text:
\section{Introduction}
Directional data analysis is concerned with statistical analysis on
various non-Euclidean manifolds, starting with circle and the sphere,
and extending to related manifolds. Comprehensive monographs are
available for statistical analysis in this setting; see,
e.g., \citet{FLE87,Mardia-Jupp00, Chikuse03}.  However, the subject of
simulation has received much less coverage, with the key contributions
scattered through the literature.

The need for effective simulation methods has grown in recent years as
directional distributions have become components in more
sophisticated statistical models, which are studied using MCMC methods.
For example, \citet{Green-Mardia06} used the matrix Fisher
distribution for random $3 \times 3$ rotation matrices in a
Bayesian model to align two unlabelled configurations of points in
$\mathbb{R}^3$, with an application to a problem of protein alignment
in bioinformatics.

In general there are suitable direct methods of simulation, especially
methods based acceptance rejection, for the simpler directional
models.  However, it is necessary to resort to cumbersome MCMC methods
for the more complicated distributions.  The purpose of this paper is
to extend availability of acceptance rejection methods to a wider
class of directional distributions.  The starting point is a new
acceptance rejection method for the Bingham distribution, which can then
be used as a building block in a wider range of applications.

The paper is organized as follows.  Following some background and
preparation in Section 2, the new acceptance rejection simulation
method for the Bingham distribution is proposed and analyzed in
Section 3.  Special cases and extensions are covered in Sections 4 and
5.  Finally Section 6 sets the results of this paper in context by
reviewing the literature and summarizing the best available methods in
different settings.

The unit sphere $S_p = \{x \in \mathbb{R}^{q}: \ x^Tx = 1\}, \ p \geq
1$, comprises the unit vectors in $\mathbb{R}^q$, where throughout the
paper $p$ and $q$ are related by $q=p+1$.  The surface area of $S_p$
is given by $\pi_q = 2 \pi^{q/2}/\Gamma(q/2)$ and the differential
element of surface area can be written as $[dx]$.  Thus the uniform
distribution on $S_p$ can be written as $\pi_q^{-1} [dx]$.  A more
explicit formula can be given using polar coordinates.  For example,
the circle $S_1$ can be parameterized by $\theta \in [0, 2 \pi)$ with
uniform measure $d \theta /(2\pi)$.  The sphere $S_2$ can be
parameterized by colatitude $\theta \in [0,\pi]$ and longitude
$[0,2\pi)$ with uniform measure
\begin{equation}
\label{eq:unif-sphere}
\sin \theta d \theta d\phi /(4\pi).
\end{equation}

Strictly speaking a probability density on a manifold is a density
with respect to an underlying measure.  In Euclidean space
$\mathbb{R}^p$ the underlying measure is usually taken to be Lebesgue
measure $dx$ without explicit comment. But on other manifolds more
care is needed.  This paper is concerned with spheres and related
compact manifolds for which there is a natural underlying uniform
measure with a finite total measure.  To avoid repeated occurences of
normalizing constants such as $\pi_q$ and differential elements such
as $[dx]$, all such probability densities will be expressed with
respect to the uniform distribution. Thus we will write the density
for the uniform distribution on $S_2$ as $f(x) = 1$ (with respect to
itself) rather than as $f(x) = 1/(4\pi)$ (with respect to $[dx]$) or
as $f(x) = \sin \theta$ (with respect to $d\theta d \phi$).

\section{Background}
Recall the acceptance-rejection method of simulation.  Consider two
densities,
\begin{equation}
\label{eq:fg}
f(x) = c_f f^*(x), \quad g(x) = c_g g^*(x)
\end{equation}
where $f^*$ and $g^*$ are known functions, but where the normalizing
constants may or may not have a known explicit form. Suppose it is
possible to simulate easily from $g$ and it is desired to simulate
observations from $f$. The key requirement is that there is a known
bound of the form
\begin{equation}
\label{eq:bound}
f^*(x) \leq M^* g^*(x) \text{  for all } x
\end{equation}
for some constant $M^*$.  The acceptance-rejection algorithm proceeds
as follows.

\begin{itemize}
\item[Step]1. Simulate $X \sim g$ independently of $W \sim \text{Unif}(0,1)$.
\item[Step]2. If $W <  f^*(X)/\{M^* g^*(X)\}$, then accept $X$.
\item[Step]3. Otherwise go back to step 1.
\end{itemize}

\noindent {\em Comments}
\begin{itemize}

\item[(a)] If we set $M=c_f M^*/c_g$, then (\ref{eq:bound}) can be
  expressed equivalently as $f(x) \leq M g(x)$ for all $x$.

\item[(b)] The bound $M$ satisfies $M \geq 1$.  The number of trials
  needed from $g$ is geometrically distributed with mean $M \geq 1$.
  The efficiency is defined by $1/M$.  For high efficiency the bound
  $M$ should be as close to 1 as possible.

\item[(c)] The algorithm can be used even if the normalizing constants
  do not have a known explicit form.  However, to compute the
  efficiency analytically, it is necessary to know the normalizing
  constants.

\item[(d)] Suppose the density $g(x) = g(x; b)$ depends on a parameter
  $b$ with corresponding bound $M^*(b)$ in (\ref{eq:bound}).  If the
  normalizing constant $c_g = c_g(b) $ has a known explicit form, then
  it is possible to maximize the efficiency with respect to $b$, even
  if $c_f$ does not have a known explicit form.
\end{itemize}

When developing acceptance-rejection simulation methods for
directional distributions, there are several issues to consider:
\begin{itemize}
\item the need for good efficiency for a wide range of
  concentration parameters for $f$, ranging from uniform to highly concentrated.
  In similar problems on $\mathbb{R}^p$, the task is simpler when
  distributions are closed under affine transformations; in such cases
  it is sufficient to consider just a single standardized form of the
  distribution for $f$.

\item the challenge in finding a tractable envelope distribution.

\item the presence of trigonometric factors in the base measure when
expressed in polar coordinates, such as in  (\ref{eq:unif-sphere}).
\end{itemize}

Next, using the concavity of the log function, we give a general
inequality which is be useful in the construction of
acceptance-rejection algorithms.  Consider the function of $u \geq 0$,
\begin{equation}
\label{eq:concave}
\phi(u) = \frac{q}{2} \log(1+2u/b) - u - \frac{q}{2} \log(1+2u_0/b) + u_0,
\end{equation}
where $q>0$ and $0<b<q$ are fixed constants and $u_0 = (q-b)/2$.  The
last two terms on the righthand side of (\ref{eq:concave}) are
constants, chosen so that $\phi(u_0) = 0$.  The value of $u_0$ is
chosen so that the function $\frac{q}{2} \log(1+2u/b)$ has slope 1 at
$u=u_0$; hence $\phi'(u_0)=0$.  Also note that $\phi''(u) < 0$ for $u
\geq 0$ so that $\phi(u)$ is a concave function.  Therefore, $\phi(u)
\leq 0$ for all $u \geq 0$.  After exponentiating, this inequality can
be re-arranged as
\begin{equation}
\label{eq:exp-bound}
e^{-u} \leq e^{-(q-b)/2} \left(\frac{q/b}{1+2u/b}\right)^{q/2}.
\end{equation}

To illustrate the usefulness of (\ref{eq:exp-bound}),
we start in Euclidean space $\mathbb{R}^p, \ p \geq 1$,and construct
an acceptance-rejection algorithm for the multivariate normal
distribution using a multivariate Cauchy envelope.  The multivariate
normal distribution $N_p(0, \Sigma)$ has density
\begin{equation}
f(x) = c_f f^*(x), \quad f^*(x) = \exp(-\frac{1}{2} x^T \Sigma^{-1} x),\quad
c_f = c_f(\Sigma) = |2 \pi \Sigma|^{-1/2},
\end{equation}
for $x \in \mathbb{R}^p$.  The multivariate Cauchy distribution   $C_p(0, \Psi)$ has density
\begin{equation}
g(x) = c_g g^*(x), \quad g^*(x) = (1+x^T \Psi^{-1} x)^{-q/2}, \quad
c_g = c_g(\Psi) = \frac{\Gamma(q/2)}{\pi^{q/2}} |\Psi|^{-1/2},
\end{equation}
where for convenience we have substituted $q=p+1$.

If we set $\Psi = b \Sigma$ so that the scatter matrix for the Cauchy
is a scalar multiple of the covariance matrix for the normal, and if we set
$u=\frac{1}{2} x^T \Sigma^{-1} x$, then the inequality (\ref{eq:exp-bound})
leads to a bound on the densities with
$$
M(b) = 2^{-p/2} q^{q/2} e^{-q/2}  \pi^{1/2} b^{-1/2} e^{b/2}/\Gamma(q/2).
$$
Mininizing over $0<b<q$ yields the optimal parameter $b=1$ with optimal bound
\begin{equation}
\label{eq:bound-n-c}
M = M(1) = \sqrt{2\pi e} \left(\frac{q}{2e}\right)^{q/2}\big/\,\Gamma(q/2),
\quad q=p+1.
\end{equation}
Table \ref{table:eucl-eff} gives collection of the efficiencies $1/M$ as a function of
dimension $p$.  For large $p$, $M \sim \sqrt{qe/2}$ by Stirling's
formula.

\begin{table}
\begin{center}
\begin{tabular}{lccccccccc}
$p$ & 1 & 2 & 3 & 4 & 5 & 10 & 50 & 100\\
eff.& 66\%& 52\%& 45\%& 40\%& 36\%& 26\%& 12\%& 9\%
\end{tabular} \end{center}
\caption{Efficiency of A/R simulation method for the multiviate normal distribution in $p$ dimensions, using a multivariate Cauchy envelope.}
\label{table:eucl-eff}
\end{table}

Note that the efficiency declines slowly with the dimension, but is
still high enough to be feasible even for dimension $p=100$.
Of course, this is just a toy example since there are better ways to
simulate the normal distribution.  However, it is important for the
next section, both as a motiviating example and as a limiting case.

\section{Simulating the Bingham distribution with an
angular central  gaussian envelope}
In this section we describe the ``BACG'' acceptance rejection method
to simulate the Bingham distribution using the angular central
Gaussian distribution as an envelope.  As before, let $q=p+1$.

The Bingham distribution, $\text{Bing}_p(A)$ on $S_p$, $p \geq 1$, where the parameter
matrix $A$ is $q \times q$ symmetric, has density
\begin{equation}
\label{eq:bingham}
f_\text{Bing}(x) = c_\text{Bing} f_\text{Bing}^*(x), \quad f_\text{Bing}^*(x) = \exp(-x^T A x).
\end{equation}
The normalizing constant $c_\text{Bing} = c_\text{Bing}(A)$ can be expressed as a
hypergerometric function of matrix argument
\citep[p. 182]{Mardia-Jupp00}, but is not sufficiently tractable to be
of interest here.  The use of a minus sign in the exponent is
unconventional but simplifies later formulae.  Since $A$ and $A + cI$
define the same distribution for any real constant $c$, we may assume
without loss of generality that the eigenvalues of $A$ satisfy
\begin{equation}
\label{eq:evals}
0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_q.
\end{equation}

The angular central Gaussian distribution, ACG($\Omega)$ on $S_p$, where the
parameter matrix $\Omega$ is $q \times q$ symmetric positive definite,
takes the form
\begin{equation}
\label{eq:acg}
f_\text{ACG}(x) = c_\text{ACG} f_\text{ACG}^*(x), \quad f_\text{ACG}^*(x) =
\left(x^T \Omega x\right)^{-q/2},
\quad c_\text{ACG} = |\Omega|^{1/2}.
\end{equation}
The angular central Gaussian distribution is simple to simulate.  If
$y \sim N_q(0, \Sigma)$, where $\Sigma$ is positive definite, then $x
= y/||y|| \sim \text{ACG}(\Omega)$ with $\Omega = \Sigma^{-1}$
\citep[e.g.,][p. 182]{Mardia-Jupp00}.

