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\begin{center}
    \Huge Joint Detection and Identification of an Unobservable Change in the Distribution of a Random Sequence\\
    \vskip 10pt
    \large
    \begin{tabular}[t]{c@{\extracolsep{4em}}c}
        Savas Dayanik and Christian Goulding  & H.\ Vincent Poor \\
        Dept. of Operations Research and Financial Engineering & School of Engineering and Applied Science \\
        Princeton University, Princeton, NJ~~08544 & Princeton University, Princeton, NJ~~08544 \\
        Email: \{sdayanik, cgouldin\}@princeton.edu & Email:  poor@princeton.edu
    \end{tabular}
    \normalsize
\end{center}

\begin{multicols}{2}{
\textbf{\emph{Abstract--}This paper examines the joint problem of
detection and identification of a sudden and unobservable change
in the probability distribution function (pdf) of a sequence of
independent and identically distributed (i.i.d.)\ random variables
to one of finitely many alternative pdf's. The objective is quick
detection of the change and accurate inference of the ensuing pdf.
Following a Bayesian approach, a new sequential decision strategy
for this problem is revealed and is proven optimal.  Geometrical
properties of this strategy are demonstrated via numerical
examples.}

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

Consider a sequence of i.i.d.\ random variables $X_1, X_2,
\ldots$, taking values in some measurable space $(E,\Em)$. The
common probability distribution of the $X$'s is initially some
known probability measure $\P_0$ on $(E,\Em)$, and then, at some
\emph{unobservable} disorder time $\theta$, the common probability
distribution changes suddenly to another probability measure
$\P_{\mu}$ for some \emph{unobservable} index $\mu\in \Mh
\triangleq \{1,\ldots,M\}$. The objective is to detect the change
as quickly as possible, and, at the same time, to identify the new
probability distribution as accurately as possible, so that the
most suitable actions can be taken with the least delay.

This problem can be viewed as the fusion of two fundamental areas
of sequential analysis: change detection and multi-hypothesis
testing.  In traditional change detection problems, there is only
one change distribution, $\P_1$; therefore, the focus is
exclusively on detecting the change time. Whereas, in traditional
sequential multi-hypothesis testing problems, there is no change
time to consider.  Instead, every observation has common
distribution $\P_\mu$ for some unknown $\mu$, and the focus is
exclusively on the inference of $\mu$. Both of these subproblems
have been studied extensively. For recent reviews of these areas,
we refer the reader to \cite{MR1210954} and \cite{MR1844531} and
the references therein.

However, the joint problem involves key trade-off decisions not
taken into account by separately applying techniques for these
subproblems. While raising an alarm as soon as the change occurs
is advantageous for the change detection task, it is undesirable
for the identification task because waiting longer provides more
observations for inferring the change distribution. Likewise, the
unknown change time complicates the identification task, and, as a
result, adaptation of existing sequential multi-hypothesis testing
algorithms is problematic.
\noindent\begin{minipage}[!b]{\columnwidth} \footnotetext{The
research of Savas Dayanik was supported by the Air Force Office of
Scientific Research, under grant AFOSR-FA9550-06-1-0496. The
research of H.\ Vincent Poor was supported in part by the U.S.\
Army Pantheon Project.}
\end{minipage}

Decision strategies for the joint problem have a wide array of
applications, such as fault detection and isolation in industrial
processes, target detection and identification in national
defense, pattern recognition and machine learning, radar and sonar
signal processing, seismology, speech and image processing,
biomedical signal processing, finance, and insurance. However, the
theory has not been broadly developed.
Nikiforov~\cite{Nikiforov1995} provides the first results for this
problem, showing asymptotic optimality for a certain non-Bayesian
approach, and Lai~\cite{Lai2000} generalizes these results through
the development of information-theoretic bounds and the
application of likelihood methods.  In this paper, we follow a
Bayesian approach to reveal a new optimal strategy for this
problem and we describe an accurate numerical scheme for its
implementation.

In Sec.\ \ref{sec:Problem-statement} we formulate precisely the
problem in a Bayesian framework, and in Sec.\
\ref{sec:Reformulation} we show that it can be reduced to an
optimal stopping of a Markov process whose state space is the
standard probability simplex.  In addition, we establish a simple
recursive formula that captures the dynamics of the process and
yields a sufficient statistic fit for online tracking.

In Sec.\ \ref{sec:dynamic-programming-solution} we use optimal
stopping theory to substantiate the optimality equation for the
value function of the optimal stopping problem.  Moreover, we
prove that this value function is bounded, concave, and continuous
on the standard probability simplex and that the optimal stopping
region consists of $M$ non-empty, convex, closed, and bounded
subsets.  Also, we consider a truncated version of the problem
that allows at most $N$ observations from the sequence of random
measurements.  We establish an explicit bound (inversely
proportional to $N$) for the approximation error associated with
this truncated problem.

