior predictive IMF for \cite{kroupa2001} }\label{subfig:kroupa_abc_imf}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/chabrier_abc_imf.pdf}
\caption{ABC posterior predictive IMF for \cite{Chabrier:2003oq, Chabrier:2003om}}\label{subfig:chabrier_abc_imf}
\end{subfigure}
\caption{The solid black lines display the IMF with 1,000 stars simulated from (a) the broken power-law model of \cite{kroupa2001} (see Eq~\eqref{eq:kroupa}) and (b) the log-normal model of \cite{Chabrier:2003oq, Chabrier:2003om}  (see Eq~\eqref{eq:chab}).  The proposed PA ABC model was used with $N = 1,000$ particles, and 95\% point-wise credible bands are displayed (blue, dashed lines) along with the posterior median (red, dotted) for each data set.  The PA model provides flexibility to approximate both existing models.
Both models have similar ABC posterior means for $\alpha$ (0.293 and 0.304, respectively).  However, the ABC posterior means for $\gamma$ are notably different.  The broken power-law model has an ABC posterior mean of 0.889 while the estimate for the log-normal model is 1.050.
	}\label{fig:otherModels}
\end{figure}

\begin{figure}[htbp]
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/kc_k_rev1.pdf}
\caption{ABC posteriors for $\lambda^{-1}$}\label{subfig:kc_k}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/kc_alpha_rev1.pdf}
\caption{ABC posteriors for $\alpha$}\label{subfig:kc_alpha}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/kc_gamma_rev1.pdf}
\caption{ABC posteriors for $\gamma$}\label{subfig:kc_gamma}
\end{subfigure} \\
\caption{Marginal ABC posteriors for data generated from the broken power-law model of \cite{kroupa2001} (thick black lines)
and for data generated from the log-normal model of \cite{Chabrier:2003oq, Chabrier:2003om} (thin blue lines).  The vertical dashed black lines indicate the ABC posterior mean for the \cite{kroupa2001} model, vertical dashed and dotted blue lines indicate the ABC posterior mean for the \cite{Chabrier:2003oq, Chabrier:2003om} model, and the dotted gray lines indicate the range of the uniform prior for the parameter.
 }
\label{fig:kroupa_chab_marginals}
\end{figure}

\subsection{Initial Mass Function to the Observed Mass Function} \label{sec:obs_challenges}
The PA model describes the formation of a star cluster at initial formation.  \remove{However, we are not generally able to observe the star cluster after initial formation due to observational uncertainties, measurement uncertainties, and aging and dynamical evolution of the cluster.  A cluster's present-day observed MF is the observed distribution of the stellar masses of a particular cluster.}
\new{In practice, we are not generally able to observe the star cluster after the initial formation because significant time is likely to have passed.  When observation of a cluster occurs, the initial cluster will have changed due to aging and dynamical evolution of the cluster.  Additionally, even if observation of the initial cluster was possible, there are observational and measurement uncertainties that would limit our capacity to get a perfect representation of the initial cluster.  The actual observed cluster is referred to as the \emph{present-day observed MF}, which describes the observed distribution of the stellar masses of a particular cluster.}

Observation limitations can be easily incorporated into the ABC framework.  For simplicity, we adopt a ``linear ramp" completeness function describing the probability of observing a star of mass $m$:
\begin{align}
	\rm{Pr}(\text{observing a star} \mid \emph{m}) = \begin{cases} 0 \text{,} & m < C_{\text{min}}\\
								\frac{m - C_{\text{min}}}{C_{\text{max}} - C_{\text{min}}}\text{,} & m \in [C_{\text{min}}, C_{\text{max}}] \\
								1 \text{,} & m > C_{\text{max}} \text{.}
								\end{cases} \label{eq:ramp}
\end{align}
We assume that the values $C_{\text{min}}$ and $C_{\text{max}}$ are known, though we note that selecting an appropriate completeness function is a difficult process which requires quantification from the observational astronomers for each set of data.
Different models for the completeness function could also be considered, including those which allow for spatially-varying observational completeness.  A benefit of ABC is the ease at which a new completeness function can be incorporated -- it amounts to a simple change in the forward model.

Due to measurement error and practical limitations in translating luminosities into masses, the masses of stars are not perfectly known.
This uncertainty can be incorporated in different ways; following \cite{weisz13}, we assume that the inferred mass of a star $m_i$ is related to its true mass $M_i$ via
\begin{align}
	\log m_i = \log M_i + \sigma_i \eta_i \text{,}
	\label{eq:masserror}
\end{align}
where $\eta_i$ is a standard normal random variable, and $\sigma_i$ is known measurement error.
The model for mass uncertainties in \eqref{eq:masserror} is simple and could be extended to account for other sources of uncertainty (e.g. redshift).

As noted previously, the lifecycle of a star depends on certain characteristics such as mass.  In the proposed algorithm, stars generated in a cluster are aged using a simple truncation of the largest masses.
That is, the distribution of stellar masses for a star cluster of age $\tau$ Myr is given by
	\begin{align}
	f_M(m \mid \theta, \tau) \propto f_M(m \mid \theta) \mathbbm{I} \{M \leq \tau^{-1/3} \times 10^{4/3}\} \text{,}
	\label{eq:age}
	\end{align}
\remove{$f_M(m \mid \theta, \tau) = f_M(m \mid \theta) \mathbbm{I} \{M \leq \tau^{-1/3} \times 10^{4/3}\}$}

\noindent corresponding to stellar lifetimes of (10$^4/M^{3}$) Myr, where $M$ is the mass of the star \citep{hansen2004, Chaisson:2011}\new{, and $f_M(m \mid \theta)$ represents some specified IMF model}.
More sophisticated models that account for effects such as binary stars and stellar wind mass loss can be inserted into this framework.

\section{Simulation study} \label{sec:sim}

We propose an ABC framework to make inferences on the IMF given a cluster's present-day observed MF.\footnote{Code for running the proposed ABC-IMF algorithm is available at \url{https://github.com/JessiCisewskiKehe/ABC_for_StellarIMF}.}  \new{Details about the proposed ABC method, including the algorithm, are presented in Appendix~\ref{methodSec:abc}.  In this section we first consider a simulation study where the data are generated from the proposed forward model with observational effects, and then we consider data from an astrophysical simulation  \citep{Bate2012, Bate2014}}.

\subsection{Simulated data with observational effects} \label{sec:sim_obs}

\remove{Next we} \new{We first} consider a suite of simulations which incorporate
aging, completeness, and measurement error, in order to analyze these effects on the resulting inference on the IMF.
The same IMF was used throughout the simulations, with
$\lambda^{-1} = 0.25$, $\alpha = 0.3$, $\gamma = 1$, and $M_{\text{tot}} = 1000$ \remove{(setting number 2 from Table)}, but we vary the range of the linear ramp completeness function of Equation~\eqref{eq:ramp}; $C_{\text{min}}$ is fixed at 0.08 $M_{\odot}$ and $C_{\text{max}} \in \{0.10, 0.25, 0.5, 0.75, 1\}$ where low values of $C_{\text{max}}$ result in fewer stars removed from the IMF and, hence, a larger number of stars in the MF.
All five sets of MFs are aged 30 Myr and have log-normal measurement error with $\sigma = 0.25$.
The observational effects, including the differing completeness function upper bounds, resulted in MF's with 800 (70.1\% of IMF stars), 659 (57.7\%), 488 (42.7\%), 415 (36.3\%), and 352 (30.8\%) stars for $C_{\text{max}} = 0.10, 0.25, 0.5, 0.75, 1$, respectively, compared to the original IMF with 1142 stars.