Setting $u = x^T A x$ in (\ref{eq:exp-bound}) and setting $\Omega =
\Omega(b) = I + 2A/b, \ b>0,$ yields the envelope inequality on the starred
densities
\begin{equation}
\label{eq:b-acg}
\begin{split}
f_\text{Bing}^*(x) &= e^{-u}\\
       &\leq e^{-(q-b)/2} \left(\frac{q/b}{1+2x^T A x/b}\right)^{q/2}\\
       &= e^{-(q-b)/2} \left(\frac{q/b}{x^T \Omega x}\right)^{q/2}\\
       &= e^{-(q-b)/2} (q/b)^{q/2} f_\text{ACG}^*(x),
\end{split} \end{equation}
using the constraint $x^Tx = 1$.
The corresponding bound $M(b)$ takes the form
\begin{equation}
\label{eq:bound-b-acg}
M(b) = c_\text{Bing} e^{-(q-b)/2} (q/b)^{q/2} |\Omega(b)|^{-1/2}.
\end{equation}
It can be checked that $\log M(b)$ is convex in $b$ with unique
minimizing value given by the solution of
\begin{equation}
\label{eq:b-find}
\sum_{i=1}^q \frac{1}{b+2 \lambda_i} = 1,
\end{equation}
where the lefthand side of (\ref{eq:b-find}) ranges between $\infty$ and 0 as $b$ ranges
between 0 and $\infty$.  Let $b_0$ denote the solution to
(\ref{eq:b-find}) and let $M(b_0)$ denote the optimal bound.

It does not seem possible to evaluate $M(b_0)$ in a more useful form
analytically, but it is possible to say what happens asymptotically.
Replace $A$ in (\ref{eq:bingham}) by $\beta A$ and think of $A$ as a
fixed matrix as $\beta>0$ gets large.  Provided the $p$ largest
eigenvalues of $A$ are strictly positive, the ACG distribution (restricted to a hemisphere about the mode)
converges to a $p$-dimensional multivariate Cauchy distribution, the
Bingham distribution converges to a $p$-dimensional multivariate
normal distribution, $b_0$ converges to 1 and the bound $M(b_0)$
converges to the bound (\ref{eq:bound-n-c}) where the dimensions $p$
and $q=p+1$ have the same meanings in both sections.

Empirically, it has been noticed that the limiting case is the worst
possible case.  For smaller values of the concentration matrix $A$,
the efficiencies will be higher.  Table \ref{table:bing-eff}
illustrates the pattern for $p=2$, i.e. $q=3$. The efficiency is never
lower than 52\%, the value from Table \ref{table:eucl-eff} for $p=2$.
This limiting value is attained in the concentrated bipolar case (when
$\lambda_2 = \lambda_3$ is large).  The girdle case ($\lambda_2 = 0$)
has higher efficiencies.  Each entry in this table has been
constructed from one million simulations, so that the standard errors
are negligible.

\begin{table}
\begin{center}
\begin{tabular}{rrr}
$\lambda_2$ & $\lambda_3$ & Efficiency \\
0 & 0 & 100\%\\
0 & 10 & 84\%\\
10 & 10 & 58\%\\
0 & 100 & 80\%\\
100 & 100 & 53\%\\
\end{tabular} \end{center}
\caption{Efficiency of the BACG A/R simulation method on $S_p$ with
  $A = \text{diag}(0, \lambda_2, \lambda_3)$ for the Bingham distribution with an
  ACG envelope.}
\label{table:bing-eff}
\end{table}

\section{Manifolds and models in directional data analysis}
In order to prepare for special cases and extensions of the Bingham
distribution, it is helpful to give a brief survey of some of the
various manifolds and models used in directional data analysis.  For each of
these manifolds there is a unique invariant measure which can be used to
define a uniform distribution.
\subsection{The sphere $S_p$ revisited}
A general model on the sphere $S_p$ is the Fisher-Bingham distribution
with density
\begin{equation}
\label{eq:fb}
f_\text{FB}^*(x) = \exp(\kappa x^T \mu_0 - x^T A x)
\end{equation}
where $\kappa \geq 0$, $\mu_0 \in S_p$ and $A (q \times q)$ is
symmetric, without loss of generality with smallest eigenvalue equal
to 0.  If $A=0$ the model reduces to the von Mises ($p=1$), the Fisher
($p=2$), or the von Mises-Fisher (any $p \geq 1$) distribution.  If
$\kappa=0$, the model reduces to the Bingham distribution considered
in Section 3.

\subsection{Real projective space $\mathbb{R}P_p$}
Real projective space is defined as the quotient space $\mathbb{R}P_p
= S_p/\{1,-1\}$ in which two antipodal points or ``directions'' $\pm
x$ are identified with one another to represent the same ``axis''.
Since the Bingham and ACG densities have the property of antipodal
symmetry, $f(x) = f(-x)$, they can also be viewed as densities on
$\mathbb{R}P_p$.

\subsection{Complex projective space $\mathbb{C}P_p$}
Another quotient space of the sphere is complex projective space,
$\mathbb{C}P_p= S_{2p+1}/S_1$.  To understand this space, suppose a
unit vector $x \in \mathbb{R}^{2q}$, $q=p+1$, is partitioned as $x^T =
(x_1^T, x_2^T)$ where $x_1$ and $x_2$ are $q$-dimensional.  The
information in $x$ can also be represented by a $q$-dimensional
complex vector $z = x_1 +ix_2$.  Then $\mathbb{C}P_p$ is obtained from
$S_{2p+1}$ by identifying the scalar multiples $e^{i\theta} z$ with
one another for all $\theta \in [0, 2\pi)$.

If the $2q \times 2q$ symmetric concentration matrix $A$ for a
$\text{Bing}_{2p+1}$ distribution can be partitioned in the form
\begin{equation}
  \notag
  A = \begin{bmatrix} A_1 & -A_2 \\ A_2 & \phantom{-} A_1 \end{bmatrix},
\end{equation}
where $A_1$ is symmetric and $A_2$ is skew symmetric, then then the
quadratic form $-x^TA x$ in the exponent of the Bingham density can be expressed
in complex notation as $-z^*A_C z$ where $A_C = A_1 + i A_2$.  In
terms of $z$, the density possesses complex symmetry, $f(z) =
f(e^{i\theta}z)$ for all $\theta \in [0, 2\pi)$.  When expressed in
complex notation this distribution is known as the complex
Bingham distribution $\text{CB}_p(A_C)$; it can also viewed as a
distribution on $\mathbb{C}P_p$ \citep{kc94b}.

\subsection{The special orthogonal group $\text{SO}(r)$}
The special orthogonal group $\text{SO}(r)$ is the space of $r \times r$
rotation matrices, $SO(r) = \{X \in \mathbb{R}^{r \times r}: X^TX=I_r,
\ |X|=1\}$.  A natural parametric distribution is given by the matrix
Fisher distribution $MF_r(F)$, with $r \times r$ parameter matrix $F$.
The density is given by
\begin{equation}
\label{eq:MF-SO}
f^*(X) = \exp\{ \text{tr}(F^TX)\}.
\end{equation}
To describe the concentration properties of this distribution, it is
helpful to give $F$ a signed singular value decomposition
\begin{equation}
\label{eq:ssvd}
 F = U \Delta V^T.
\end{equation}
The adjective ``signed'' means that $U$ and $V$ are $r \times r$
rotation matrices and the elements of the diagonal matrix $\Delta$
satisfy $\delta_1 \geq \cdots \geq \delta_{r-1} \geq |\delta_r|$,
where the final element is negative if and only if $|F|<0$.

\subsection{The Stiefel manifold $\mathcal{V}_{r,q}$}
Let $1 \leq r \leq q$ and define the the Stiefel manifold $V_{r,q}=
\{X_1 \in \mathbb{R}^{q \times r}: X_1^TX_1 = I_r\}$ to be the space
of $q \times r$ column orthonormal matrices $X_1$, say.

The matrix Fisher-Bingham distribution on $V_{r,q}$, denoted
$MFB(F_1,A,C)$, with parameter matrices $F_1 (q \times r)$, $A (q \times
q \text{ symmetric})$ and $C (q \times q \text{ symmetric})$, is
defined by the density
\begin{equation}
\label{eq:MFB-stiefel}
f_{\text{MFB}}(X) \propto \text{etr}(F_1^TX_1-C X_1^TAX_1).
\end{equation}
Special cases include the matrix Fisher distribution, denoted
$MF(F_1)$, with density
\begin{equation}
\label{eq:MF-stiefel}
f_{\text{MF}}(X) \propto \text{etr}(F_1^TX_1),
\end{equation}
the ``full'' matrix Bingham distribution, denoted
$MB(A,C)$, with density
\begin{equation}
\label{eq:MB-stiefel-full}
f_{\text{MB}}(X) \propto \text{etr}(-C X_1^TAX_1),
\end{equation}
and the ``balanced'' matrix Bingham distribution, denoted
$MB(A)$, with density
\begin{equation}
\label{eq:MB-stiefel-balanced}
f_{\text{MB-bal}}(X) \propto \text{etr}(-X_1^TAX_1).
\end{equation}
If $r=q$ the balanced matrix Bingham distribution reduces to the
uniform distribution.  The reason is that $X_1$ is an orthogonal matrix
in this case, so that $X_1X_1^T = I_q$ and $\text{tr}(X_1^TAX_1) = \text{tr}(X_1
X_1^TA) = \text{tr}(A)$ is constant in (\ref{eq:MB-stiefel-balanced}).

If $r=q-1$, it is possible to extend $X_1$ uniquely by adding an extra
column to form a $q \times q$ rotation matrix $X$, say.  Hence
$V_{q-1,q}$ can be identified with $SO(q)$.  However, the version of
the matrix Fisher distribution in (\ref{eq:MF-SO}), with an $r \times
r$ parameter matrix $F$, is more general than that in
(\ref{eq:MF-stiefel}), with an $r \times (r-1)$ parameter matrix
$F_1$.

If $r=q$, then $V_{q,q}$ is the same as the orthogonal group $O(q)$,
which is twice the size of the special orthogonal group $SO(q)$, since
in $O(q)$ a matrix $X$ is allowed to have determinant $\pm 1$.
Although the two densities (\ref{eq:MF-SO}) and (\ref{eq:MF-stiefel})
formally look the same, they live on different spaces.

\subsection{The Grassmann manifold $\mathcal{G}_{r,q}$}
Let $1 \leq r < q$.  The Grassmann manifold $\mathcal{G}_{r,q}$ is
defined to be the set of all $r$-dimensional subspaces of
$\mathbb{R}^q$.  It can be described as a quotient space of a Stiefel
manifold $\mathcal{G}_{r,q} = V_{r,q}/O(r)$, in which a $q \times r$
column orthormal matrix $X_1$ is identified with $X_1R$ for all $r
\times r$ orthogonal matrices $R$.  It should be noted that the
notation for this manifold is not standardized; some authors write
$\mathcal{G}_{r,q-r}$ instead of $\mathcal{G}_{r,q}$.