In Sec.\ \ref{sec:Special-cases} we show that the separate
problems of change detection and sequential multi-hypothesis
testing are solved as special cases of the overall joint solution.
We illustrate some geometrical properties of the optimal method
and demonstrate its implementation by numerical examples for the
special cases $M=2$ and $M=3$.  Specifically, we show instances in
which the $M$ convex subsets comprising the optimal stopping
region are connected and instances in which they are not.
Likewise, we show that the continuation region (i.e., the
complement of the stopping region) need not be connected. We refer
the reader to \cite{DGP06} for complete proofs of the results.

\section{Problem statement}
\label{sec:Problem-statement}

Let $(\Omega,\F, \P)$ be a probability space hosting random
variables $\theta:\Omega\mapsto\{0,1,\ldots\}$ and
$\mu:\Omega\mapsto \Mh \triangleq \{1,\ldots,M\}$ and a process $X
= (X_n)_{n\geq1}$ taking values in some measurable space
$(E,\Em)$. Suppose that for every $t\ge 1$, $i \in \Mh$, $n\ge 1$,
and $(E_k)^n_{k=1}\subseteq \Em $ we have
\begin{multline*}
  \P\left\{\theta=t, \mu=i, X_1 \in E_1,\ldots, X_n \in E_n \right\}  \\
  = (1-p_0) (1-p)^{t-1} p \nu_i \prod_{k=1}^{(t-1) \land n}
  \P_0(E_k) \prod_{\ell = t\lor 1}^{n} \P_i(E_{\ell})
\end{multline*}
for some given probability measures $\P_0,\P_1,\ldots,\P_M$ on
$(E,\Em)$, known constants $p_0\in[0,1]$, $p\in(0,1)$, and
$\nu_i>0,i\in \Mh$ such that $\nu_1+\cdots+\nu_M = 1$, where
$x\wedge y\triangleq \min\{x,y\}$ and $x\vee y\triangleq
\max\{x,y\}$.  Namely, $\theta$ is independent of $\mu$; it has a
zero-modified geometric distribution with parameters $p_0$ and $p$
in the terminology of
\cite[Sec.\
3.6]{MR1490300}, which reduces to the standard geometric
distribution when $p_0=0$.

Conditionally on $\theta$ and $\mu$, the random variables $X_n$, $n\ge
1$ are independent;  $X_1,\ldots, X_{\theta-1}$ and $X_{\theta},
X_{\theta+1},\ldots$ are identically distributed with common
distributions $\P_0$ and $\P_{\mu}$, respectively.  The probability
measures $\P_0,\P_1,\ldots,\P_M$ always admit densities with respect
to some sigma-finite measure $m$ on $(E,\Em)$; for example, we can
take $m = \P_0+\P_1\cdots+\P_M$.  So, we fix $m$ and denote the
corresponding densities by $f_0, f_1,\ldots,f_M$, respectively.

Suppose now that we observe sequentially the random variables
$X_n$, $n\ge 1$. Their common pdf $f_0$ changes at stage $\theta$
to some other pdf $f_{\mu}$, $\mu\in \Mh$. Our objective is to
detect the change time $\theta$ as quickly as possible \emph{and}
to identify the change index $\mu$ as accurately as possible. More
precisely, given costs associated with detection delay, false
alarm, and false identification of the change index, we seek a
strategy that minimizes the expected total change detection
\emph{and} identification cost.

Let $\mathbb{F} = (\F_n)_{n\geq0}$ denote the natural filtration
of the observation process $X$, where
\begin{align*}
  \F_0=\{\varnothing,\Omega\}\quad\text{and}\quad
  \F_n=\sigma(X_1,\ldots,X_n),\quad n\geq1.
\end{align*}
A \emph{strategy} $\delta=(\tau, d)$ is a pair consisting of a
\emph{stopping time} $\tau$ of the filtration $\mathbb{F}$ and a
\emph{terminal decision rule} $d: \Omega \mapsto \Mh$ measurable
with respect to the history $\F_{\tau}=\sigma(X_{n\wedge\tau};
n\geq1)$ of observation process $X$ through stage $\tau$. Applying
a strategy $\delta=(\tau,d)$ consists of announcing at the end of
stage $\tau$ that the common pdf has changed from $f_0$ to $f_d$
at or before stage $\tau$. Let
\begin{align*}
  \Delta \triangleq \{(\tau,d) \mid \tau\in\mathbb{F}, \text{ and
    $d\in\F_\tau$ is an $\Mh$-valued r.\ v.}\}
\end{align*}
denote the collection of all such sequential decision strategies.