We are interested in the differences among the ABC posteriors and predictive IMFs among the varying $C_{\text{max}}$ values.
The marginal ABC posteriors are displayed in Figure~\ref{fig:abc_pa_posterior_obs}, which also includes the analogous ABC marginal posteriors without observational effects \remove{from Figure}.
Except for the marginal posteriors of $M_{\text{tot}}$, for $C_{\text{max}} = 0.1 \text{ and } 0.25$, the posteriors get broader, which is expected because larger $C_{\text{max}}$ results in fewer observations and greater uncertainty.
However, the marginal posteriors for $C_{\text{max}} = 0.5, 0.75$, and $1$ are quite similar. The marginal posteriors of $M_{\text{tot}}$ in Figure~\ref{subfig:marg_mtot_obs} are all similar and significantly broader than the case without observational effects.  Hence, the observational effects appear to have a profound impact on inference for $M_{\text{tot}}$.  The pairwise joint ABC posteriors are displayed in Figure~\ref{fig:abc_pa_joints_obs} as a reference, and seem to follow the same general patterns noted for the marginals (i.e., they are broader as $C_{\text{max}}$ increases).

Finally, the posterior predictive IMFs are combined into a single plot displayed in Figure~\ref{fig:abc_pa_pred_obs}.  As in the previous section, the posterior predictive IMFs are the pointwise medians of 1000 independent draws from the ABC posteriors of Figure~\ref{fig:abc_pa_posterior_obs}.  Also included in the figures are 95\% credible bands based on the 2.5 and 97.5 percentiles of the 1000 posterior draws.  The true IMF is plotted as a thick yellow line and the corresponding ABC posterior predictive IMF without observational effects \remove{(the red dashed line of Figure)} is also displayed.
The posterior predictive IMFs for $C_{\text{max}} = 0.1$ and $0.25$ overlap well with the true IMF and the posterior predictive IMF without observational effects, but with wider 95\% credible bands.
The posterior predictive IMFs for $C_{\text{max}} = 0.5, 0.75,$ and $1$ have similar shapes and 95\% credible bands.
Their posterior predictive IMFs peak at a higher mass than the others.  These differences are not surprising given that there are far fewer stars below, for example, $-0.5 \log_{10}(M_{\odot})$:
46, 61, and 96 stars for $C_{\text{max}} = 1, 0.75, \text{ and } 0.5$ compared to 225 and 368 stars for $C_{\text{max}} = 0.25, \text{ and } 0.1$, respectively, and 685 stars in that range for the original IMF.

The conclusion drawn from these simulations is that the completeness function affects the resulting inference -- when more stars are missing from the original IMF due to the completeness function, the resulting ABC posteriors tend to be broader to reflect the increased uncertainty.

\begin{figure}[htbp]
   \centering
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/marg_k_obs.pdf}
\caption{ABC marginal posterior for $\lambda^{-1}$}\label{subfig:marg_k_obs}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/marg_alpha_obs.pdf}
\caption{ABC marginal posterior for  $\alpha$}\label{subfig:marg_alpha_obs}
\end{subfigure} \\
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/marg_gamma_obs.pdf}
\caption{ABC marginal posterior for $\gamma$}\label{subfig:marg_gamma_obs}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/marg_mtot_obs.pdf}
\caption{ABC marginal posterior for  $M_{\text{tot}}$}\label{subfig:marg_mtot_obs}
\end{subfigure}
 \caption{Marginal ABC posteriors for the simulation setting of Section~\ref{sec:sim_obs}.  The different color and types of lines indicate the differing upper limits of the linear ramp completeness function of Equation~\eqref{eq:ramp}, $C_{\text{max}}$, corresponding to the weighted kernel density estimates of the marginal ABC posteriors for (a) $\lambda^{-1}$, (b) $\alpha$, (c) $\gamma$, and (d) $M_{\text{tot}}$.
The lower limit, $C_{\text{min}}$, is fixed at 0.08 $M_{\odot}$.
All five datasets started with the same IMF using $\lambda^{-1} = 0.25$, $\alpha = 0.3$, $\gamma = 1$, and $M_{\text{tot}} = 1000$, were aged 30 Myr, and had log-normal measurement error applied with $\sigma = 0.25$.
The solid magenta line is the ABC marginal posterior using an identical IMF, but with no observational effects applied \remove{(i.e. the same ABC marginal posteriors as the red dashed lines in Figure)}, and is included for comparison; note that the vertical axis of (d) does not extend to the full range of this ABC marginal posterior for $M_{\text{tot}}$.
The vertical dotted gray lines indicate the range of the priors for (a), (b), and (c).
   }
   \label{fig:abc_pa_posterior_obs}
\end{figure}

\begin{figure}[htbp]
   \centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_k_alpha_obs.pdf}
\caption{ABC Joint $(\lambda^{-1}, \alpha)$}\label{subfig:joint_alpha_k_obs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_k_gamma_obs.pdf}
\caption{ABC Joint $(\lambda^{-1}, \gamma)$}\label{subfig:joint_gamma_k_obs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_k_mtot_obs.pdf}
\caption{ABC Joint $(\lambda^{-1}, M_{\text{tot}})$}\label{subfig:joint_mtot_k_obs}
\end{subfigure} \\
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_alpha_gamma_obs.pdf}
\caption{ABC Joint $(\alpha, \gamma)$}\label{subfig:joint_gamma_alpha_obs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_alpha_mtot_obs.pdf}
\caption{ABC Joint $(\alpha, M_{\text{tot}})$}\label{subfig:joint_mtot_alpha_obs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/joint_gamma_mtot_obs.pdf}
\caption{ABC Joint $(\alpha, \gamma)$}\label{subfig:joint_mtot_gamma_obs}
\end{subfigure} \\
\caption{Pairwise ABC posterior particles samples of $(\lambda^{-1}, \alpha, \gamma, M_{\text{tot}})$ for the simulation settings of Section~\ref{sec:sim_obs}.  The different color and types of points indicate the $C_{\text{max}}$ values of the linear ramp completeness function, and the size of the plot symbol is scaled with the particle weight.
The lower limit, $C_{\text{min}}$, is fixed at 0.08 $M_{\odot}$.
All five datasets started with the same IMF using $\lambda^{-1} = 0.25$, $\alpha = 0.3$, $\gamma = 1$, and $M_{\text{tot}} = 1000$, were aged 30 Myr, and had log-normal measurement error applied with $\sigma = 0.25$.
The magenta upside-down triangles are the ABC marginal posterior using an identical IMF, but with no observational effects applied \remove{(i.e. the same ABC joint posteriors as the red triangles in Figure)}, and is included for comparison.
}
\label{fig:abc_pa_joints_obs}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=.85\textwidth]{figures/obs_pred_imf.pdf}
 \caption{Posterior predictive IMF for the simulation settings of Section~\ref{sec:sim_obs}.
The thick yellow line is the true IMF, the thicker lines of varying line type and color are the ABC posterior predictive median IMF for the different values of $C_{\text{max}}$,
the thinner lines of the same type and color define a point-wise 95\% credible band  for the $C_{\text{max}}$ with the matching color, and
the shaded regions are the different ranges of completeness (all starting at $C_{\text{min}} = 0.08 M_{\odot}$ indicated by the left end of the arrows).
The solid magenta line is the ABC posterior predictive IMF using an identical IMF, but with no observational effects applied \remove{(i.e. the same ABC posterior predictive IMFs as the red dashed line in Figure)}, and is included for comparison.
The posterior predictive IMFs are based off of 1000 independent draws from the final ABC posteriors.
} \label{fig:abc_pa_pred_obs}
\end{figure}