Since $\text{tr}(X_1^TAX_1) = \text{tr}(R^TX_1^TAX_1R)$, the balanced matrix
Bingham distribution (\ref{eq:MB-stiefel-balanced}) on the Stiefel
manifold $\mathcal{V}_{r,q}$ can also be viewed as a distribution on
the Grassmannian manifold $\mathcal{G}_{r,q}$.

For every $r$-dimensional subspace in $\mathbb{R}^q$, there is a
unique complementary $(q-r)$-dimensional subspace.  If $X_1$ and $X_2$
are column orthonormal matrices, whose columns are bases of these
subspaces, then $X=(X_1 \ X_2)$ is a $q \times q$ orthogonal matrix.
Further $X_1$ follows a balanced matrix Bingham distribution on
$\mathcal{G}_{r,q}$ with parameter matrix $A$ if and only if $X_2$
follows a balanced matrix Bingham distribution on
$\mathcal{G}_{q-r,q}$ with parameter matrix $-A$ (but be warned that
the eigenvalues of $-A$ will not have the standardized form in
(\ref{eq:evals})).  Hence for simulation purposes, we may without loss
of generality suppose that $r \leq q/2$.

\section{Accidental isomorphisms}
The two quotient manifolds $\mathbb{R}P_p$ and $\mathbb{C}P_p$ of direct
interest for the Bingham distribution due to the existence of
``accidental isomorphisms'' in which the quotient manifold becomes
identified with another familiar manifold through a quadratic mapping.
These isomorphisms are called ``accidental'' because there does not
seem to be any systematic pattern.  In each case the uniform
distribution on the quotient manifold maps to the uniform distribution on
the new manifold, and the Bingham distribution maps to a distribution
related to the von Mises-Fisher distribution on the new manifold.  The
implications for simulation are laid out in the next subsections.
\subsection{$\mathbb{R}P_1 = S_1$}
Euclidean coordinates on the circle can be represented in polar
coordinates by $x = (x_1, x_2)^T$ where $x_1 = \cos \theta, x_2 = \sin
\theta, \ \theta \in [0,2\pi)$.  Consider a two-to-one map to a new
circle defined by $\phi = 2\theta$, with Euclidean coordinates
$y=(y_1,y_2)^T$ where $y_1 = \cos \phi = x_1^2-x_2^2, \ y_2 = \sin
\phi = 2 x_1 x_2$.  Note that the antipodal directions $\theta, \
\theta+\pi$ map to the same value of $\phi$, so that the map is in
fact a one-to-one map between $\mathbb{R}P_1$ and $S_1$.  A quadratic
form in $x$ can be rewritten as
$$
x^T A x = \frac{1}{2}(a_{11}-a_{22})y_1 + a_{12}y_2 + \frac{1}{2}(a_{11}+a_{22}),
$$
which is a linear function of $y$.  Hence a Bingham distribution,
whose density is quadratic in $x$ on $\mathbb{R}P_1$ can be
identified with a von Mises distribution, whose density is linear in
$y$, on $S_1$.

Similarly, in the ACG density the quadratic form $x^T\Omega x$ becomes
a linear function of $y$, so the density in $y$ reduces to the wrapped
Cauchy density \citep[p. 52]{Mardia-Jupp00}.

Suppose $A$ is diagonal, $A=\text{diag}(0, \lambda)$.  In
this case the dominant axis of the Bingham distribution is the
$x_1$-axis.  The corresponding von Mises density takes the form
$$
f_{\text{VM}}(y) \propto \exp(\kappa y_1), \quad \kappa=\lambda/2,
$$
so that the corresponding von Mises density has its mode in the
$y_1$-direction.  The corresponding wrapped Cauchy density, with
$\Omega= I + 2A/b$, takes the form,
$$
f_{\text{WC}}(y) = \frac{(1-\rho^2)}{1+\rho^2-2\rho y_1}
$$
where $\rho=(\beta-1)/(\beta+1)$ \citep[p. 52]{Mardia-Jupp00}.

Hence the simulation method for the Bingham distribution with an ACG
envelope can be recast as a simulation for the von Mises distribution
with a wrapped Cauchy envelope.  It turns out that this latter method
is identical to the proposal of \citet{Best-Fisher79}, even up to the
choice of the optimal tuning constant $b$.

\subsection{$\mathbb{C}P_1 = S_2$}
The complex projective space $\mathbb{C}P_{k-2}$ arises in the study
of shape for configurations of $k$ landmarks in the plane, and the
identification with $S_1$ when $k=3$ was used to visualize the shape space
for triangles of landmarks \citep{Kendall84}.  \citet{kc94b} showed
that the complex Bingham distribution on $\mathbb{C}P_1$ can be
identified with the Fisher distribution on $S_2$.

Motivated by this accidental isomorphism, \citet{kc06a} developed a
complex Bingham quartic (CBQ) distribution on $\mathbb{C}P_p, \ p \geq
1$.  When $p=1$, this distribution reduces to the $\text{FB}_5$
distribution.  \citet{Ganeiber12} developed an effective and
reasonably efficient simulation method for the CBQ distribution for $p>1$.
However, since the technique is not based on an angular central
Gaussian envelope, details will not be given here.

\subsection{$\mathbb{R}P_3 = SO(3)$}

There is a quadratic mapping taking an unsigned 4-dimensional unit
vector $\pm x$ to a $3 \times 3$ rotation matrix $X=M(x)=M(-x)$, say.
More specifically
\begin{equation}
\label{S3toSO3}
M(x) = \begin{bmatrix}
x_1^2+x_2^2-x_3^3-x_4^2 & -2(x_1x_4-x_2x_3) & 2(x_1x_3+x_2x_4)\\
2(x_1x_4+x_2x_3) & x_1^2+x_3^2-x_2^2-x_4^2 & -2(x_1x_2-x_3x_4)\\
-2(x_1x_3-x_2x_4) & 2(x_1x_2+x_3x_4) & x_1^2+x_4^2-x_2^2-x_3^2
\end{bmatrix}
\end{equation}
\citep[p. 285]{Mardia-Jupp00}.  Further a random axis $\pm x$ on
$\mathbb{R}P_3$ follows a Bingham distribution if and only if the
corresponding random matrix $M(x)$ follows a matrix Fisher
distribution.  In particular, if $A=\Lambda$ is diagonal, then
$F=\Delta$ in (\ref{eq:ssvd}) is also diagonal with the parameters related by
\begin{equation}
\label{eq:B-MF}
\lambda_1=0, \quad
\lambda_2 = 2(\delta_2+\delta_3), \quad
\lambda_3 = 2(\delta_1+\delta_3), \quad
\lambda_4 = 2(\delta_1+\delta_2).
\end{equation}
\citet{kb12b} gives some further details.

A simple way to simulate a rotation matrix from the matrix Fisher
distribution $MF_3(F)$ for a general parameter matrix $F$ with signed
singular value decomposition (\ref{eq:ssvd}) is given as follows.
Using the BACG method simulate $x$ from $\text{Bing}_3(\Lambda)$ with
$\Lambda$ given by (\ref{eq:B-MF}), and let $M(x)$ denote the
corresponding rotation matrix using (\ref{S3toSO3}).  Then $U M(x)
V^T$ follows the matrix Fisher distribution $MF_3(F)$.  From Table
\ref{table:eucl-eff}, the efficiency will be at least 45\%.

\section{Simulating the Fisher-Bingham distribution
with an angular  central gaussian envelope}
The von Mises-Fisher density on $S_p$ takes the form (\ref{eq:fb})
with $A=0$.  The elementary inequality $(1-y)^2 \geq 0$, with $y = x^T
\mu_0$ can be re-arranged to give
\begin{equation}
\label{eq:vmf-b}
\begin{split}
f_\text{F}^*(x) &\leq \exp\left[\left(\kappa/2\right)\left\{\left(x^T \mu_0\right)^2
+ 1\right\}\right]\\
&= \exp\left\{\kappa - \left(\kappa/2 \right)x^T A x \right\} \\
&= e^\kappa f_\text{Bing}^*(x),
\end{split} \end{equation} where $A = I_q - \mu_0 \mu_0^T$.  Hence an
acceptance rejection simulation method for the von Mises-Fisher
distribution can be constructed using a Bingham envelope.

The two sides of (\ref{eq:vmf-b}) match when $x=\mu_0$ so that it is
not possible to get a tighter bound.  In relative terms, the two
starred densities are maximally different when $x=-\mu_0$.  This
difference matters most when $\kappa$ is large, when the efficiency of
acceptance-rejection with a Bingham envelope drops to $50\%$; the
efficiency rises to 1 as $\kappa \rightarrow 0$.  Empirically the
efficiency lies between these two extremes for intermediate values of
$\kappa$.

The inequality (\ref{eq:vmf-b}) can be combined with Section 3 to
provide a method to simulate the von Mises-Fisher distribution with an
ACG envelope.  Of course there is no need for a new method for the von
Mises-Fisher distribution.  Good methods are already available; see
the Section 8 for a discussion.  However, the bounds of this
section can be combined with the previous section to simulate the
Fisher-Bingham distribution with an ACG envelope.

The Fisher-Bingham distribution on $S_p$ takes the form (\ref{eq:fb})
and can be bounded by a Bingham density
$$
f_\text{FB}^*(x) \leq
\exp(\kappa- x^T A^{(1)} x),
$$
where $A^{(1)} = A +(\kappa/2)(I-\mu_0\mu_0^T)$.  Then Section 3
can be used to bound this Bingham density  by an ACG density.

Say the Fisher-Bingham distribution is ``aligned'' if $\mu_0$ is an
eigenvector of $A$ and if the density has its mode at $x=\mu_0$.  In
this case the Bingham envelope usually has an efficiency of at least
50\%, with the efficiency falling below this level only when the
density excessively flat at its mode and the concentration is high.
Under high concentration this situation corresponds to the case where
the limiting normal distribution would have a singular covariance
matrix.

\section{The balanced matrix Bingham distribtion}
The density for the balanced matrix Bingham distribution was given by
(\ref{eq:MB-stiefel-balanced}).  It can be viewed as either a density
on the Stiefel manifold $\mathcal{V}_{r,q}$ (which is invariant under
multiplication on the right by $r \times r$ orthogonal matrices, or on
the Grassmannian manifold $\mathcal{G}_{r,q}$.  The $q \times q$
concentration matrix $A$ has the same form as for the Bingham
distribution in Section 3.

The matrix ACG distribution, denoted $MACG_{r,q}(\Omega)$, where $\Omega$ is
a positive definite symmetric $q \times q$ matrix, is also lies on
$\mathcal{V}_{r,q}$.  It  is also invariant  under rotation on the right
and hence can also be viewed as a distribution on the Grassmann
manifold $\mathcal{G}_{r,q}$.  The density takes the form
$$
g_{MACG}^*(X) = |X^T \Omega X|^{-q/2}, \quad c_g = |\Omega|^{r/2}
$$
\citep[e.g.][p. 40]{Chikuse03}.  Simulations from this distribution
can be constructed as follows.  Let $Y$ be a $q \times r$ matrix whose
columns are independently normally distributed, $N_q(0, \Omega^{-1})$.
Set $X=X(X^TX)^{-1/2}$ using the symmetric square root of a positive
definite matrix.  Then $X \sim MACG_{r,q}(\Omega)$.