For every strategy $\delta=(\tau, d)\in\Delta$, we define a
\emph{Bayes risk function}
\begin{align}
  R(\delta) = c\,\E[(\tau-\theta)^+] + \E[a_{0
    d}\II{\tau<\theta}+a_{\mu
    d}\II{\theta\leq\tau<\infty}]\label{E:BayesRiskUnderP}
\end{align}
\noindent as the expected diagnosis cost: the sum of the expected
detection delay cost and the expected terminal decision cost upon
alarm, where $c>0$ and $a_{ij}\ge 0, i\in\{0\}\cup\Mh,j\in\Mh$ are
known constants satisfying $a_{ii}=0, i\in\Mh$ (i.e., no cost for
a correct terminal decision), and $(x)^+\triangleq\max\{x,0\}$.

The problem is to find a sequential decision strategy
$\delta=(\tau,d)\in\Delta$ (if it exists) with the \emph{minimum
Bayes
  risk}
\begin{align}
  R^* \triangleq \inf_{\delta\in\Delta} R(\delta).\label{E:UDef1}
\end{align}

\section{Posterior analysis and formulation as an optimal stopping
  problem}
\label{sec:Reformulation}

In this section we show that the Bayes risk function in
(\ref{E:BayesRiskUnderP}) can be written as the expected value of the
running and terminal costs driven by a certain Markov process.  We use
this fact to recast the minimum Bayes risk in (\ref{E:UDef1}) as a
Markov optimal stopping problem.

Let us introduce the posterior probability processes
\begin{align*}
  \Pi_n^{(0)} &\triangleq
  \P\{\theta>n\,|\,\F_n\}\quad\text{and}\qua
  \Pi_n^{(i)} \triangleq \P\{\theta\leq n,\mu = i\,|\,\F_n\}
\end{align*}
for $i\in \Mh, n\geq 0$. Having observed the first $n$
observations, $\Pi_n^{(0)}$ is the posterior probability that the
change \emph{has not} yet occurred at or before stage $n$, while
$\Pi_n^{(i)}$ is the posterior joint probability that the change
\emph{has} occurred by stage $n$ and that the hypothesis $\mu=i$
is correct.  The connection of these posterior probabilities to
the loss structure for our problem is established in
the next proposition.

\begin{proposition}\label{P:BayesRiskInTermsOfPi}
  For every sequential decision strategy $\delta\in\Delta$, the Bayes
  risk function (\ref{E:BayesRiskUnderP}) can be expressed in terms of
  the process $\bPi\triangleq\{ \bPi_n= (\Pi_n^{(0)}, \ldots,
  \Pi_n^{(M)})\}_{n\geq 0}$ as
{
  \begin{align*}
    R(\delta) &= \E\!\left[ \sum_{n=0}^{\tau-1}c\,(1\!-\!\Pi_n^{(0)})
      +\II{\tau<\infty}\!\sum_{j=1}^{M}\II{d=j}\!\sum_{i=0}^{M}
      a_{ij}\Pi_{\tau}^{(i)}\right]\!\!.
  \end{align*}}
\end{proposition}

While our original formulation of the Bayes risk function
(\ref{E:BayesRiskUnderP}) was in terms of the values of the
unobservable random variables $\theta$ and $\mu$, Proposition
\ref{P:BayesRiskInTermsOfPi} gives us an equivalent version of the
Bayes risk function in terms of the posterior distributions for
$\theta$ and $\mu$.  This is particularly effective in light of
Proposition \ref{P:PiProperties}, which we state with the aid of some
additional notation that is referred to throughout the paper.  Let
\begin{align*}
  S^M \triangleq
  \left\{\bpi=(\pi_0,\pi_1,\ldots,\pi_M)\in[0,1]^{M+1}\,\bigm|\,
    {\textstyle\sum_{i=0}^M}\pi_i = 1 \right\}
\end{align*}
denote the standard $M$-dimensional probability simplex. Define
the mappings $D_i:S^M \times E \mapsto [0,1], i\in \Mh$ and $D:S^M
\times E \mapsto [0,1]$ by
\begin{align*}

  D_{i}(\bpi,x) &\triangleq \left\{
    \begin{aligned}
      &(1-p)\pi_0 f_0(x), && i=0\\
      &(\pi_i+\pi_0\,p\nu_i) f_i(x), && i\in \Mh
    \end{aligned}
  \right\}
\end{align*}
and $D(\bpi,x)\triangleq\sum_{i=0}^{M}D_{i}(\bpi,x)$, and the
operator $\mathbb{T}$ on the collection of bounded functions $f:S^M
\mapsto\R$ by
\begin{align}
  \label{E:T-operator}
  (\mathbb{T} f)(\bpi) &\triangleq\!\int_{E}
  m(dx)\,D(\bpi,x)\,f\!\left({\textstyle\frac{D_0(\bpi,x)}{D(\bpi,x)},\ldots,
    \frac{D_M(\bpi,x)}{D(\bpi,x)}}\right)
\end{align}
for every $\bpi\in S^M$.

\begin{proposition}\label{P:PiProperties}
  (a) The process
    $\bPi^{(0)}\triangleq\{\Pi_n^{(0)},\F_n\}_{n\geq 0}$ is a
    supermartingale, and $\E\,\Pi_n^{(0)} \leq (1-p)^n$ for every
    $n\geq 0$.