\subsection{Astrophysical simulation data} \label{sec:bate}

Next we consider a star cluster generated from the radiation hydrodynamical simulation presented in \cite{Bate2012} and published in \cite{Bate2014}.\footnote{The astrophysical data is available at \url{https://ore.exeter.ac.uk/repository/handle/10871/14881}}
This simulation resulted in 183 stars and brown dwarfs with a total mass of the resulting objects of about 88.68 $M_{\odot}$ formed from a 500 $M_{\odot}$ molecular cloud of uniform density.
Understanding that simulations are only an approximation of reality, this astrophysical simulation was implemented to include realistic physics of star cluster formation such as radiative feedback.
The technical details of the simulation are beyond the scope of this work, but can be found in \cite{Bate2012}.
Figure~\ref{fig:bate1} displays the resulting IMF as a density and histogram.

In \cite{Bate2012}, validation of the simulated cluster was carried out by comparing its IMF with the model of \cite{chabrier2005}, and was not able to statistically differentiate them using a Kolmogorov-Smirnov test.  The \cite{chabrier2005} IMF is displayed in Figure~\ref{fig:bate1} as a comparison to the simulation data.  While the general shape does appear to match well, the \cite{Bate2012} data has a small second mode around $1M_{\odot}$.  The \cite{Bate2012} data seems to have more objects on the lower mass end and fewer between 0.5 and 1$M_{\odot}$ than expected with the \cite{chabrier2005} IMF model.
Additionally, because the shape of the low-mass end of the IMF is not well-constrained observationally,  \cite{Bate2012} compares the ratio of number of brown dwarfs to number of stars with masses $<1M_{\odot}$ and finds acceptable agreement with observations.
\cite{Bate2012} also carryout an analysis of the mechanism(s) behind the shape of the IMF.
They found that larger objects have had longer accretion times, while lower mass objects tended to have a dynamical encounter that result in the accretion terminating; hence there ended up being, as \cite{Bate2012} described, a ``competition between accretion and dynamical encounters.''  This competition for material seems consistent with the ideas underlying the proposed PA model.

The 183 objects were used as the observations in the proposed ABC algorithm using 1000 particles, 5 sequential time steps, a $kN$ of $10^4$ (for adaptively initializing the algorithm), and the 25th percentile for shrinking the sequential tolerances based on the empirical distribution of the retained distances from the preceding time step.  The resulting ABC marginal posteriors are displayed in Figure~\ref{fig:abc_bate_posterior}, the pairwise ABC joint posteriors in Figure~\ref{fig:abc_bate_joints}, and the posterior predictive IMF in Figure~\ref{fig:abc_bate_pred}.
The ABC posterior means for $\lambda^{-1}$, $\alpha$, and $\gamma$ are 0.260, 0.537, and 1.091, respectively.  The ABC posterior mean of $\alpha$ is notably higher than the ABC posterior means of $\alpha$ for the \cite{kroupa2001} (0.293) and \cite{Chabrier:2003oq, Chabrier:2003om} (0.304) simulated data discussed in Section~\ref{sec:pa} (see Figure~\ref{fig:otherModels}).  The ABC posterior mean of $\gamma$ is also slightly higher than the 1.050 posterior mean of the \cite{Chabrier:2003oq, Chabrier:2003om} data.
Though the IMF has a slightly irregular shape with a small second mode around 1 $M_{\odot}$ as noted previously, the proposed ABC method's  posterior predictive median and
95\% predictive bands generally fit the IMF shape well.

\begin{figure}[htbp]
\centering
\includegraphics[width = .5\textwidth]{figures/bate_imf.pdf}
\caption{Astrophysical simulation IMF.  The black curve displays the IMF of the 183 stars and brown dwarfs simulated from \cite{Bate2012} and the red dotted curve is the IMF of \cite{chabrier2005}.  The right axis provides the number of stars (and brown dwarfs) for the histogram (plotted in blue).
   }
   \label{fig:bate1}
\end{figure}

\begin{figure}[htbp]
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/bate_marginal_k.pdf}
\caption{ABC posterior for $\lambda^{-1}$}\label{subfig:bate_k}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/bate_marginal_alpha.pdf}
\caption{ABC posterior for $\alpha$}\label{subfig:bate_alpha}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = .95\textwidth]{figures/bate_marginal_gamma.pdf}
\caption{ABC posterior for $\gamma$}\label{subfig:bate_gamma}
\end{subfigure}
\caption{Marginal ABC posteriors for astrophysical simulation data from \cite{Bate2012}.  The vertical dashed red lines indicate the ABC posterior mean, and the dotted gray lines indicate the range of the uniform prior for the parameter.
 }
   \label{fig:abc_bate_posterior}
\end{figure}

\begin{figure}[htbp]
   \centering
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/bate_joint_k_alpha.pdf}
\caption{ABC Joint $(\lambda^{-1}, \alpha)$}\label{subfig:joint_alpha_k_bate}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/bate_joint_k_gamma.pdf}
\caption{ABC Joint $(\lambda^{-1}, \gamma)$}\label{subfig:joint_gamma_k_bate}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\includegraphics[width = \textwidth]{figures/bate_joint_alpha_gamma.pdf}
\caption{ABC Joint $(\alpha, \gamma)$}\label{subfig:joint_gamma_alpha_bate}
\end{subfigure} \\
\caption{
Pairwise joint ABC posteriors for astrophysical simulation data from \cite{Bate2012}.  Pairwise ABC posterior particles samples of (a) $(\lambda^{-1}, \alpha)$, (b) $(\lambda^{-1}, \gamma)$, and (c) $(\alpha, \gamma)$ for the astrophysical simulation data from \cite{Bate2012}.  The size of the plot symbol is scaled with the particle weight.
}
\label{fig:abc_bate_joints}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{figures/bate_pred_imf.pdf}
 \caption{Posterior predictive IMF for astrophysical simulation data from \cite{Bate2012}.
The median of the posterior predictive IMF (solid black) with a corresponding 95\% point-wise predictive band (dashed black) compared to the true IMF (blue dotted).  For the posterior predictive IMF, 1000 draws were made from the final ABC posteriors and then 1000 cluster samples were drawn from the proposed forward model.
} \label{fig:abc_bate_pred}
\end{figure}