If $\Omega$ is related to $A$ by $\Omega=\Omega(b) = I_q + 2A/b$ as in
Section 3, then the balanced matrix Bingham density can be bounded by
the matrix ACG density by using the inequality in (\ref{eq:exp-bound})
$r$ times.  Namely, let the eigenvalues of $X^TAX$ be denoted $u_1
\geq \cdots \geq u_r \geq 0$.  Since $f_{\text{MB-bal}}^*(X) = \exp(\sum u_i)$
and $f_{\text{MACG}}^*(X) = \{\prod (1+2u_i/b)\}^{-q/2}$, applying
(\ref{eq:exp-bound}) $r$ times yields the envelope bound
\begin{equation}
\label{eq:bound-mb-macg}
M(b) = c_\text{MB-bal} \{e^{-(q-b)/2} (q/b)^{q/2} |\Omega(b)|^{-1/2}\}^r.
\end{equation}
Optimizing over $b$ yields the same equation (\ref{eq:b-find}) as
before with the same value for the optimal value $b_0$.

The efficiency is expected to decline as $r$ increases.  However, as
noted before, we may restrict attention to the case $r \leq q/2$.
More numerical investigation is needed of the efficiency in this setting.

\section{Review and commentary on different simulation
methods}
Since the simulation literature for directional distributions is
widely scattered, it is useful to summarize the best simulation
methods for various distributions of interest.  Table \ref{table:recc}
lists several common distributions on different spaces, together with the
recommended method of simulation.

Recently, some MCMC simulation methods on manifolds have been proposed
by \citet{Kume-Walker09} (Fisher-Bingham on $S_p$), \citet{Habeck09}
(matrix Fisher on SO(3)), \citet{Hoff09} (matrix Fisher-Bingham
distributions on Stiefel and Grassmann manifolds) and
\citet{Byrne-Girolami13} (more general distributions).  However, there
is still an ongoing investigation into the efficiency of different
MCMC methods, so that the table entry will just state ``MCMC'' when
there is not a simpler more specific recommendation.  Further details
are given in the following subsections.

\begin{table}[t!]
\label{table:recc}
\caption{Recommended simulation methods various distributions on different
  directional spaces}
\begin{tabular}{lll}
Distribution & Space & Simulation method \vspace{.2cm} \\
von Mises-Fisher & $S_p$ & \citet{Wood87}\\
Bingham & $S_p$ or $\mathbb{R}P_p$ & BACG\\
complex Bingham & $S_{2p+1}$ or $\mathbb{C }P_p$ & \citet{kc04a} \\
complex Bingham quartic & $S_{2p+1}$ or $\mathbb{C }P_p$ & \citet{kc06a}\\
aligned Fisher-Bingham &  $S_p$ & BACG-based\\
general Fisher-Bingham &  $S_p$ & MCMC\\
matrix Fisher & $\mathcal{V}_{r,q}$ & MCMC\\
matrix Fisher & $SO(3)$ & BACG\\
matrix Fisher & $SO(p), \ p>3$ & MCMC\\
balanced matrix Bingham & $\mathcal{V}_{r,q}$ or  $\mathcal{G}_{r,q}$ &
BACG\\
general matrix Bingham & $\mathcal{V}_{r,q}$ & MCMC\\
matrix Fisher-Bingham & $\mathcal{V}_{r,q}$ & MCMC\\
\end{tabular}
\end{table}

\subsection{Uniform distribution on $S_p$}
The simplest general method to simulate a uniform distribution on the
unit sphere $S_p$ is to set $x = u /||u||$ where $u \sim N_q(0,I_q)$, \
$q=p+1$.  In low dimensions there are sometimes simpler methods using
polar coordinates.  E.g. on the circle $S_1$, let $\theta \sim
\text{Unif}(0,2\pi)$.  On the sphere $S_2$ with colatitude $\theta$ and
longitude $\phi$, let $\cos \theta \sim \text{Unif}(-1,1)$ independently of
$\phi \sim \text{Unif}(0,2\pi)$.

On the Stiefel manifold the easiest approach is to simulate $U_1 (q
\times r)$ with independent $N(0,1)$ entries, and set $X_1= U_1
(U_1^TU_1)^{-1/2}$ using the symmetric square root of a positive
definite matrix.

\subsection{von Mises-Fisher distribution $\text{F}_p(\kappa, \mu_0)$ on
  $S_p$}
For general $p \geq 1$, the recommended method of simulation is an
acceptance/rejection method due to \citet{Ulrich84}, as modified by
\citet{Wood94}.  This method uses a fractional linear transformation
of a beta variate to provide an envelope for $u = x^T \mu_0$.  It
gives good efficiency across the whole range of values for $\kappa$.
In particular, for large $\kappa$ the distribution of $2(1-u)$ is
approximately the squared radial part of a multivariate normal distribution
under the von Mises-Fisher model and of a
multivariate Cauchy distribution under the envelope model, mimicking
the efficiency calculations in (\ref{eq:bound-n-c}).

Once the distribution of $u \in [0,1]$ has been simulated, it is
straightforward to whole von Mises-Fisher distribution by
incorporating a uniformly distributed random direction $y$, say, on
$S_{p-1}$ (so $y$ is a $p$-vector). More specifically, if $R = [R_1 \
\mu_0]$ is any $q \times q$ rotatation matrix whose last column equals
$\mu_0$, let $x = u \mu_0 + (1-u^2)^{1/2} R_1^T y$.

For $p=1$ the Ulrich-Wood method is essentially identical to the
\citet{Best-Fisher79} method.  One small exception to the
recommendation to use the Ulrich-Wood method is the case $p=2$ dimensions
when $u$ follows a truncated exponential distribution and can be
simulated more simply by the inverse method without any need for
rejection \citep[p. 59]{FLE87}.

\subsection{Bingham distribution $\text{Bing}_p(A)$ on $S_p$ or
$\mathbb{R}P_p$}
The BACG method developed in this paper is the first general-purpose
acceptance/rejection simulaion method for the Bingham distribution.
However, earlier methods have been discussed in the literature for
some special cases.  In particular if $p=1$, the BACG method reduces
to \citet{Best-Fisher79} method for the von Mises distribution as
noted in Section 6.

If $p=2$ and either $0=\lambda_1 <\lambda_2 = \lambda_3$ (bipolar
case) or $0=\lambda_1 = \lambda_2 < \lambda_3$ (girdle case), the
simulation problem can be reduced to a one-dimensional problem.
\citet{Best-Fisher86} developed effective envelopes in these cases,
with efficiencies broadly comparable to the BACG method.

If the eigenvalues appear in pairs then the methods for the complex
Bingham can be used.  \citet{kc04a} developed several simulation
methods that sometimes are better than BACG.

The BACG method here supersedes the MCMC method of \citet{Kume-Walker06}.

\subsection{Fisher-Bingham distribution $\text{FB}(\kappa,\mu_0,A)$
on $S_p$}
The Kent (or $\text{FB}_5$) distribution on $S_2$ is a special case of an
aligned Fisher-Bingham distribution.  An efficient simulation
algorithm for $\text{FB}_5$ was developed by \citet{kb05a}.  Otherwise, for
other aligned $\text{FB}$ distributions when $p=2$ or for other values of
$p$, the BACG-based method developed in Section 6 is the recommended
method.

In particular, these methods supersede earlier acceptance-rejection
methods developed by \citet{Wood87} for various special
types of aligned Fisher-Bingham distribution on $S_2$.  They also
supersede the acceptance rejection method of \citet[Appendix
A4]{Scealy-Welsh11} for a higher-dimensional version of the Kent
distribution, for which the efficiency drops to 0 under high
concentration when $p>2$.  In addition they supersede the MCMC
method of \citet{Kume-Walker09} in the aligned case.

For non-aligned Fisher-Bingham distributions, it is difficult to make
any firm theoretical statements about the behvaviour of the algorithm
in Section 6.  However, under moderate concentration it is still
likely to be preferable to the MCMC methods of \citet{Kume-Walker09}.

\subsection {Matrix Fisher distribution $\text{MF}(F)$ on $SO(r)$}
When $r=2$, $SO(2)$ is the same as $S_1$ and the matrix Fisher on
SO(2) is identical to the von Mises distribution on $S_1$, so no new
methodology is needed.

When $r=3$ the accidental isomorphism in Section 5.3 reduces this case
to the Bingham distribution on $S_2$, which can be simulated by the
BACG method.

Earlier methods to simulate the matrix Fisher distribution on SO(3),
now superseded by BACG, were based on MCMC algorithms.  These include
\citet{Green-Mardia06} and \citet{Habeck09}.

The cases $r>3$ are at least partly covered by the next subsection.

\subsection{Matrix Fisher-Bingham distribution $\text{MFB}(F,A,C)$ on
$\mathcal{V}_{r,q}$}

For the general matrix Fisher-Bingham distribution $\text{MFB}(F,A,C)$ on
$\mathcal{V}_{r,q}$, there is not yet a convenient and efficient A/R
algorithm other than for the balanced matrix Bingham case, where a
solution was given in Section 7.  However, the recent MCMC algorithms
of \citet{Hoff09} and \citet{Byrne-Girolami13} can deal this this
case.

\subsection{Product manifolds}
Finally, the main setting not covered in this paper is the class of
product manifolds where multivariate versions of directional models
can be defined.  There are a few special cases where acceptance
rejection methods are available (e.g. \citet[supplementary material]
{Mardia-etal06} for the
sine and cosine versions of a bivariate version of the von Mises
distribution on the torus), but in general MCMC methods are needed.
\bibliographystyle{dcu}

redpajama_set_name: RedPajamaArXiv


text:
Louis Lozowick
Bridge in Shadow (Williamsburg Bridge)
Angry Skies (Adante Cantabile)
Abandoned Quarry (Rockport Quarries)
Self-Portrait in Spring
Backyards of Broadway
Madison Ave
High Voltage Cos Cob
Train and Factory
Still Life # 2 With Apples
Summer Home
Lynching (Lynch Law)
Skyscrapers (with butterflies)
Louis Lozowick (1892-1973) was a leading Precisionist artist, born in the Ukrainian town of Ludvinovka. He studied at the Kiev Art School from 1903 to 1906, before immigrating to the United States and returning to school in 1912 at the National Academy of Design. He continued his studies at Ohio State University, graduating in 1918, and subsequently entered into the Army Medical Corps, where he served until 1919. Upon his discharge, Lozowick spent five years abroad, traveling to Paris, Moscow, and Berlin.
Mary Ryan Gallery is the exclusive representative of Lozowick's prints.
Lozowick began exhibiting his Machine Ornament drawings and abstract paintings based upon American cities during a formative period spent in Europe, making his first lithograph in Berlin in 1923. He returned to New York in 1924, where he continued to explore lithography, and participated in organizing the 1927 Machine Age Exposition. He held his first solo exhibition of lithographs depicting primarily soaring urban and industrial scenes at the Weyhe Gallery in 1929. A prominent Precisionist, his drawings and paintings of the 1920s are directly influenced by Russian avant-garde artists, such as El Lissitzky. He championed the hard-edged, linear style of Constructivism, which rooted his shift to a more social realist sensibility seen in his work during the Great Depression.
During his time in New York, Lozowick lectured on modern Russian art for the Société Anonyme, took on various design commissions, and, in 1926, joined the executive board of the radical journal New Masses. He sat as secretary to the American Artists' Congress and was an active member of the John Reed Club. He also held various artistic offices of the WPA from 1935-40. Under federal sponsorship, Lozowick was able to create prints in various media and paintings, including a pair of murals for the New York City General Post Office in the late 1930s. Lozowick moved to South Orange, New Jersey, in 1943 with his family, where he continued to teach, lecture, travel, and produce lithographs until his death in 1973.
Lozowick exhibited widely during his lifetime and even after his death. Recent solo and group exhibitions include the Palmer Museum of Art (2019), Brooklyn Museum of Art (2015), Whitney Museum of American Art (2015), National Gallery of Art, Washington D.C. (2015), British Museum (2008), de Young Museum (2007) and the Hirshhorn Museum and Sculpture Garden (2001). Mary Ryan Gallery represents the estate of Louis Lozowick and has held major exhibitions featuring his work in 2004, 2001, 1999, and 1997.
The Whitney Museum of American Art holds the largest collection of Lozowick's paintings and drawings, and it is the repository of his personal library. The Smithsonian American Art Museum holds the largest collection of the artist's lithographs.
His work is in numerous prominent museum collections, including the Brooklyn Museum, NY; Carnegie Museum of Art, PA; Cleveland Museum of Art, OH; Crystal Bridges Museum of American Art; AR; Fine Arts Museums San Francisco, CA; Hirshhorn Museum and Sculpture Garden, Washington D.C.; Jewish Museum, NY; Metropolitan Museum of Art, NY; National Gallery of Art, Washington D.C.; Philadelphia Museum of Art, PA; Rhode Island School of Design, RI; Smithsonian American Art Museum, Washington D.C.; Walker Art Center, Minneapolis, MN and the Whitney Museum of American Art, NY.
In 1982, Hudson Hills Press published a catalogue raisoné of Lozowick's prints, edited by Janet Flint.
Palmer Museum announces intriguing exhibition on American lithography
Penn State News
Louis Lozowick, James E. Allen
A Revolution Indeed, and Bathed in Smoke
by Benjamin Genocchio