  (b) The process
    $\bPi^{(i)}\triangleq\{\Pi_n^{(i)},\F_n\}_{n\geq 0}$ is a
    submartingale for every $i\in \Mh$.

  (c) The process
    $\bPi=\{(\Pi_n^{(0)},\ldots,\Pi_n^{(M)})\}_{n\geq 0}$ is a Markov
    process, and
    \begin{align}
      \Pi_{n+1}^{(i)} =
      \frac{D_i(\bPi_n,X_{n+1})}{D(\bPi_n,X_{n+1})},\quad i\in
      \{0\}\cup\Mh ,\quad n\geq 0,\label{E:Pi-Dynamics}
    \end{align}
    with initial state $\Pi_{0}^{(0)} = 1-p_0$ and
    $\Pi_{0}^{(i)}=p_0\nu_i$, $i\in \Mh.$
    Moreover, for every bounded function $f:S^M\mapsto\R$ and $n\geq
    0$, we have $\E[f(\bPi_{n+1})|\bPi_n] = (\mathbb{T} f)(\bPi_n)$.
\end{proposition}

\begin{remark}\label{R:PiProperties}
  Since $\bPi$ is uniformly bounded, the limit
  $\lim_{n\rightarrow\infty}\bPi_n$ exists by the martingale
  convergence theorem.  Moreover,
  $\lim_{n\rightarrow\infty}\Pi_n^{(0)}=0$ a.s.\ by Proposition
  \ref{P:PiProperties}(a) since $p\in(0,1)$.
\end{remark}

Now, let the functions $h, h_1,\ldots,h_M$ from $S^M$ into $\R_+$ be
defined by
\begin{align*}
  h(\bpi)\triangleq \min_{j\in \Mh} h_j(\bpi) \quad \text{and} \quad
  h_j(\bpi) \triangleq \sum_{i=0}^{M} \pi_i\, a_{ij},\quad j\in \Mh,
\end{align*}
respectively.  Then, we note that for every $\delta=(\tau,d)\in
\Delta$, we have
\begin{align*}
  R(\tau, d) &= \E\left[ \sum_{n=0}^{\tau-1}c(1-\Pi_n^{(0)})
    +\II{\tau<\infty}\sum_{j=1}^{M}\II{d=j}h_j(\Pi_{\tau})\right]\\
  &\geq \E\left[ \sum_{n=0}^{\tau-1}c(1-\Pi_n^{(0)})
    +\II{\tau<\infty}h(\Pi_{\tau})\right] = R(\tau,\tilde{d})
\end{align*}
where we define on the event $\{\tau<\infty\}$ the terminal decision
rule $\tilde{d}$ to be any index satisfying
$h_{\tilde{d}}(\Pi_{\tau})=h(\Pi_{\tau})$.  In other words, an optimal
terminal decision depends only upon the value of the $\bPi$ process at
the stage in which we stop.  Note also that the functions $h$ and
$h_1,\ldots,h_M$ are bounded on $S^M$.  Therefore, we have the
following:

\begin{lemma}\label{L:OSP1}
  The minimum Bayes risk (\ref{E:UDef1}) reduces to the following
  optimal stopping of the Markov process $\bPi$:
  \begin{align*}
    R^* &= \inf_{(\tau,d)\in\Delta}R(\tau,d) =
    \inf_{(\tau,\tilde{d})\in\Delta}R(\tau,\tilde{d})\\
    &= \inf_{\tau\in\mathbb{F}}
    \,\E\left[\sum_{n=0}^{\tau-1}c\,(1-\Pi_n^{(0)})+\II{\tau<\infty}h(\Pi_\tau)\right].
  \end{align*}
\end{lemma}

We simplify this formulation further by showing that it is enough to
take the infimum over
\begin{align}
  C \triangleq \{\tau\in\mathbb{F}\,|\,\tau<\infty \text{ a.s. and }
  \E Y_\tau^-<\infty\},\label{E:C}
\end{align}
where we define
\begin{align*}
  -Y_n \triangleq \sum_{k=0}^{n-1}c\,(1-\Pi_k^{(0)})+h(\Pi_n),\quad
  n\geq
\end{align*}
as the minimum \emph{partial risk} obtained by making the best
terminal decision on $\{\tau=n\}$. Since $h(\cdot)$ is bounded on
$S^M$, the process $\{Y_n, \F_n; n\ge 0\}$ consists of integrable
random variables. So the expectation $\E Y_\tau$ exists for every
$\tau\in\mathbb{F}$, and our problem becomes
\begin{align}
  -R^*=\sup_{\tau\in\mathbb{F}}\E
  Y_\tau.\label{E:OptimizationProblemTau}
\end{align}