\section{Discussion}
\label{discussionSec}

Accounting for the complex dependence structure in observable data, such as the initial masses of stars formed from a molecular cloud,
is a challenging statistical modeling problem.  A possible, but unsatisfactory, resolution is to proceed as though the dependency
is sufficiently weak that an independence assumption is acceptable. Such approximations can be reasonable at small sample sizes, but
are often revealed to be insufficient by modern data sets.
Instead, we draw on PA models, proposing a new forward model for star formation.
Though the new generative model was motivated by inference on stellar IMFs, the general concept is
generalizable to other applications. Simulation-based approaches to inference, including ABC, allow for inference
with such models.

The new generative model starts with the total mass of the system and stochastically builds individuals stars of particular mass at a sub-linear, linear, or super-linear rate.
A goal of the proposed model and algorithm is to begin making a statistical connection between the observed stellar MF and the formation mechanism of the cluster, not that the proposed model shape is superior to the standard IMF models.  Rather, the proposed model is more general in the sense that it
captures the dependencies among the masses of the stars by connecting the star masses to a possible cluster formation mechanism, and also can accommodate standard models proposed in the astronomical literature.
Additionally, by coupling the proposed model with ABC, observational limitations such as the aging and completeness of the observed cluster can easily be accounted for.
Code for running the proposed ABC-IMF algorithm is available at \url{https://github.com/JessiCisewskiKehe/ABC_for_StellarIMF}.

\new{In agreement with other studies that have implemented ABC algorithms (e.g. \citealt{WeyantEtAl2013,IshidaEtAl2015}), we found the selection of informative summary statistics to be a crucial, but challenging step in the algorithm development.  In the IMF setting, we had initially considered a number of different possible summary statistics, but it became clear that matching the shape of the IMF was important to constrain the parameters (along with the number of stars generated in the cluster).
To assess the similarity between the observed and simulated IMFs, the $L_2$ distance was effective, but we imagine that other functional distances could also work well.  In future applications of ABC, practitioners may find it useful to consider functional summaries and distances if the setting allows for it.
To reach these conclusions, it required us to create a simplified setting where the true posterior was available; when possible, we suggest others consider this when trying to select useful summary statistics and distance functions.
}

While the proposed model is able to account for a particular dependency among the masses during cluster formation, there are several extensions that would be scientifically and statistically interesting.   First, the generative model could be extended to capture the spatial dependency among the observations.
Intuitively, such an approach could account for a mechanism that limits the formation of multiple very massive stars relatively near to each other.
Understanding the spatial distribution of masses of stars during formation would help advance our understanding of stellar formation and evolution.
Other effects that could be incorporated into the generative model include accounting for binary and other multiple star systems, the possible disturbances to the observed MF as stars die (beyond the censoring of the most massive stars), or spatial completeness functions (i.e. a completeness function that depends on not only the mass of the object, but also its location in the cluster).

Hence, the proposed generative model, used in conjunction with ABC, provides a useful framework for
dealing with complex physical processes that are otherwise difficult to work with in a
statistically rigorous fashion. As increasing computational resources allow for greater model
complexity in astronomy and other fields, the proposed and other ABC
algorithms may open new opportunities for Bayesian inference in challenging problems. There appears to
be significant potential to extend this approach to even more complicated situations.

redpajama_set_name: RedPajamaArXiv


Fairhaven's newest music CD is in production. The finished product is due for release at our Preaching Conference and College Days in April. The CD, entitled Throw Out the Lifeline, features the Fairhaven men's quartets and is sure to become a "must have" in your music library!


"Herps" is the commonly used term for reptiles and amphibians which comes from the word, herpetology, which is the study of these two groups. Whether scared, curious, or uncertain about these "creeping" animals, you may encounter them and want to know exactly what it is that you've found.
Is a turtle laying eggs in your yard? Is that a lizard or a salamander in my garden? Is that snake dangerous? All these questions are pretty common and we are here to help! We have several field guides available in our online Herp Shop that are useful for identification.
Need to learn about our six venomous snake species in NC? The NC Venomous Snake Booklet is available as a free PDF download.
Once you identify the herps you found in North Carolina, please help us out by entering them into HerpMapper.
Download the mobile app for HerpMapper and log your finds wherever you go.
Still unsure? Fill out the form below with as many details as possible and we will get back to you soon.


It was raining cats and dogs in Manila last July 8 so I decided to stay home. At #Doctors20 I had announced that I might revive my Facebook page after a long pause as internet.org now provides free access via one major Philippine telco.
I'd read a lot of tips on how to engage with more people on Facebook pages. There's one tip I had hesitated to try – Invite questions. I figured I had the time then because of the bad weather and did just that. Wow, I received about a hundred questions!
I used Canva to make the poster. I'd discovered it a few days earlier thanks to a Facebook post by Remo (@bonedoc) about building a social media empire.
Dr. Willie Ong shared a few of my posts and my reach exploded. I call that the #DocWillieOng effect. Dr. Ong is a cardiologist and author of several health books. He's a columnist who also shares health tips on his Facebook page, which as of this writing has 689K likes! Because of his help, I peaked at 1.4 million post reach last week. I'm also indebted to him for tips on how to manage a Facebook page.
What are my top posts so far?
No. 1 is an infographic about what happens to the body with one sip of cola.
No. 2 is a post talking about the risk factors for diabetes. I write all my posts in Filipino. The images are copyright and royalty free from Pixabay. Some photos are available on Canva for $1 each single use.
No. 3 is a post explaining why blood sugar rises in diabetes. This was challenging not only to write in Filipino but also to explain in simple terms concepts such as insulin resistance and insulin deficiency.
No. 4 is a post enumerating the symptoms of diabetes. I was able to use the old photos I had when I started this Facebook page in 2012.
No. 5 is a post about the symptoms of hypothyroidism. There were many questions asked here that I could hardly keep up and answer everyone.
I try to post at least once a day. I'm learning how to answer questions, which improves my communication skills. Someday, I'd like to write a paper about the questions the patients ask. I think we can learn a lot from these. Often the patients ask about things that they were unable to ask their own doctors. They also needed clarification about what the doctor explained to them that they could not understand. Many questions were from overseas Filipinos who did not have access to healthcare.
I also had to set my own limits about what queries I can or cannot answer as this was NOT by any means an online consultation. I learned to be patient with those who shared my posts to advertise products with no therapeutic claims. I weeded out comments sharing links to dubious products. I even had to ask one group selling herbal supplements to desist from using my posts on their page.
The most difficult question which I leave unanswered on my Facebook page is where my clinic is. I've asked some friends about this. They advised me to reply by private message about my clinic location and hours. But I rarely do so. Sometime ago I wrote about the ARK, Act of Random Kindness. I'd like to think that this FB page is my daily ARK. #ParaSaBayan. Though people may think otherwise, I did not set up this page to gain more patients. The lines at my clinic are more than long enough. I've said it before, I haven't solved long waiting times yet.
I am grateful to those who have left messages of thanks and encouragement for me to continue with this.
Thanks again for your great job and please please never stop doing it!!!
Lots of kisses from Spain and hope to see you in another event!!!
Thanks so much Nuria! You have no idea how your simple message has warmed my heart and given me strength to continue 🙂 yes, hope to see you again!