redpajama_set_name: RedPajamaCommonCrawl


de·te·ri·o·rate v. i. To grow worse; to be impaired in quality; to degenerate.
Under such conditions, the mind rapidly deteriorates. --Goldsmith.
De·te·ri·o·rate v. t. [imp. & p. p. Deteriorated p. pr. & vb. n. Deteriorating ] To make worse; to make inferior in quality or value; to impair; as, to deteriorate the mind.
The art of war . . . was greatly deteriorated. --Southey.
v 1: become worse or disintegrate; "His mind deteriorated"


text:
Charles Faucette, Jr.  (born October 7, 1963) is a former American football linebacker and former St. Louis Rams strength and conditioning coach. He played two seasons for the San Diego Chargers, appearing in ten games. Faucette attended the University of Maryland, College Park from 1983 to 1986 and left as the Terrapins second-leading all-time tackler with 466, behind only Eric Wilson (481). , Faucette still ranks third on Maryland's all-time list, only behind Wilson and D'Qwell Jackson (473).  He is currently the Head Football Coach at Ida S. Baker High School in Cape Coral, Florida.

Raised in Willingboro Township, New Jersey, Faucette played both baseball and football at Willingboro High School.

Faucette helped the Terrapins to three-straight Atlantic Coast Conference championships (1983–85), serving as team captain in his senior year. He was a two-time All-ACC selection and a three-time honorable mention All-American. At  and , Faucette had a prototypical physique. However, due to the unusual college defense in which he played, a wide-tackle-six, pro teams were not sure how he would fit into their more standard 3-4 or 4-3 defenses. He was drafted in the tenth round of the 1987 NFL Draft by the New York Giants.

After two seasons with the Chargers, he retired from the NFL after he suffered a broken neck, and started his coaching career as the head coach at Crawford High School in San Diego in 1990. Faucette went on to coach the linebackers for the Hamilton Tiger-Cats of the Canadian Football League from 1990 to 1992, alongside serving as the team's head strength coach. Between 1996 and 1999, he served as the boys' athletic director and head football coach at St. Pius X High School in Houston, Texas, leading his team to a 10–3 record and a state runner-up finish in 1998.

Entering the collegiate level, Faucette became an assistant strength & conditioning coach at Texas in 1999. He left for Southern Methodist University after two seasons to become the Mustangs' head strength & conditioning coach. Faucette returned to the Longhorns for a second stint in 2006–07. In 2008, he was hired by the St. Louis Rams as strength and conditioning coach. In June 2008, Faucette received the highest honor from the Collegiate Strength and Conditioning Coaches association (CSCCa) as he was named Master Strength and Conditioning coach. On February 5, he was retained by new Rams head coach Steve Spagnuolo.  On January 7, 2010, Coach Spagnuolo elected not to renew his contract  He was the head coach from 2010 to 2016 at Lutheran High School South in St. Louis.  On February 26, 2016, out of a candidate pool of over 150 coaches, Bishop Lynch High School in Dallas, Texas named Faucette as its Head Football Coach. In his first season, Faucette led Bishop Lynch to the 2016 TAPPS Division I state championship, the school's first football state title since 2003.

Faucette was originally drafted in the 12th round (290th pick) out of high school to play baseball for the Toronto Blue Jays. In 1981, at 17, he played outfield for their rookie-league team in Bradenton, Florida. In 1982, he played in Florence, South Carolina and Medicine Hat, Alberta before going to play college football for the University of Maryland. His daughter, Juliann, is a Professional volleyball player, attended the University of Texas, and was a three time All-America. His brother Patrick Faucette is an actor.

References

1963 births
Living people
American strength and conditioning coaches
American football linebackers
Maryland Terrapins football players
Texas Longhorns football coaches
SMU Mustangs football coaches
St. Louis Rams coaches
People from Levittown, Pennsylvania
People from Willingboro Township, New Jersey
Players of American football from New Jersey
Players of American football from Pennsylvania
Sportspeople from Burlington County, New Jersey
Willingboro High School alumni

redpajama_set_name: RedPajamaWikipedia


Our Xbox One Review of Rico I always love a good shooter. Whether that`s a FPS or a TPS I normally find myself indulged in any of them and on the whole quite enjoy them. There have been some that I have really enjoyed, both in the form of ID and AAA and some that ..
I always love a good shooter. Whether that`s a FPS or a TPS I normally find myself indulged in any of them and on the whole quite enjoy them. There have been some that I have really enjoyed, both in the form of ID and AAA and some that I have not enjoyed, again both ID and AAA. Over the past few weeks, I`ve had the chance to get my hands on a new fast-paced FPS game, and that comes in the form of Rico. How was it going to fare? Was it going to be something I was going to enjoy? Or would it be let down for a shooter? I must say I`ve quite enjoyed it overall, but I do fear for some, it will feel to repetitive.
Rico is developed by Ground Shatter and published by Rising Star Games, and is described as a action movie FPS. Across a variety of modes and cases, the aim of each round is going to be quite simple. You are a police detective and your aim is to move through each level (Which is designed to be a massive raid on a building) and clear each room of enemies. You will have several rooms to clear throughout, and the quicker you complete the level and clear all the enemies, the better that is. In addition, you`ll need to collect ammo and health in order to survive and not be wiped out by one of several enemies.
Straight from the off you`re able to go into a Tutorial level which will show you the basics of what to do – think of it as a massive crash course where you`ll need to go through each room, and shoot the pop up enemies that appear in record time, before moving onto the real thing. In addition, you have the ability to play through a case mode, which will be where the main focus and story of the game is, and also, be able to play online in Multiplayer games, and lockdown mode, which is a take on Survival, which I quite enjoyed. There is a good variety in the modes to play, and something to cater for people.
I like the style of the game. I think it`s good, and I enjoy it. However, there is a fear that what you play is going to become rather competitive after a while, and even with the different game modes, the emphasis is still the same throughout, and for some people, that`s not going to be good. Some people will play through and enjoy it throughout, but I fall in the category that I was finding myself turning it off after a while just to have a break from playing the same thing again. I like variety, and although I quite like the mode variety I have, I`d have like to have seen some variety in the actual gameplay.
I always like a game that`s easy to pick up and play, and you get that with Rico. If you`ve ever played any other shooter game then you`ll be pleased to know that it`s near enough the same configuration that every shooter seems to follow with. If you`re new to shooter games, then you will be taught in the tutorial.
Graphically I love it. I think the artistic style that was taken by the development team was good, and it looks a bit different from what I`ve played when it comes to shooters, and think it looks good. The digital comic book style is something that we don't see too often in video games, and mixed with the fast-paced action, made it look good. It`s not going to be everyone`s cup of tea, but for me, I like what they did with it.
Ultimately Rico is a title that`s not going to be for everyone, and I think the repetitiveness throughout it is going to be the problem. There is a good variety in the modes that are on offer throughout and I personally quite enjoyed playing the story mode, and I like the options that you get to be able to play online with (Admittedly when it works- I can`t say I had awful lots of success when it came to getting many games online) but for some people, it`s going to be a game that offers a bit of play value before the need to turn it off and switch to something else that`s going to offer something varied in the game.
However, Rico is a game that I`m enjoying my time with. I like the artistic style that was taken by the devs, and being a bit of a stickler for shooters I enjoyed it. But perhaps I`m being a bit biased there as a shooter genre lover. If someone who was new to the shooter genre played it, I`m not sure how long they`d stick with it for. But if you can take a look past the fact that you`re not going to be getting much variety in how the game plays and your aim as you wipe out rooms of enemies, you do get a game that offers good fun, and good spells of shooting your way through to the quickest time, and I`ve quite enjoyed my time so far.


YOKOTA AIR BASE, Japan -- A U.S. Marine suspected in the death of a transgender in the Philippines has been transferred from custody on an American ship to a Philippine military base.
Local police have filed a murder complaint against the Marine, who they have identified as Pfc. Joseph Scott Pemberton, who was seen entering the Celzone Lodge in Olongapo City on Oct. 11 with Jeffrey Laude, 26, also known as Jennifer. Laude's naked body was discovered in a hotel bathroom a short time later.
In a statement Wednesday, U.S. Marine Corps Forces Pacific -- which has yet to confirm Pemberton's identity pending official charges -- announced that a suspect had been transferred from the USS Peleliu to a holding location at Camp Aguinaldo, the military headquarters of the Armed Forces of the Philippines in Quezon City.
"The Marine will remain in the custody of the United States pursuant to the Visiting Forces Agreement (VFA) with the Republic of the Philippines," Marine Corps spokesman Col. Brad Bartelt said in the statement.
Custody of American servicemembers accused of crimes has long been a sticking point in the Philippines, and is a major issue for leftist and student groups that have been protesting outside the U.S. Embassy since the slaying took place and are looking for any signs of favoritism in the handling of the case.
The slaying has been widely played in the Philippine media, though it is unclear whether there will be any long-term impact on the two countries' recent agreement to bolster the U.S. military's presence in the Philippines, a former American colony.
As part of the Pacific pivot toward the Asia-Pacific region as the wars in Iraq and Afghanistan have wound down, the U.S. has been moving toward prepositioning rotational forces, materiel and disaster aid in the Philippines. U.S. bases here were closed down in 1991 after the Philippine Senate refused to renew their leases.
A joint investigation of Laude's death by the Naval Criminal Investigative Service and Philippine National Police is ongoing, Bartelt said.
The suspect, who is assigned to the 2nd Battalion, 9th Marine Regiment, 2nd Marine Division, based at Camp Lejeune, N.C., will remain in the Philippines during the investigation of the death and any judicial proceedings, he said.
"The Marine Corps is working closely and cooperating fully with Philippine authorities to ensure a thorough investigation is completed and due process of law is followed," Bartelt said.
With the transfer of the suspect, the commander of U.S. Pacific Command, Adm. Samuel J. Locklear III, has authorized the Peleliu to depart Subic Bay, he said. The ship was among five Navy vessels to port at Subic Bay following joint military exercises in the Philippines. The other four were allowed to leave last week.