Observe that $\E \tau <\infty$ for every $\tau\in C$ because $\infty >
(1/c)\E Y_\tau^- \geq \E (\tau-\theta)^+ \geq \E (\tau-\theta) \geq \E
\tau -\E \theta \ge \E\tau - (1/p)$.  In fact, we have $\E
Y_\tau>-\infty \Leftrightarrow \E Y_\tau^-<\infty \Leftrightarrow \E
\tau<\infty$ for every $\tau\in\mathbb{F}$.  Since
$\sup_{\tau\in\mathbb{F}}\E Y_{\tau} \geq \E Y_0 > -h(\Pi_0) >
-\infty$, it is enough to consider $\tau\in\mathbb{F}$ such that
$\E\tau <\infty$. Namely, (\ref{E:OptimizationProblemTau}) reduces to
\begin{align}
  -R^*=\sup_{\tau\in C} \E Y_\tau.\label{E:OptimizationProblemC}
\end{align}

\section{Solution via optimal stopping theory}
\label{sec:dynamic-programming-solution}

In this section we derive an optimal solution for the problem
in~\eqref{E:UDef1} by building on the formulation
of~\eqref{E:OptimizationProblemC} via the tools of optimal
stopping theory, which are detailed in \cite{MR0331675}.

\mysubsection{The optimality equation}\label{sec:Derive-Opt-Eqn}

We begin by applying the method of truncation with a view of
passing to the limit to arrive at the final result.  Define for
every pair of integers $n, N$ satisfying $0 \le n \le N$ the
sub-collections
\begin{align*}
  C_n &\triangleq \{\tau \vee n\,|\,\tau\in C\}\quad\text{and}\quad
  C_n^N \triangleq \{\tau \wedge N\,|\,\tau\in C_n\}
\end{align*}
of stopping times in $C$ of \eqref{E:C} and the families of
(truncated) optimal stopping problems
\begin{align}
  \label{E:Vn-and-VnN}
  -V_n \triangleq \sup_{\tau\in C_n}\E Y_\tau
  \quad\text{and}\quad
  -V_n^N \triangleq \sup_{\tau\in C_n^N}\E
  Y_\tau
\end{align}
corresponding to $(C_n)_{n\geq 0}$ and $(C_n^N)_{0\leq n\leq N}$,
respectively. Note that $C\equiv C_0$ and $R^*\equiv V_0$.

To investigate these optimal stopping problems, we introduce
versions of the \emph{Snell envelope} of $(Y_n)_{n\geq 0}$ (i.e.,
the smallest regular supermartingale dominating $(Y_n)_{n\geq
  0}$) corresponding to $(C_n)_{n\geq 0}$ and $(C_n^N)_{0\leq n\leq
  N}$, respectively, defined by
\begin{align*}
  \gamma_n &\triangleq \esup_{\tau\in C_n} \E [Y_\tau\,|\,\F_n]
  \quad\text{and}\quad \gamma_n^N \triangleq \esup_{\substack{\tau\in
      C_n^N}} \E [Y_\tau\,|\,\F_n].
\end{align*}
Then through the following series of lemmas we point out several
useful properties of these Snell envelopes. Finally, we extend
these results to an arbitrary initial state vector and establish
the optimality equation.  Note that each of the ensuing
(in)equalities between random variables are in the $\P$-almost
sure sense.

First, these Snell envelopes provide the following alternative
expressions for the optimal stopping problems introduced in
\eqref{E:Vn-and-VnN} above.

\begin{lemma}\label{L:Vn-equal-expected-gamma}
  For every $N\ge 0$ and $0\le n\le N$, we have $-V_n = \E \gamma_n$
  and $-V_n^N = \E \gamma_n^N$.
\end{lemma}

Second, we have the following backward-induction equations.

\begin{lemma}\label{L:backward-induction-eqns}
  We have $\gamma_n = \max\{Y_n, \E [\gamma_{n+1}\,|\,\F_n]\}$ for
  every $n\ge 0$. For every $N\ge 1$ and $0\le n \le N-1$, we have
  $\gamma_N^N = Y_N$ and $\gamma_n^N = \max\{Y_n, \E
  [\gamma_{n+1}^N\,|\,\F_n]\}$.
\end{lemma}