Click on the available unit types to assemble your army and hit any key to send them off to battle! Units change color during battle to indicate how well they match up against whichever opponent they come in contact with. Disclaimer: Comes with no sound and only one level due to the hurried nature of production; I had about 8 hours to put this one together. But hey, it works!
I know it was an accident, but i'm afraid you won't get too many ratings having compiled an installer. Most people won't install jam games.
I did play this however, and it appears to be an interesting concept. I didn't understand how it related to the theme though. The UI also looked very stretched and hard to read the giant pixels. Clean this up and it could be a decent strategy game!
Definitely don't use an installer next time, it's much easier as a player to use the straight executable.
I'm guessing you had far loftier goals than you were able to pull off in the time you had, as the gameplay is pretty limited. I tried out a few different combinations of units/equipment but the combat happens so quickly it really wasn't clear what was going on, nor what the colors really mean. A narrower focus (more directly related to the theme) might have helped you achieve more in the limited time!
I still installed your game, though. I don't really see how this game has anything to do with the theme, there is no music, no sounds, not even any gameplay at all - You just watch a couple of people fight and have no influence on the battle at all.
As for the rest, could only do so much in the 8 hours or less I had.
Combine your arms correctly to win the battle!


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Queen: L 120 in. / D 86 in. / H 42 in.
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Other wood and stain options available – please inquire.


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All paper card pieces come pre-cut, make the mask by yourself with the installation instructions.
Need a tape or gluing tabs to build it (not including).
Robust polygon construction to survive the wildest of parties. Ideal for festivals and parties.
Easy assembly in 2 hours or less.
All paper card pieces come pre-cut, assemble them by yourself with the installation instructions.
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text:
Laser Periscope Detection
NAVAIR - Ms. Donna Attick - donna.moore@navy.mil
Opens: January 8, 2018 - Closes: February 7, 2018 (8:00 PM ET)
N181-022
TITLE: Laser Periscope Detection
TECHNOLOGY AREA(S): Air Platform, Ground/Sea Vehicles, Information Systems
ACQUISITION PROGRAM: PMA 299 (ASW) H-60 Helicopter Program
OBJECTIVE: Develop a technology for long-range detection of periscopes in maritime environments using a high-power Short Wave Infrared (SWIR) laser to detect submarine periscopes and submarine optics masts from aircraft at both low and high altitudes.
DESCRIPTION: The need exists for improved periscope detection and discrimination between periscopes and other objects. An innovative laser periscope detection system would be complementary to the existing radar periscope detection capability to provide cueing and reduce false alarms. Combining data from the current radar detection algorithm with data from an optical detection algorithm will increase the probability of detection, and reduce the probability of false alarms during a mission. The laser periscope detection system should have a standalone mode for aircraft without radar for detecting periscopes. The laser periscope detection system should be integrated with existing or planned future laser systems, including Electro-Optical and Infrared (EO/IR) systems, to maximize capability while reducing the space, weight, and power (SWaP) of the combined system compared to the total for separate systems. Integration may include housing both systems in the same pod, utilizing the same environmental systems and using the same scanner and receivers where possible, data recorders, and displays.
The system should be designed to be rugged, compact, and lightweight enough to be used in naval aircraft, both fixed- and rotary-wing platforms such as the P8 and the MH-60 ROMEO and SIERRA in accordance with the applicable Military Standards (specified and included in the references). It is therefore the goal of this program to seek the development of a power-scalable laser system solution that will meet the size, weight, performance, and reliability requirements below while considering component costs for future production of the system. The proposer should consider this development as the innovative advancement and combination of laser and supporting technologies towards the goals stated below.
The performance objectives of the laser solution are:
1. Repetition rate, Threshold: 100 hertz
2. High peak power, Threshold: 40 milli-joules/pulse, 5-7 nanosecond pulse width, (low power should be used during design)
3. Wavelength: (1-2 micro-meters)
4. Line width: less than or equal to 0.1 nanometer
5. Laser beam quality M-squared less than 3.
6. Lightweight. (Total weight including the laser head, cooling system, power supply, and control system) Threshold: less than 100 pounds, Objective: less than 60 pounds.
7. Small volume. (Total volume for the cooling system, power supply, control system and laser head) Threshold: less than 3 cubic feet, Objective: less than 2 cubic feet.
8. Ability to be ruggedized and packaged to withstand the shock, vibration, pressure, temperature, humidity, electrical power conditions, etc. encountered in a system built for airborne use.
9. Reliability: Mean time between equipment failure—300 operating hours.
10. Full Rate Production Cost: Threshold <$50,000; Objective <$15,000 (based on 1,000 units)
11. Detection Range: >15 Km (Threshold), >25 Km (Objective)
12. Range Accuracy: 10m
13 Azimuth Accuracy at 15Km: 100m
14. Power: <400 W
15. Wall plug efficiency: Threshold >3%, Objective >5%
16. Field of View: 40 degrees (Threshold), 140 degrees (Objective)
17. Photo receiver Sensitivity: TBD
18. Beam Divergence: 100 mrad < Theta < 215 mrad
19. Probability of detection at 25Km: Threshold: 0.8 Objective 0.9
20. Probability of detection at 10Km: Threshold: 0.9 Objective 0.95
21. Probability of false alarm at 25Km: Threshold: once per day
Due to SWaP, ruggedization requirements and restricted use of hazardous material in airborne applications, the following will not be accepted: argon ion lasers, chemical lasers, and dye lasers. Furthermore, systems using cryogenic cooling will also be discounted.
PHASE I: Determine and design a viable and robust laser system solution consisting of a single laser that meets or exceeds the requirements specified. Identify technological and reliability challenges of the design approach, and propose viable risk mitigation strategies. The Phase I effort will include the development of prototype plans for Phase II.
PHASE II: Design, fabricate, and demonstrate a laser system prototype based on the design from Phase I. Test and fully characterize the system prototype.
PHASE III DUAL USE APPLICATIONS: Finalize the design and fabricate a shock resistant laser system solution and assist to obtain certification for flight on a NAVAIR R&D aircraft. High-power, pulsed lasers have applications in manufacturing and lithography. Oceanographic bathymetry systems for survey and exploration work would benefit greatly from this laser system solution.
1. Saleh, B.E.A. Fundamentals of Photonics. John Wiley & Sons, Inc., 1991. https://www.researchgate.net/profile/Bahaa_Saleh2/publication/228109642_Fundamental_of_Photonics/links/0046352bada6cf0ab6000000/Fundamental-of-Photonics.pdf
2. MIL-STD 1399-300A: Electric Power, Alternating Current (metric). http://everyspec.com/MIL-STD/MIL-STD-1300-1399/MIL_STD_1399_300A_647/
3. MIL-STD 461 E: Requirements for the Control of Electromagnetic Interference Characteristics of Subsystems and Equipment.https://snebulos.mit.edu/projects/reference/MIL-STD/MIL-STD-461E.pdf
4. MIL-STD 810G: Environmental Engineering Considerations and Laboratory Tests.http://everyspec.com/MIL-STD/MIL-STD-0800-0899/MIL-STD-810G_12306/
5. MIL-STD 464 A: Interface Electromagnetic Environmental. http://everyspec.com/MIL-STD/MIL-STD-0300-0499/MIL-STD-464A_21938/
6. MIL-STD 8591: Airborne Stores, Suspension Equipment and Aircraft-store Interface. http://everyspec.com/MIL-SPECS/MIL-SPECS-MIL-A/MIL-A-8591H_10997/
KEYWORDS: Airborne; Anti-Surface Warfare; Periscope Detection; Maritime Environment; Anti-Submarine Warfare (ASW); Laser Detection
** TOPIC NOTICE **
These Navy Topics are part of the overall DoD 2018.1 SBIR BAA. The DoD issued its 2018.1 BAA SBIR pre-release on November 29, 2017, which opens to receive proposals on January 8, 2018, and closes February 7, 2018 at 8:00 PM ET.
Between November 29, 2017 and January 7, 2018 you may talk directly with the Topic Authors (TPOC) to ask technical questions about the topics. During these dates, their contact information is listed above. For reasons of competitive fairness, direct communication between proposers and topic authors is not allowed starting January 8, 2018 when DoD begins accepting proposals for this BAA.
However, until January 24, 2018, proposers may still submit written questions about solicitation topics through the DoD's SBIR/STTR Interactive Topic Information System (SITIS), in which the questioner and respondent remain anonymous and all questions and answers are posted electronically for general viewing until the solicitation closes. All proposers are advised to monitor SITIS during the Open BAA period for questions and answers and other significant information relevant to their SBIR/STTR topics of interest.
Topics Search Engine: Visit the DoD Topic Search Tool at sbir.defensebusiness.org/topics/ to find topics by keyword across all DoD Components participating in this BAA.
Proposal Submission: All SBIR/STTR Proposals must be submitted electronically through the DoD SBIR/STTR Electronic Submission Website, as described in the Proposal Preparation and Submission of Proposal sections of the program Announcement.
Help: If you have general questions about DoD SBIR program, please contact the DoD SBIR Help Desk at 800-348-0787 or via email at sbirhelp@bytecubed.com