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Ben Willey - Pastor
"Most likely to get caught singing 80's throwbacks."
Over the course of our campus' young life, Ben has been a (sometimes) focused leader, thoughtful visionary, and a tender shepherd to our community. His heart is for Jesus, and for people. No one walks away from a conversation with him without feeling listened to and cared for.
Ben is a deeply relational person. He has a genuine interest in and concern for the life of everyone he comes in contact with. This concern stems from a gentleness of spirit that overflows from his intense love of the gospel.
Ben is competitive by nature, and bleeds Eagles green through and through. If he's not trash talking during the week, he's screaming at his tv during the weekend (like a true Philadelphia fan). He has a passion for reading, playing soccer, goofing around with his kids, and undertaking the occasional carpentry project.
Ben and his wife Adriana live in Barrington and love having land for their two sons, Atticus and Solomon, to play on.
(We decided that none of us wanted to write our own bio because it feels weird. Each bio was written by someone else at Fellowship Collingswood)
He can be reached at bwilley@fellowshipsj.org


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Jaguar-Land Rover to invest in EV production in the UK
July 08, 2019 United Kingdom
This past Friday in London, Jaguar Land Rover announced it will modernize one of its existing assembly plants in central England so that it can eventually produce a range of new electrified vehicles. Them ove is also designed to kick-start a moribund British auto industry that has been hit hard in recent years by global consumer trends as well as uncertainty over Brexit.
The plant in question is the facility in Castle Bromwich; the plan is for it to produce a number of different electrified models in the coming years, including the next-generation Jaguar XJ. The model was to have gone entirely electric, but the automaker has confirmed that gasoline-powered versions would also be offered during the new generation's lifespan.
The investments amount to several hundred million pounds sterling, and it will allow Jaguar-Land Rover to maintain jobs within Britain. Despite now being owned by an Indian conglomerate, the company has very much held on to its British roots, so that is a crucial motivator. As well, the move is in line with the company's oft-stated goal of moving speedily towards the electrification of its product range.
The company also confirmed, moreover, that a battery-production facility will be up and running as soon as 2020. The plant will have a capacity of 150,000 units per year. In subsequent years, the Castle Bromwich will become operational, and new Jaguar-Land Rover EVs will begin to see the light of day.
Source: www.auto123.com


Low cost airlines offer a way to travel on the cheap — provided you understand how the system works! From baggage fees to reward credit cards, our Military TravelingMom, a frequent flyer on budget airlines, tells you what you need to know before you book on Frontier Airlines, Allegiant Airlines and Spirit Airlines.
Editor's note: This post contains affliate links. If you click through and buy, TravelingMom gets a small commission.
As a military family, we travel frequently, especially now that my husband is stationed on the other side of the country. Because flying between California and Florida once a month is spendy, I'm always looking for cheap flights. Usually that means low cost airlines with no extras.
Those budget airlines, done right, can be cheap. Done wrong, they can be as expensive, or even more expensive, than the major carriers.
With the tagline "low fares done right," Frontier prides itself on offering the lowest fares while providing top notch customer service. This rapidly growing low cost carrier serves 95 cities throughout the United States, Mexico and the Dominican Republic.
Frontier Airlines is known for having incredible sales, including non-stop flights starting at just $20! Flights that cheap entice many to drop everything to travel, even last minute.
What's Good About Frontier Airlines?
MyFrontier – Free Frontier miles membership that allows travelers to earn miles for each flight.
Discount Den Membership – For just $49.99 a year, Discount Den Members are the first to hear about sales, earn Frontier miles with each flight, and have access to the lowest fares. The low fares apply to up to six people on one booking.
There are quite a few things about Frontier Airlines that I really like. Earning free flights can be quick and easy with one-way flights starting as low as 10,000 miles! That means the credit card bonus of 40,000 miles could get you up to four one-way tickets, or two round trip tickets.
What's Not So Good About Frontier Airlines?
No complimentary food or beverages – Plan to buy them at the airport or on the plane or bring your own snacks and a refillable water bottle you can fill after passing through TSA security.
No reclining seats – Makes it hard to get comfortable on long flights.
Little to no padding on seats – Your bum can get a little sore after so long.
Hefty fee for overweight luggage ($75 for anything over 50 lbs) – I've had them waive the fee if it was over by a little bit, like a pound or so.
No WiFi available – Just plan ahead and download shows or movies or bring a book to read.
Giving up the creature comforts of padded and reclining seats can make longer flights extremely uncomfortable. It can also be difficult when trying to save on bag fees to make sure you stay within the allowable weight limit. Paying the overweight fee can be shocking to the budget.
Family Miles Pooling – Elite 20k members and Frontier Airlines cardholders can choose to pool their miles with up to 8 people. This allows "family" members to combine miles to be able to earn free flights quicker. Best part: "You define your family." That means you can pool miles with aunts and uncles, friends or even co-workers.
Low pet travel fee – Pets are allowed on most flights for just $75 each way.
Assigned seats – Even if you don't pay to choose your seating assignment, you will be assigned a seat at check in.
Bundle & Save option – Frontier Airlines offers the option to bundle bags and seat assignments, which can save you some money on the extra fees along with giving you perks like priority boarding.
Automatic credit vouchers issued after a delay of more than three hours – How much you get depends on the length of delay. I've gotten $25 for a three hour delay and $100 for a six hour delay.
Military Discounts – Frontier Airlines offers military discount fares for active duty service members, but the biggest perk is that active duty members with valid military ID get two free checked bags and a free carry-on. Unfortunately, this perk isn't extended to dependents.
Does Frontier Airlines Work for Families?
Frontier Airlines does everything it can to make sure families are seated together. When booking, you have the option to pay a fee to select assigned seats. That ensures you don't have to be separated from your loved ones. If you end up with separate seats, ask at the gate whether there are any seats available together. Another family friendly benefit is that families with young ones are given priority/zone 1 boarding to allow a little extra time to get the kiddos settled in.
Also, check the tail of the plane to see if you're flying with Griz the Bear, Flo the Flamingo or Otto the Owl. If you are, be sure to ask your flight attendant for a free collectable trading card!
TravelingMom Tip: If you're paying for seat assignments, consider the stretch seating, usually the first few rows. These rows give you some extra leg room and allow full recline. For a little more month, another option for more legroom is an emergency exit row (unless you're traveling with young kids; kids 14 and youngers are not allowed to sit in exit rows on planes).
Allegiant Airlines isn't as well known or as large as the other two budget airlines. Focusing on no frills, Allegiant only offers non-stop flights and a limited flight schedule to destinations within the U.S., mostly in and out of lesser known airports.
What's Good About Allegiant Airlines?
Non-stop flights – No hassle of changing planes.
Fly in and out of smaller airports – Hometown airports are usually smaller, less crowded and sometimes even offer free long term parking.
Allegiant World Mastercard – Allegiant offers a 3, 2, 1 structure. Earn 3x the points on all Allegiant purchases, 2x when used for dining costs and 1x on everything else.
Assigned seats – You'll be given a random seat assignment upon check in.
Cheaper baggage fees – Baggage fees vary, but in most cases they are less than the competitors.
Unlike other airline credit cards, Allegiant World Mastercard offers an opportunity to earn a free companion flight. Earn that by booking a vacation that includes at least four nights in a hotel or a week long car rental.
What's Not So Good About Allegiant Airlines?
Limited flight schedule – To keep costs down Allegiant doesn't fly in and out of destinations every day.
No frequent flyer program – The only way to earn points is to get the Allegiant Mastercard.
Baggage weight limits – Oversized baggage fees start at 40 lbs and the fees are hefty, starting at $50.
No free refreshments – Plan to buy them at the airport or on the plane or bring your own snacks and a refillable water bottle you can fill after passing through TSA security.
Call center booking fee – There is a charge of $14.99 to book your flight using the call center.
Boarding pass printing fee – For a fee of $5 you can print your boarding pass at the airport.
Seasonal flights – Some of Allegiant's routes are seasonal and don't operate all year.
The limited flight schedule might work when booking, but what happens when there is a cancellation? Unfortunately, you have to wait until the next scheduled day. That means you could be stranded for up to a week or so. Not good at all!
Read More: Why is Travel Insurance So Important?
It's also important to know where you're going on Allegiant Airlines. For example, when you book online for a flight to Orlando for your family vacation at Disney World, you won't be booking a flight into Orlando International Airport. You'll be flying into its other, lesser known airport, Sanford. That means no access to Disney's Magical Express. You'll need to add in the cost of a shuttle or rental car when calculating the total price for your trip.
Buy one get one – The opportunity to earn BOGO flights as a Allegiant card holder is a great way to save.
Easy to use app – Allegiant's app is available for Apple and Android products.
Smaller airports – Smaller crowds, quicker TSA lines and possibly closer to home, smaller airports have their benefits.
Non-stop flights – No chance of missing your connecting flight. And you don't have to pack up and run to another gate.
Military discounts – Three free checked bag weighing up to 100 lbs, free carry on and more.
Allegiant Airlines offers the best military discounts of the three airlines. Besides the free baggage, military members are also eligible for free seat assignments and boarding pass printing. And the discounts are extended to spouses and dependents! But the perk that makes Allegiant shine: it allows one free pet in cabin for military.
Does Allegiant Airlines Work for Families?
Allegiant tries to work with families, but doesn't make any promises. To guarantee everyone sits together, you'll have to pay for seat assignments.
Families can check strollers and car seats for free at the time of check in or at the gate.
TravelingMom Tip: Due to the limited flights, use Allegiant to visit family, so you have a free place to stay if your flight is cancelled. Or buy travel insurance that covers the cost of travel delays.
After a rocky start, Spirit Airlines became a pioneer in the low cost airline travel business. It was the first to start charging for carry on bags under the "pay for the amenities you want" business model we see among low cost carriers today.
Flying to more than 60 destinations in the United States, Latin America and the Caribbean, Spirit Airlines has more than 400 daily flights to choose from.
What's Good About Spirit Airlines?
Fit Fleet – Spirit prides itself on having one of the youngest fleets in the industry. The planes are top in fuel efficiency which in turn saves you money. Newer planes also means the risk for maintenance issues is lower, so less chance of delays.
Free Spirit – Spirit Airlines frequent flyer program allows you to earn miles with each trip.
Unique destinations – Out of the three low cost carriers, Spirit is the only one that travels to destinations like Jamaica, Panama, and Guatemala.
Spirit Airlines Mastercard – Offered through Bank of America, Spirit's credit card lets users earn miles 2x faster and the current sign up bonus of 15,000 miles can be used for up to three round trip tickets.
The great thing about Spirit's frequent flyer program is that Spirit has partnered with a variety of companies so that you can earn miles on purchases like dining out and hotels. And with flights starting as low as 2,500 miles you'll be flying free in no time!
What's Not So Good About Spirit Airlines?
Baggage limits – Oversized baggage fees start at anything over 40 lbs.
No complimentary refreshments on board – Better pack your own snacks or plan to purchase some.
No reclining seats – It can be rough sitting straight up for hours on end.
Charges to print boarding pass – There are charges to print boarding passes at the airport kiosks or by a desk agent.
No WiFi – Download shows or movies or add a new book to your kindle before hand.
No padding on seats – After a long flight or even a short one your bottom can end up a wee bit sore.
More rows in each plane – Added rows means less leg room.
Spirit Airlines doesn't want you to focus on what you're missing out when flying with them. The company sells the "more rows on each plane" as a way to pass on lower costs, which translates into cheaper flights. But no matter how you sell it, less leg room isn't a good thing.
$9 Fare Club – For $69.99 a year, members can cash in on flights as low at $9 one way.
Fabulous destinations – Flying to locations like Haiti, Puerto Rico or Colombia on the cheap is awesome!
Assigned seats – Even if you don't pay for your seat when you check in you're given an assigned seat.
Free Spirit Dining – In connection with the Free Spirit miles membership you can earn miles when dining out, even locally.
The Big Front Seat – Wider seats and added leg room for a slight cost.
Flight Flex – Up to 24 hours before departure you can modify your itinerary for free one time. If there is a difference in fare you are responsible for the change in cost.
Military Discounts – Spirit Airlines allows military members with valid military ID to check two free bags and one free carry on.
No other airline's discount program can compare to Spirit's $9 Fare Club. Being able to fly for $18 (plus taxes and fees) round trip is unheard of and even if you have to pay baggage fees, the flight is cheaper than anyone else.
Does Spirit Airlines Work for Families?
Spirit Airlines does its best to accommodate families. If flights are booked at the same time, Spirit tries to ensure that you are seated together if you chose not to pay to choose your seat. But to be guaranteed seats together, I recommend paying the small cost for peace of mind.
Spirit allows one car seat and/or stroller per child to be checked for free.
TravelingMom Tip: Take advantage of the $9 Fare Club – pack a backpack with snacks, a carry on with clothes and take a weekend getaway on the cheap!
"You get what you pay for" is a phrase that is thrown around all the time when discussing budget airlines. And it's true. If you go in expecting first class you'll be greatly disappointed. Keep an open mind, remember you're saving possibly hundreds, plan ahead and have a fantastic trip!
Low cost airlines are always a nice option to travel! It sure can be quite cheap but once you add up things like heavy baggage, booked seats, foods… not that cheap. So you always have to be careful when thinking about low cost.
How about traveling frontier specifically with child seats? My kids are 6 and 8 and need boosters. Will I have to pay extra to bring those along?
I do suggest calling Frontier to double check, but you should be able to gate check your booster seats for free. If you plan on using them during the flight they have fit within Frontier's dimensions and cannot be used during takeoff and landing.