We also have that these versions of the Snell envelopes coincide in
the limit as $N\rightarrow\infty$.  That is,

\begin{lemma}\label{L:gamma-equals-gamma-prime}
  For every $n\geq 0$, we have $\gamma_n = \lim_{N\rightarrow\infty}
  \gamma_n^N$.
\end{lemma}

Next, recall from \eqref{E:T-operator} and Proposition
\ref{P:PiProperties}(c) the operator $\mathbb{T}$ and let us introduce the
operator $\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)$ on the collection of bounded functions $f:S^M
\mapsto \R_+$ defined by
\begin{align*}
  (\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) f)(\bpi) \triangleq
  \min\{h(\bpi),c(1-\pi_0)+(\mathbb{T} f)(\bpi)\},\quad\bpi\in S^M.
\end{align*}
Observe that $0\leq \mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) f \leq h$.  That is, $\bpi\mapsto(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) f)(\bpi)$
is a nonnegative bounded function. Therefore, $\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^2 f\equiv \mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) f)$
is well-defined.  If $f$ is nonnegative and bounded, then $\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^n
f\equiv \mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^{n-1} f)$ is defined for every $n\ge 1$, with
$\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^0 f\equiv f$ by definition.  Using operator $\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)$, we can express
$(\gamma_n^N)_{0\leq n\leq N}$ in terms of the process $\bPi$ as
stated in the following lemma.

\begin{lemma}\label{L:pg-36}
  For every $N\ge 0$, and $0\le n \le N$, we have
    $\gamma_n^N = -c\sum_{k=0}^{n-1}(1-\Pi_k^{(0)})-(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^{N-n}h)(\Pi_n)$.
\end{lemma}

The next lemma shows how the optimal stopping problems can be
rewritten in terms of the operator $\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)$. It also conveys the
connection between the truncated optimal stopping problems and the
initial state $\bPi_0$ of the $\bPi$ process.

\begin{lemma}\label{L:pg-38}
  We have
  (a) $V_0^N=(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^N h)(\bPi_0)$ for every $N\geq 0$, and
  (b) $V_0={\displaystyle\lim_{N\rightarrow\infty}(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^N
      h)(\bPi_0)}$.
\end{lemma}

Observe that since $\bPi_0\in\F_0=\{\varnothing,\Omega\}$, we have
$\P\{\bPi_0=\bpi\}=1$ for some $\bpi\in S^M$.  On the other hand, for
every $\bpi\in S^M$ we can construct a probability space
$(\Omega,\F,\P_{\bpi})$ hosting a Markov process $\bPi$ with the same
dynamics as in \eqref{E:Pi-Dynamics} and $\P_{\bpi}\{\bPi_0=\bpi\}=1$.
Moreover, on such a probability space, the preceding results remain
valid.  So, let us denote by $\E_{\bpi}$ the expectation with respect
to $\P_{\bpi}$ and rewrite \eqref{E:Vn-and-VnN} as
\begin{align*}
  -V_n(\bpi) \triangleq \sup_{\tau\in C_n}\E_{\bpi} Y_\tau
  \quad \text{and} \quad -V_n^N(\bpi) \triangleq \sup_{\tau\in
    C_n^N}\E_{\bpi} Y_\tau
\end{align*}
for every $\bpi\in S^M$.  Then Lemma \ref{L:pg-38} implies that
\begin{align}
\label{eq:value-functions}
  V_0^N\!(\bpi)=(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^N h)(\bpi)\!\!\quad\text{ and
  }\quad\!\!V_0(\bpi)=\lim_{N\rightarrow\infty}(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^N h)(\bpi)
\end{align}
for every $\bpi\in S^M$.  Taking limits as $N\rightarrow\infty$ of
both sides in $(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^{N+1}h)(\bpi) = \mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi)^N h)(\bpi)$ and applying
the monotone convergence theorem on the right-hand side yields
$V_0(\bpi) = (\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) V_0)(\bpi)$.  Hence, we have shown the following
result.

\begin{proposition}[Optimality equation]\label{P:Dyn-prog-eqn}
  For every $\bpi\in S^M$,
  \begin{align}
    V_0(\bpi)\!=\!(\mathbb{M}}                     % Operator (Mf)(pi) = min{h(pi), c(1 - pi_0) + (Tf)(pi) V_0)(\bpi) \equiv
    \min\{h(\bpi),c(1\!-\!\pi_0)\!+\!(\mathbb{T} V_0)(\bpi)\}.\label{E:Dyn-prog-eqn}
  \end{align}
\end{proposition}

\begin{remark}
  By solving $V_0(\bpi)$ for any initial state $\bpi\in S^M$, we
  capture the solution to the original problem since property (c) of
  Proposition \ref{P:PiProperties} and \eqref{E:OptimizationProblemC}
  imply that
    $R^* = V_0(1-p_0,p_0\nu_1,\ldots,p_0\nu_M)$.
\end{remark}