redpajama_set_name: RedPajamaCommonCrawl


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Home News World was 'lucky' with swine flu: WHO chief
World was 'lucky' with swine flu: WHO chief
World Health Organisation chief Margaret Chan said Monday that the world had been "lucky" with the swine flu pandemic after the virus failed to mutate to a more deadly form.
Chan told representatives of the WHO's 193 member states that "good news" in public health terms was normally driven by political commitment, resources and the ability to deliver care.
"Sometimes, though, we are just plain lucky. This has been the case with the H1N1 influenza pandemic," she said at the opening of the WHO's annual assembly.
"The virus did not mutate to a more lethal form," while the vaccine worked and it showed little resistance to anti-viral drugs, she added.
Chan acknowledged that "few health systems were overwhelmed and the effects were usually short-lived," with less disruption to everyday lives or travel than feared.
"Had things gone wrong in any of these areas, we would have a very different agenda before us today," she added.
Some health experts and politicians have criticised the WHO-led global effort against 2009 H1N1 flu after the new virus was uncovered in Mexico and the United States in April 2009, claiming there was an over-reaction.
Swine flu was declared a pandemic as it rapidly spread around the world, prompting massive spending on specially developed but largely unused swine flu vaccine and emergency precautions.
Pandemic flu has left 18,030 people dead since the virus was uncovered, according to latest WHO data.
A year ago, the WHO chief was warning about the threat of a growth in severe cases of the "subtle, sneaky" new swine flu virus, especially in developing nations, just weeks before she followed scientific advice and declared a pandemic.
An independent committee of experts set up by the WHO is investigating the international response to the swine flu pandemic. It is due to deliver its findings by the beginning of next year.
Chan reiterated on Tuesday that she welcomed the review and its examination of criticisms that have been levelled at the WHO.
"We want to know what went wrong and, ideally, why. We want to know what can be done better and, ideally, how," she added.
The WHO assembly runs until Friday, and is due to examine an array of global health issues.
SWISS pilots approve new collective labour agreement
Switzerland's largest 'open air' solar farm planned at Bern Airport
UN urges charges dropped in Greek migrant rescue trial
Greenpeace blasts number of private jets heading to WEF


LINK is a fun and addictive word game for iOS and Android. Connect the letters to form words until you fill the entire grid. Create puzzles and share them with your friends. Includes hundreds of built-in puzzles for hours of fun.