The skiing in Jasper is fantastic for a small mountain - it's very diverse and advanced skiers can get a lot out of it in just a few days.
Unique, uncrowded and hugely welcoming, Jasper and Marmot Basin have a reputation as the world's friendliest ski town and resort.
Canada's northernmost resort, Jasper's scenic ski area (Marmot Basin) is relatively compact, famously uncrowded and great for all abilities. With a great on-piste vibe, Marmot Basin is known as The Big Friendly. The backcountry was previously limited to the bowls and tree-lined gullies within this area but new terrain has now opened to put Marmot firmly on the map.
Open to skiers and snowboarders for the 2017-18 season, Tres Hombres is a large, north-facing area previously beyond Marmot Basin's outer limits. Without any trees, and measuring around 45 acres, it is providing advanced and expert skiers and snowboarders with incredible, wide-open off-piste terrain including powder-filled bowls and chutes that descend 367 vertical metres - Marmot Basin's longest, uninterrupted fall-line.
On the banks of the Athabasca River, Jasper town is a 15-minute shuttle from the ski area. This 19th century fur-trading post has a wonderful sense of community - the locals go out of their way to make you feel welcome.
Facilities are excellent, with plenty of restaurants and shops. Après activities range from drinks at Caribou Chalet Lodge, to swimming at the local aquatic centre, and guided tours to frozen waterfalls. If you're lucky, you may even catch a glimpse of a moose or elk by the river during your Jasper ski holiday.
Make the most of your Jasper ski holiday with our insider tips.
With no accommodation in the ski area, Jasper's hotels are all located downtown or in the surrounding area. There are several options for each budget, and you can choose between full-service resort hotels, secluded log cabins and comfortable, good value inns.
We've stayed at or visited all our Jasper accommodation options, so call us for advice and we'll recommend the best ones for you.
This superb property has elegant cedar chalets and log cabins arranged in the style of an alpine village in a secluded location on the edge of Lac Beauvert.
The Jasper Inn is a 10 minute walk from the town centre and offers reasonably priced accommodation.
See our Jasper offers for some excellent savings and superb package prices.
Nice town with a variety of eating places. We did a Maligne Canyon ice walk with Jasper Hikes and Tours which was amazing and superb value. The mountain itself has groomed slopes and was suitable for intermediate boarders. It is a small mountain so a couple of days boarding was perfect.
Very friendly town and resort. The ski bus service would be better if it ran a little more. Lots of things to do other than ski. lots of walking with potential for wildlife spotting. Grocery stores, good a well stocked. Ski resort small compared to some but very easy to navigate and try different runs or trees knowing you'll end up safely at the base. recommend The Raven restaurant...amazing.
The drive over to Jasper is beautiful and we loved the area. We found the town quiet (even by our tame standards!) and the ski area was quite small. we enjoyed it but as avalanche risk closed some of the more adventurous areas we were ready to move on. We finished skiing mid afternoon and enjoyed exploring the local area, especially Maligne Canyon (stunning!).


MADE BY HOLLAND - On 17 February 1867 the first vessel sailed through the Suez Canal, shortening the sailing time between Europe and the Far East tremendously. Around 17,000 vessels make use of the canal each year, making it one of Egypt's main sources of income. In 2014 the country announced they wanted to expand the canal to almost double the number of vessel passing each day.


Excursions - Positano's Concierge - We provide you the best services!
- Our unparalleled ancient Roman archaeological remains and museums including - Pompeii, Herculaneum, the National Museum in Naples and The Villa Tiberius on Capri.
- Our wonderful local cuisine & drinks - there are popular wineries where our locally produced red and white wines can be samples and the makers of our famous lemoncello are happy to welcome special visitors.
- Our natural wonders including the Blue Grotto and stunning ravines and mountain valleys, and of course, the majestic Madame Vesuvius herself ( slumbering quietly of course !).
- Our legacy from the early invaders from Ancient Greece - including the Temple of Hera and the archaeological site at Paestum.
- Our world famous musical events particularly the moonlight opera and concerts hosted at Ravello.
- Our array of art galleries, boutiques and craft exhibitions.
- Our coastline "up close and personal" in a small boat which can whisk you to the most private of coves and beaches, or take you round the islands which have been the homes of the rich and famous for centuries.
And when all the sight-seeing is over, there will be time to sit back and relax over a fine meal and watch the magnificent sunsets over the Amalfi coast, to prepare oneself for the next day in Paradise.


This image gallery named as Carport Into Garage Modern Garage And Shed Denver might be ideal solution for your decoration. As the author has insert this image into Canopy Garage Door post section, would ease you to find the perfect match of you decor desire.
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Q: How can I toggle individual
* elements using only basic javascript and DOM manipulation to change their styling? I am trying to toggle each item on a list to change its style through CSS, but I keep getting an error. Can someone explain what is wrong with my code?
HTML
    <button id="enter">Enter</button>

Notebook

Apple

Mango

Banana

BlackBerry

Watermelon

CSS
.change {
  text-decoration: line-through;
}

Javascript
var li = document.querySelectorAll("li");

li.addEventListener("click" , function () {
    li.classList.toggle(".change");
})

Uncaught TypeError: li.addEventListener is not a function

A: You need to loop through all the elements using forEach(). And also remove . before change

var li = document.querySelectorAll("li");

for(let i = 0;i<li.length;i++){
  li[i].addEventListener('click',function(){
    li[i].classList.toggle("change");
  })
}
.change {
  text-decoration: line-through;
}
<button id="enter">Enter</button>

Notebook

Apple

Mango

Banana

BlackBerry

Watermelon

redpajama_set_name: RedPajamaStackExchange


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Star Wars The Mandalorian TV Series
Update: Robert Rodriguez Directed Parts for 'The Mandalorian' Season 2; Bryce Dallas Howard Returns; James Mangold Denies Any Involvement
February 29, 2020 October 31, 2021 John Hoey #Star Wars, Disney Plus, The Mandalorian
Filming for The Mandalorian Season 2 started filming in the fall, and as principal photography approaches its end, word is that James Mangold and Robert Rodriguez visited the set and directed parts for the show. In addition to this, Bryce Dallas Howard has returned to direct, and the guest stars for this season are big. Here are the details of what we know to this point.
UPDATE: Ron Howard tweeted IGN's version of this story which in effect confirms part of the report, that his daughter Bryce Dallas Howard is indeed returning to direct an episode for season 2 of The Mandalorian.
The Mandalorian Season 2: James Mangold, Robert Rodriguez to Direct Parts, Bryce Dallas Howard to Return – IGN https://t.co/AdSVCik3Oo
— Ron Howard (@RealRonHoward) February 29, 2020
What makes this update interesting is a reply Ron Howard received under his tweet, by none other than James Mangold himself, who denied any involvement with The Mandalorian.
Hey Ron. I'm so pleased if it's true about Bryce directing another part of Mandalorian, but I can tell you its not true about me. Not doing it, never discussed it. And I suggest fans should look at who's "reporting" this & make a note that these sites present conjecture as fact.
— Mangold (@mang0ld) February 29, 2020
Remember this report was never confirmed by Lucasfilm, and until that happens these should always be treated as rumors (unless Ron Howard confirms it I guess). Speaking of which, Mangold hasn't commented on the way more publicized and bigger rumor that he is taking over directing Indiana Jones 5. Interesting.
The information comes from our friends Black Series Rebels, who talk about it on their latest episode (which can be viewed in its entirety below). The guys have a laid back and fun show, but when it comes to knowing their information, they know their information.
Season 2 of The Mandalorian will premiere on Disney Plus in October, with the show currently bucking the trend of recent Star Wars content as it has been universally praised by critics and fans alike. With that praise comes a desire for talent to want to get involved. The draw to the show stems from a low-risk, high-reward mentality, as BSR's Stephen Ellis says, "Talent is lining up to get a chance to play in Star Wars without any of the pressure."
BSR has been told that Robert Rodriguez (El Mariachi, Desperado, Planet Terror) and Academy Award nominee James Mangold (Cop Land, 3:10 to Yuma, Logan, Ford v Ferrari), and other unnamed directors, have directed sequences for an episode of The Mandalorian season 2. On what scale is uncertain but they have at least each directed sequences for an episode, with Rodriguez's contribution being called "great".
Robert Rodriguez's experience and love for modern Westerns makes this news come as no surprise. If your friend was the showrunner for a Western-style smash hit Star Wars series and invited you to come to the set, offering the chance to direct, what would you say? Exactly.
James Mangold's involvement feels like a bit of a closure and redemption. Not that a director with Mangold's resume needs help or work, but as you may remember, Mangold was set to write and direct a Boba Fett standalone film, but the project never made it past the early development stages. Well, he has finally directed at least some Star Wars, it just happens to be a different character wearing Mandalorian armor this time. In addition to this, it was reported earlier today that Mangold is in line to take over directing the upcoming Indiana Jones movie.
Rodriguez and Mangold directing parts for The Mandalorian makes me curious what the future may hold. As we know, there are (currently unfounded) rumors that claim Taika Waititi has been approached about directing one of the next theatrical Star Wars films, off the heels of directing the final episode of The Mandalorian Season 1. Though Waititi has brushed aside those rumors, I think there could be some truth to them. Is it possible Mangold gets another crack at a Star Wars film? He is reportedly directing Lucasfilm's Indiana Jones movie, so who knows? That's just wistful looking ahead to the future, to the horizon, on my part. Okay sorry, back to where I was, what I was doing.
BSR was also told Bryce Dallas Howard is returning to direct for Season 2. I imagine just as she did with Season 1, Howard is directing her own episode. Howard directed "Chapter 4: Sanctuary" for season 1, so when you share your GIFs of "The Child" sipping bone broth, you can thank her for that.
The last bit Alex and Steve talked about comes with no names attached, but apparently the guest stars for Season 2 are "big", and we will update you if we find out more information on this.
Here is Black Series Rebels' full episode where they talk about this story and more Star Wars goodness. The details from this story can be found starting at the 35:00 minute mark. Our buddies Alex, Steve, and Cisco also want you to know that starting with episode 101, they will be expanding BSR beyond Star Wars, to talk about other fun things in movies and pop culture beyond the galaxy far, far away, so keep up with them in their new expanded show!
https://www.youtube.com/watch?v=KWZuQKilIGU&feature=emb_logo
The Mandalorian season 2 will debut on Disney Plus in October, and it's shaping up to be bigger and better than the first. I have spoken.
John Hoey
John Hoey is the Lead Editor and Senior Writer for Star Wars News Net and the host of The Resistance Broadcast podcast
"For my ally is the Force, and a powerful ally it is."
https://www.starwarsnewsnet.com/author/john-hoey
The Resistance Broadcast - Will Cad Bane Appear in 'The Book of Boba Fett' Chapter 6?
The Mando Fan Show: 'The Book of Boba Fett' Chapter 3 Review (Video and Audio)
The Resistance Broadcast - Will 'The Book of Boba Fett' Finale Have a Post-Credits Scene?
Temuera Morrison Teases "Wow" Level Surprises Coming in 'The Book of Boba Fett' Finale
← The Clone Wars Fan Show: Episode 2 – 'A Distant Echo' Rating and Review (VIDEO)
Why Last Night's Episode Of 'The Clone Wars' Removed A Joke About Padmé →
John Hoey is the Lead Editor and Senior Writer for Star Wars News Net and the host of The Resistance Broadcast podcast"For my ally is the Force, and a powerful ally it is."
The Best of the New Star Wars Comics, So Far, PART TWO
February 5, 2016 October 25, 2021 Kyle Larson
Star Wars: Episode I Racer and Jedi Knight: Jedi Academy Heading to Nintendo Switch And PS4
March 27, 2020 October 25, 2021 Darby Harn
The Resistance Broadcast – Episode 82: Don't BB-H8! Appreciate!!! With GMA's Jeff Lowe.
September 11, 2017 November 4, 2021 Val Trichkov (Viral Hide)