\mysubsection{Some properties of the value
function}\label{sec:V-properties}

Now, we reveal some important properties of the value function
$V_0(\cdot)$ of (\ref{eq:value-functions}).  These results help us
to establish an optimal solution for $V_0(\cdot)$, and hence an
optimal solution for $R^*$, in the next subsection.

\begin{lemma}\label{L:V-concave}
    If $g:S^M \mapsto \R$ is a bounded concave function, then so is $\mathbb{T} g$.
\end{lemma}

\begin{proposition}\label{P:V-concave}
  The mappings $\bpi \mapsto V_0^N(\bpi), N\geq 0$ and $\bpi \mapsto
  V_0(\bpi)$ are concave.
\end{proposition}

\begin{proposition}\label{P:V-convergence-rate}
    For every $N\ge 1$ and $\bpi\in S^M$, we have
        \begin{align*}
            V_0(\bpi)\leq V_0^N(\bpi) \leq
            V_0(\bpi)+\left(\frac{\|h\|^2}{c}+\frac{\|h\|}{p}\right)\frac{1}{N}.
        \end{align*}
        Since $\|h\|\triangleq \sup_{\bpi\in S^M} |h(\bpi)|<\infty$,
        $\lim_{N\rightarrow\infty} \downarrow V_0^N(\bpi) = V_0(\bpi)$
        uniformly in $\bpi\in S^M$.
\end{proposition}

\begin{proposition}\label{P:V0N-continuous}
  For every $N\ge 0$, the function $V_0^N:S^M\mapsto\R_+$ is
  continuous.
\end{proposition}

\begin{corollary}\label{C:V-continuous}
  The function $V_0:S^M \mapsto \R_+$ is continuous.
\end{corollary}

Note that $S^M$ is a compact subset of $\R^{M+1}$, so while continuity
of $V_0(\cdot)$ on the interior of $S^M$ follows from the concavity of
$V_0(\cdot)$ by Proposition \ref{L:V-concave}, Corollary
\ref{C:V-continuous} establishes continuity on all of $S^M$, including
its boundary.

\mysubsection{An optimal sequential decision
  strategy}\label{sec:optimal-soln}

Finally, we describe the optimal stopping region in $S^M$ implied
by the value function $V_0(\cdot)$, and we present an optimal
sequential decision strategy for our problem. Let us define for
every $N\ge 0$,
\begin{align*}
  \Gamma_N &\triangleq \{\bpi\in S^M\,|\, V_0^N(\bpi)=h(\bpi)\},\\
  \Gamma_N^{(j)} &\triangleq \Gamma_N \cap \{\bpi\in S^M\,|\,
  h(\bpi)=h_j(\bpi)\}, \; j\in \Mh, \\
  \Gamma &\triangleq \{\bpi\in S^M\,|\, V_0(\bpi)=h(\bpi)\},\\
  \Gamma^{(j)} &\triangleq \Gamma \cap \{\bpi\in S^M\,|\,
  h(\bpi)=h_j(\bpi)\}, \; j\in \Mh.
\end{align*}
For each $j\in \{0\}\cup \Mh$, let $e_j\in S^M$ denote the unit
vector consisting of zero in every component except for the $j$th
component, which is equal to one. Note that
$e_0,\ldots,e_M$ are the extreme points of the closed
convex set $S^M$, and any vector $\bpi=(\pi_0,\ldots,\pi_M)\in S^M$
can be expressed in terms of $e_0,\ldots,e_M$ as $\bpi =
\sum_{j=0}^{M}\pi_je_j$.

\begin{theorem}\label{T:Gamma-decreasing-subsets}
  For every $j\in \Mh$, $(\Gamma_N^{(j)})_{N\geq 0}$ is a decreasing
  sequence of non-empty, closed, convex subsets of $S^M$.  Moreover,
  \begin{gather*}
    \Gamma_0^{(j)} \supseteq \Gamma_1^{(j)} \supseteq \cdots \supseteq
    \Gamma^{(j)},\\
    \Gamma^{(j)}\supseteq \left\{\bpi\in S^M
      \,|\,h_j(\bpi)\leq\min\{h(\bpi),c(1-\pi_0)\}\right\} \ni
    e_j,\\
    \Gamma = \bigcap_{N=1}^{\infty}\Gamma_N =
    \bigcup_{j=1}^{M}\Gamma^{(j)},\quad\text{and}\quad
    \Gamma^{(j)}=\bigcap_{N=1}^{\infty}\Gamma_N^{(j)},\quad
    j\in \Mh.
  \end{gather*}
  Furthermore, $S^M = \Gamma_0 \supseteq \Gamma_1 \supseteq \cdots
  \supseteq \Gamma \supsetneqq \{e_1,\ldots,e_M\}$.
\end{theorem}

\begin{lemma}\label{L:gamma-n-V}
  For every $n\geq 0$, we have $\gamma_n =
  -c\sum_{k=0}^{n-1}(1-\Pi_k^{(0)})-V_0(\Pi_n).$
\end{lemma}

\begin{theorem}\label{T:sigma-properties}
  Let $\sigma \triangleq \inf\{n\geq 0 \,|\, \bPi_n \in\Gamma\}$.
  (a) The stopped process $\{\gamma_{n \wedge\sigma}, \F_n;
    n\geq 0\}$ is a martingale.