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NBA Trade Scenario
Understanding the 76ers' Strategy
The Doug Collins Affect
Best Minds in Basketball
ESPN and Their Philly Hate
More Pessimism…More Roster…More Review
A Pessimistic Roster Review
Bill Belichick: The NFL's Barry Bonds
Did DeSean Jackson Make LeSean McCoy?
The Phillies Maintain Their Disarray
DH for Life
Why Angelo Cataldi and the Old School Way of Thinking is Outdated
Back into Your Fantasy Life!
Break Time: The Walking Dead!
Fantasy Team!
A Fan's Responsibility
Where does responsibility lie.
I am going to use a comparison to start off this post, but I swear it will be quick and painless. I'm prefacing my comparison because, well, if you're anything like me, you would rather explain why your girlfriend's pants make her look incredibly fat and kick back during the ensuing hailstorm of fury than hear anything more about the Dolphin's bullying story. But it works so nicely for what I'm about to say and it is so insanely relevant that I just couldn't resist.
For those of you who have been living under a sports rock for the past two weeks or so, while I envy you, I'll recap the story quickly. Richie Incognito is an offensive guard for the Miami Dolphins who has a REALLY bad reputation around the league as just not a good guy. Jonathan Martin is a second year offensive tackle who also plays for the Dolphins and has some confidence problems (I'm assuming that, but it's subjective anyways so let's just roll with it). Every veteran in the league makes rookies carry their pads and pay for insanely large dinner bills for the team, as a way of initiating the rookies into the league. Incognito, being the truly not good person that he is, (again, subjective, get off my case), took the whole rookie hazing thing WAY too far. Martin wasn't even a rookie any more and Incognito was still hazing him and sending him horrible and racist texts that, even if he was friends with Incognito, are despicable. Just imagine, sending anyone you work with a racist and horribly degrading text about that person's family, and tell me just how good of friends you two are after that text. Well, Martin left the team because, apparently, he can stand up for himself without immediately resorting to violence against someone he works with. Incognito got removed from the team, and he is pretty upset that his "friend" Martin is letting Incognito take all this blame for being a so called bully. Alright, that is about as much as I can stand, hope your caught up.
All of this is just background story to the real story. Whose fault is this bullying case? The obvious answer is Incognito, the guy who actually bullied the younger player. But then again, Martin should be held responsible for this whole situation since he never stood up for himself and just fought back. I mean, honestly, why didn't he just punch Incognito in the face?
Yes, this Dolphin's argument is really stupid. So, I'm going to quickly switch over to what actually matters, the Eagles. Our Birds are going into the Redskins game without a home win in over a year. Now, most sane people would say that the team needs to play better at home. However, when one actually looks into the situation, doesn't it make more sense to blame the fans. I mean, granted, the fans pay for their tickets which support the team (which that support is only a fraction of the percent to which we tirelessly offer our heart and soul to the success and failure of this team), but it really is our fault that the Eagles haven't won at home in over a year. Yes, every game has been sold out. Yes, we have been there in rain, snow, freezing cold, for hours on end during seemingly endless games which the Birds might as well have not shown up in order to spare us the misery of yet another blowout loss. These aren't the reasons why the Eagles can't win at home. No, it's because we boo. When something goes wrong, when our offense goes three and out early in the game or when the defense can't get off the field, the players are worried that we are going to start booing them. The players feel like they aren't only playing against the opponent, but also their own fans. It's the fans who are to blame for this home losing streak, we have been putting too much pressure on the players. (As I'm reading over this, I am really fighting the urge to go on a tangent and explain how stupid that argument is since our road record, where the team actually has to face the other team and the crowd, is pretty good)
Am I the only person who sees this as insulting. We don't play the games, we don't scout the other teams (well, not professionally), and we pay to be at the game instead of the other way around. Oh, and by the way, everything that happens on Sunday is our escape from our reality. Sunday is where we as a city get to forget our jobs, our personal problems, our concerns, and come together as a city. One city, bonded through the passion for one team. (I sound like a Sixer's commercial under Doug Collins)
When did we start to blame the victim for what the bully did to them because the victim should have just stood up for themselves? When did winning become the fans responsibility? Yea, I get that this isn't the cleanest of comparisons, but I hope your starting to get the main picture behind all of this. I know that most of the players are not blaming the fans for the losing streak, just as I know most people don't blame the kid getting picked on. The point, in both of these situations, is that there are some people who do truly believe that. I don't think I will ever understand that side of the argument.
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When photographing movement, you have a couple of options on how to capture it: either freeze the motion completely, or allow the subject to show a bit of motion blur.
So, this week's blog post is all about how to capture movement with motion blur, as usual, broken down step by step.
Before we dig in, make sure you download the free motion blur challenge that goes along with this blog post. Remember, taking action and putting into practice what you are learning will give you far more bang for your buck than simply reading about ti! Click on the image below to download.
Since you will be letting in a lot of light through your shutter by slowing it down, you will probably need to use a higher aperture number (thereby giving you a smaller aperture to let in less light) and a low ISO number (ditto).
If that's not enough to help you get a slow enough shutter speed, you will need to either seek out lower light situations, or use a polarising filter which helps cut down on the amount of light hitting the sensor - but you can normally get your shutter speed low enough without too much problems, unless it is a bright sunny day for example.
Alrighty, since we have our Aperture and ISO set, the next thing we need to do is set our shutter speed.
How fast your subject is moving, and how much motion blur you want, will determine which shutter speed you should use when you want to capture motion blur.
For example, if you were to shoot a moving car at 1/200 you would probably get some motion blur, but if you used that exact same shutter speed on a person walking you probably wouldn't.
Secondly, it also depends on how much blur you want the moving object to have.
For example, do you want just the hint of blur around the edges? Or so much that they are a complete blur with no distinguishable features?
Remember, it's your photograph, so you get to decide!
Because there are a few factors to it, unfortunately there is no "magic" number I can give you for shutter speeds to use. However, as I never want to leave you without at least some guidelines to get you started, so here are some suggested speeds to start with.
Please note you will definitely, 100%, need to adjust up or down from this depending on how you want the scene to look, and the speed of your subject.
For this way of capturing motion to work well you need to make absolutely sure that you keep your camera perfectly still, otherwise the whole scene will be blurred and not just the moving subject, and that's not what we want!
That's why you pretty much absolutely need to use a tripod for capturing motion blur, so if you have one, now is the time to dig it out.
If you don't have one, it can be a bit more tricky, but still doable. The trick is to make sure that your camera is absolutely stationary, maybe by having it sit on a still object like a table.
If neither of those are possible, make like a stealth ninja and aim to keep yourself perfectly still, and try not to use shutter speeds under 1/60 (which means not getting things like light trails), or it won't work.
As I really prefer my posts to be actionable if I possibly can, I have a little assignment for you.
Find a moving object that you can practise on: this could be a person walking or running, a motorbike on a road, a cyclist, a spinning wheel - anything that has motion.
Next, try to capture intentional motion blur of the moving subject using the guidelines above.
Remember, you only want the moving subject to be blurred, not the rest of the scene - this is why a tripod is so incredibly useful!
Experiment with different shutter speeds to see the different effects you can get when you slow your shutter speed down completely, compared to it being not quite just fast enough.
I've made a little download for you that has a challenge for you with motion blur - just click on the box below to get it delivered to your inbox!


Looking for tips on how to feed your cattle during a drought? Are your hay supplies adequate during this dry season? Want to estimate your cattle rate-of-grain? Interested in alternative feeds for fall?
Click here to read our Fall Feed Guide from Purina.
WASHINGTON — The Agriculture Department said Friday that consumers could still expect higher food prices next year, but the expected 3 percent to 4 percent increase was unchanged from last month, even as extreme heat in the Farm Belt continued to reduce the grain harvest and increase feed prices for livestock.
"The data out this morning shows that nothing much has changed," said Ephraim Leibtag, deputy director of research at the department's Economic Research Service.
The prices of beef and veal will go up the most next year, the report said, almost entirely because of higher costs for feed, which is made from corn and other grains. Beef and veal prices are expected to increase 4 percent to 5 percent.
The most immediate impact of the drought will be seen this year in poultry prices, the government predicted.
Government forecasters reiterated their projection of a 3.5 percent to 4.5 percent jump in chicken and turkey prices this year. The prices are expected to rise a bit less next year, in the range of 3 percent to 4 percent, the government said.
The increase in feed prices has caused many cattle producers to sell off or cull their herds. Cattle sales are up nearly 30 percent over last year, said Sara Baker, a spokeswoman for Sageworks, a financial information company based in New York that tracks the sales.
The Agriculture Department said the reduction in the number of cows could result in a temporary decline in beef prices this year as a surplus of cattle is sold and more meat enters the market.
Cheese and milk products are expected to rise 3.5 percent to 4.5 percent in price next year, unchanged from last month's forecast. Egg prices will also be affected, with prices forecast to rise 3 percent to 4 percent next year.
About 40 percent of the corn crop is used to feed livestock. Feed costs account for 50 percent to 70 percent of livestock and poultry production, according to the American Feed Industry Association, a trade group.
A government estimate released this month said that because of worsening drought conditions, farmers would produce about 10.6 billion bushels of corn this year, down from what had been projected at the beginning of the year to be a record 15 billion bushels. The reduction in corn and soybean supplies has pushed up their prices to record levels.
The Agriculture Department said this week that 85 percent of this year's corn crop was in drought-affected areas, as was 83 percent of the soybean crop.
As of Monday, just 23 percent of the corn crop was rated in good or excellent condition, while 31 percent of soybeans were rated good or excellent.
An earlier version of this article misspelled the given name of a deputy director of research at the Agriculture Department. He is Ephraim, not Ephrain, Leibtag.
Don't forget to visit Eagle Hardware for for license and everything else you need before the season begins.
We will take orders for Rye grass seed starting on September 1. We'll have Jumbo, Oregon, and TAM seed this year. We expect the demand to be quite high this year due to the drought. Place your order early to ensure that you receive the quantity and type of seed you want. Questions? Give us a call at 972-635-7878.