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Q; Have you ever humiliated yourself so bad you just want to die? I sent the pastor of my church a card telling him how I feel about him, that I like him, etc. and now I regret it. He was feeling low and I wanted to cheer him up. All my words sounded really stupid. There's no telling what he thinks of me. I can't face him. I've thought of suicide to end this misery I'm in. That's how mortified I am. What can I do?
Since you mentioned suicide, what you should do is immediately call a good therapist in your area!
This reason for feeling so bad is NOT the biggest problem. You could contact the pastor to find out how he is feeling, etc…. The "thought of suicide to end this misery" is the problem!


text:
Why Hillary Clinton Refuses to Say Whether Merrick Garland Will Be Her SCOTUS Pick
The Democratic hopeful stays silent on renominating Obama's stalled candidate.
Damon Root | 9.9.2016 12:50 PM
White House / Flickr.com
Democratic presidential candidate Hillary Clinton has had lots of things to say about the future of the U.S. Supreme Court. "I have a bunch of litmus tests" for potential SCOTUS picks, Clinton has said. Foremost among those tests is the willingness of her judicial nominees to overturn the Supreme Court's 2010 defense of free speech in Citizens United v. Federal Election Commission.
But there is one major future-of-SCOTUS issue on which Clinton has been conspicuously silent. As Bloomberg's Greg Stohr puts it, "there's still a big question she hasn't answered: Would she re-nominate Merrick Garland to the open seat on the Supreme Court?"
Merrick Garland, who most recently served as chief judge of the U.S. Court of Appeals for the District of Columbia Circuit, was of course nominated by President Barack Obama back in March to replace the late Justice Antonin Scalia. But Garland's nomination has languished in the Senate ever since, with Republican lawmakers refusing to hold hearings or bring his name to a vote. And Clinton, Stohr observes, "has studiously avoided saying whether she would renominate Garland for the vacancy if it is still pending next year."
Why the silence? One possible explanation is that Garland is not at all the sort of liberal jurist that many of Clinton's supporters would like to see replace Scalia. Remember that the news of Garland's nomination did not exactly set progressive hearts aflutter. The president of the National Organization for Women questioned Garland's commitment to abortion rights, for example, while left-wing pundits complained that it was "extremely disappointing" to see the nomination of more "white men."
Garland's nomination surely came as a disappointment to that segment of the Democratic big tent that still cares about civil liberties. That's because Garland's record, as I've previously described it, "reflects a version of legal liberalism that tends to line up in favor of broad judicial deference to law enforcement and wartime executive power." In other words, Garland is not quite the dream candidate of the ACLU; nor is he the dream candidate of the Black Lives Matter movement.
So it's not really that much of a surprise to find Hillary Clinton keeping quiet on whether or not she'll re-up Merrick Garland's nomination if she wins the election. Clinton probably sees no upside in reoffending the various constituencies that found Garland to be such a disappointing judicial pick in the first place.
NEXT: Student Expelled Without Fair Hearing After University Misrepresented His Inoffensive Comment
Damon Root is a senior editor at Reason and the author of A Glorious Liberty: Frederick Douglass and the Fight for an Antislavery Constitution (Potomac Books).
Hillary Clinton Supreme Court Election 2016


Experienced cataract surgeons know that, despite having performed thousands of procedures over the years, every case is unique, and challenges can present suddenly. When a complexity is recognized preoperatively, the surgeon can prepare for it. The most difficult cases are those that involve a surprise. Like a good pilot, a surgeon desires an uneventful course, but he or she is always on the lookout for signs of danger. The surprises I encountered during surgery made the case I describe herein particularly difficult.
A local retinal specialist referred a woman with macular degeneration to me. His view of the retina was becoming increasingly blurred due to the development of a cataract. Upon initial examination at my clinic, everything seemed normal. The patient's biometric measurements were normal for her emmetropic eye, her pupils dilated well, and she had a cataract of moderate density in the right eye. With the goal of a plano result, I selected a single-piece monofocal IOL. The patient was nervous, because the surgery was to be performed on her better-seeing eye, on which she relied for performing daily activities. I wanted to perform an elegant and uneventful surgery to make the patient and the referring retinal physician happy.
After the topical and intracameral administration of anesthesia, I created a 0.8-mm paracentesis incision and a 2.8-mm temporal phaco incision with diamond blades. I carefully created a 5-mm round capsulorhexis that I centered on the visual axis. I extracted the nucleus during phacoemulsification and performed coaxial I/A to remove the cortex. A small amount of subincisional cortical material remained, which I planned to clean up after the IOL's insertion (Figure 1A). By implanting the IOL first, I reasoned, the posterior capsule would be protected during aspiration of the remaining cortical fragment.
After confirmation of its power, the IOL was loaded into the injector. The implant began to unfold as I inserted it directly into the capsular bag (Figure 1B and 1C). I used the chopper to center the lens, and I watched as the haptics unfolded into position (Figure 1D). Something was not right. The IOL began to descend posteriorly toward the retina (Figure 2A). The posterior capsule should have been securely holding the lens in place, but it was not. I had to think fast.
I used the chopper to bring the IOL up toward the anterior chamber, and I injected viscoelastic under the optic to help support it (Figure 2B). Viscoelastic is also helpful to compartmentalize the eye and prevent vitreous prolapse. At this point, I was tempted simply to dial the IOL into the ciliary sulcus, but I knew I had to avoid this approach, because the thick, nonangulated haptics of a single-piece acrylic IOL placed in the sulcus will eventually induce iris chafing, uveitis, and possibly glaucoma and hyphema.1 I decided to explant the IOL and replace it with a three-piece lens.
Using a microscissors and forceps, I grasped the IOL and cut it to facilitate its explantation. The technique that I prefer is to bisect most of the optic, pull it toward the incision, and rotate the optic out of the eye (Figure 2C). This method allows an IOL with a 6-mm optic to be removed through an incision that is half that size. In this case, however, I was in for another surprise: when I was removing the IOL, I inadvertently nicked an iris vessel and induced bleeding (Figure 2D). I injected more viscoelastic to push the iris away from the IOL, administered intracameral epinephrine to stop the bleeding, enlarged the incision, and finally removed the entire IOL.
I carefully examined the eye and studied the open posterior capsule. Rather than a small rent or break, a large opening extended the entire width of the posterior capsule. I can only speculate as to the cause of this capsular break, because even after repeatedly watching the video, I was unable to identify the exact moment when the capsule ruptured. It seems most logical that the IOL itself may have had a sharp or irregular edge that literally cut the capsule as the lens was being inserted. Because posterior capsules are less than 10 μm thick, they are susceptible to damage.
Fortunately, there was only a small amount of prolapsed vitreous in the anterior segment. To remove it and the last piece of subincisional cortical material, I used a 23-gauge bimanual vitrectomy system through two limbal paracentesis incisions (Figure 3A and B). I placed a three-piece IOL in the ciliary sulcus and oriented the lens vertically at the 12- and 6-o'clock meridians to achieve centration and a good fit (Figure 3C). Using the existing 5-mm capsulorhexis, I captured the 6-mm optic via a buttonhole technique to maximize long-term stability. With the bimanual 23-gauge vitrectomy system, I removed the viscolelastic. I then placed a 10–0 nylon suture to ensure a watertight closure of the main incision, which was particularly important for this case (Figure 3D). If the incision leaked and the anterior chamber collapsed, the IOL could become dislodged, causing more vitreous to prolapse forward.
Cases of posterior capsular rupture carry an increased risk of complications such as endophthalmitis, cystoid macular edema, and retinal detachment. My patient's postoperative course was prolonged, but the healing progressed nicely. The extended surgical procedure induced a higher degree of inflammation than usual, which required a longer course of postoperative steroids and nonsteroidal anti-inflammatory drugs. Care was coordinated with the patient's retinal specialist, who examined her retinal periphery in detail to ensure that the IOL exchange and anterior vitrectomy did not induce a retinal break or detachment. The patient recovered good vision and was very happy.
This case taught me several important lessons. First, surgeons must remain focused and vigilant even during seemingly routine procedures, because complications can present as surprises. For this case, everything looked great until the IOL started to sink into the vitreous. Second, ophthalmologists must do what is best for the patient in the long term. I am happy that I did not take the shortcut of simply dialing the existing single-piece IOL into the sulcus, which would have led to future problems and the need for another surgery. Third, more than one surprise can arise during surgery. When I encountered the ruptured posterior capsule, I did not expect also to face a bleeding iris during explantation of the IOL. Lastly, surgeons must take their time to resolve problems. The question I ask myself is, what would I want done if this were my eye?
Chang DF, Masket S, Miller KM, et al. Complications of sulcus placement of single-piece acrylic intraocular lenses: recommendations for backup IOL implantation following posterior capsule rupture. J Cataract Refract Surg. 2009;35(8):1445-1458.


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Why aren't you watching Casual?
Maybe you don't have a Hulu account. I would allow that, but really, Hulu has enough network shows that you should dump your cable TV subscription and replace it with $7.99 a month at Hulu. No, I wasn't paid to say that [although if someone at Hulu does want to pay me to shill for them, you can reach me on Twitter @adversioned].
If you do have a Hulu account though, no excuses. You should be watching Casual.