In the current economic climate, SaaS customers often deliberately delay payment of invoices. In order to protect your SaaS business and improve your cash flow, you should consider including the following in the terms of your SaaS agreement.
If you use agents in relation to your SaaS agreements, when you terminate the agency agreement under the Commercial Agents Regulations 1993 you must pay the agent compensation, regardless of the reason for the termination.


Dating Pisces zodiac sign is a daunting task. They are passionate hungry and concealed individuals. They crave love but will not directly ask for it. They live in a reality but they bubble far away from the domain of facts. Impressing them can be a tough task. Getting involved with a Pisces emotionally requires one to have a large heart and patience.
Proved hopeless romantics, Pisces might pull an active front in rejecting one, but deep inside they are dying for affection and rescue. Impressing such a woman or man requires tremendous attraction. They have a knack for inventiveness. Pisces star sign are good when escaping the reality and venturing into their wild imaginations.
Pisces zodiac sign are often artists, writers or painters. They perform well in tasks that are always evolving compared to same procedure of different day jobs. They are faithful and devoted partners. Pisces in love enjoy long serious relationships. In the event of a breakup, they do not drown with the next option. They are also faithful and tend to stick faithfully to one partner once they fall in love. Dating Pisces man or dating Pisces woman can be an amazing experience.
Pisces are weak willed. They give up too quickly. When they direct their efforts and heart into a venture that seems to fail, they give up and get upset. They are weak over things that makes them to feel pain quickly. This pain will be reverted to the relationship since they end up hiding it deep instead of sharing with their spouses.
Above all, Pisces are suckers for romantic gestures. If you are wondering how to date Pisces, a typical romantic gesture should be sincere. For instance flowers, candle, wine or poetry will do the trick. Reciting a romantic poem or singing a favorite song should also do the trick while dating the Pisces woman.
So, Why Date A Pisces?
They are highly adaptable. Pisces get acquitted with new situations and areas very fast. They are comfortable in unfamiliar surroundings, unlike the other star signs. They are functional and prosper out of their comfort zones.
*Pisces display a high sense of devotion to family and their spouses. They are ride or die partners once you know how to get a Pisces to commit*.
This sun sign will rarely cheat or leave one for someone 'better'. Additionally, they sincerely care about people they love and will do anything in their power to sustain the relationship. They are compassionate. Pisces will give more than they get from a relationship. If you are dating Pisces man, they are good supportive partners who will take on your problem as theirs. They will be involved in your suffering until you arrive at a profitable solution.
When building a conversation with a first date with Pisces, try to revolve around dreams and fantasies. They are idealistic dreamers who value talk about the plans and goals regardless of how idealistic or unreal they are. Additionally, discussion on mystical and spiritual power could interest them profoundly. Talk artistically, they are into arts. Pisces are great lovers of drawing, music and literature. These are their most significant pass time.
Pisces enjoy deep fictions conversations. They are believers of aliens, space, the universe and strange ocean animals. Recipe and sports discussion will not cut the edge when it comes to dating them. They enjoy mystery. Additionally, when dressing for the first date with a Pisces, try something creative but also comfortable. The venue for the first date with Pisces should either be a concert, a museum or a literature café.
Pisces have a tendency of being over-sentimental. They get emotionally worked out when dealing with relationship issues. This overflow of emotions might project to a miscommunication or a disagreement. Therefore, if you are wondering how to date Pisces man or date Pisces woman, yous should be supportive to the core. Pisces are lazy and lack motivation. They have a laid back approach to dealing with issues that affect their lives. They have a hard time going the extra mile to make things work.
When dating Pisces, you should be a good decision maker. They are terribly indecisive and have a hard time making up their minds. They are the biggest pessimists. They emit negative attitude even in their relationship, especially when having a bad day. They have a 'half glass' approach to life. Therefore, caution should be taken to shake this feeling off.
When dating Pisces, a selfless partner is hard to get. Building a relationship requires selflessness. This act requires putting other people before your need; Pisces are good at this. They will sacrifice their time to be there for you in your darkest hour. Read the below dating tips on how to get the Pisces to commit to the core.
Pisces zodiac sign enjoy fictional and fantasies. When making a conversation with them, try to avoid disapproving them. Although at first, they might sound like a bunch of conspiracy fanatics, they are fun to be around. Furthermore, they are indecisive. Try being supportive in making relationship decisions instead of laying out criticism.
When on a quest to win a Pisces heart, try connecting with them first. They are big in conversation, being an active listener would do one great. Not only should you listen to their stories but also chip in a contribution once or twice. Most importantly, avoid interjecting his or her ideas that should help build a dialogue that lasts.
Pisces in love are shy individuals. They might crave all the attention and love in the world, but they value their space. The little art, writing or painting is the center of their lives. One should avoid being possessive of their time and space. Moreover, before she or he trusts you, avoid stalking.
Pisces are intuitive and will always trust their guts. When dating Pisces, always give them their space to think things. They might have problems making decisions. However, you should support them. Pisces are emotional. It is wise not to jeopardize how they feel by hurting them. They show a high tendency to begin crying. A Pisces is a daydreamer. They have a lazy streak in life courtesy of sometimes sleeping till the afternoon. Embracing positive criticism could go a long way to saving the Pisces compatibility.
Pisces do not operate well under pressure. With the slightest sign of failure they usually give up. They quickly give up good projects or pursuits over feelings of helplessness. They are furthermore afraid of the extra mile. Engaging them in an outdoor activity will make them explore nature; something that might be new to them. Furthermore, they are good outside their comfort zones.
In the event you cheat on him or her, they will walk without looking behind. They are highly intuitive and can sense rejection quickly. Despite having a soft spot for open displays of romance, Pisces are also good at reciprocating the love back. They make excellent and loving sexual partners in bed.


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redpajama_set_name: RedPajamaGithub