  (b) The random variable $\sigma$ is an optimal stopping time
    for $V_0$, and

  (c) $\E\,\sigma<\infty$.
\end{theorem}

Therefore, the pair $(\sigma, d^*)$ is an optimal sequential
decision strategy for \eqref{E:UDef1}, where the optimal stopping
rule $\sigma$ is given by Theorem~\ref{T:sigma-properties}, and,
as in the proof of Lemma~\ref{L:OSP1}, the optimal terminal
decision rule $d^*$ is given by
\begin{align*}
  d^* = j\! \quad\! \text{ on the event}\! \quad\! \{\sigma=n, \bPi_n\in
  \Gamma^{(j)}\}\! \quad\! \text{ for every } n\geq 0.
\end{align*}
Accordingly, the set $\Gamma$ is called the \emph{stopping region}
implied by $V_0(\cdot)$, and
Theorem~\ref{T:Gamma-decreasing-subsets} reveals its basic
structure.  We demonstrate the use of these results in the
numerical examples of Sec.\ \ref{sec:Special-cases}.

Note that we can take a similar approach to prove that the
stopping rules $\sigma_N\triangleq\inf\{n\geq 0\,|\, \bPi_n \in
\Gamma_{N-n}\}, N\geq 0$ are optimal for the truncated problems
$V_0^N(\cdot), N\geq 0$ in (\ref{eq:value-functions}).  Thus, for
each $N\geq 0$, the set $\Gamma_{N}$ is called the stopping region
for $V_0^N(\cdot)$: it is optimal to terminate the experiments in
$\Gamma_N$ if $N$ stages are left before truncation.

\section{Special cases and examples}\label{sec:Special-cases}

\mysubsection{A.\ N.\ Shiryaev's sequential change detection
problem}

Set $a_{0j}=1$ for $j\in \Mh$ and $a_{ij}=0$ for $i,j\in \Mh$, then
the Bayes risk function \eqref{E:BayesRiskUnderP} becomes
  $R(\delta) = \P\{\tau<\theta\} + c\,\E[(\tau-\theta)^+]$.
This is the Bayes risk studied by Shiryaev
\cite{MR0155708,MR0468067} to solve the sequential change
detection problem.

\mysubsection{Sequential multi-hypothesis testing}

Set $p_0=1$, then $\theta = 0$ a.s.\ and thus the Bayes risk
function \eqref{E:BayesRiskUnderP} becomes
  $R(\delta) = \E[c\tau + a_{\mu d}\II{\tau<\infty}]$.
This gives the sequential multi-hypothesis testing problem studied
by Wald and Wolfowitz \cite{MR0034005} and Arrow, Blackwell, and
Girshick \cite{MR0032173}; see also \cite{MR597146}.

\mysubsection{Two alternatives after the change}

In this subsection we consider the special case $M=2$ in which we
have only two possible change distributions, $f_1(\cdot)$ and
$f_2(\cdot)$. We describe a graphical representation of the
stopping and continuation regions for an arbitrary instance of the
special case $M=2$.  Then we use this representation to illustrate
geometrical properties of the optimal method (Sec.\
\ref{sec:dynamic-programming-solution}.\ref{sec:optimal-soln}) via
model instances for certain choices of the model parameters $p_0$,
$p$, $\nu_1$, $\nu_2$, $f_0(\cdot)$, $f_1(\cdot)$, $f_2(\cdot)$,
$a_{01}$, $a_{02}$, $a_{12}$, $a_{21}$, and $c$.

Let the linear mapping $L:\R^3\mapsto\R^2$ be defined by
$L(\pi_0,\pi_1,\pi_2)\triangleq(\tfrac{2}{\sqrt{3}}\pi_1
+\tfrac{1}{\sqrt{3}}\pi_2,\pi_2)$. Since $\pi_0=1-\pi_1-\pi_2$ for
every $\bpi=(\pi_0,\pi_1,\pi_2)\in S^2\subset\R^3$, we can recover
the preimage $\bpi$ of any point $L(\bpi)\in L(S^2)\subset\R^2$.
For every point $\bpi=(\pi_0,\pi_1,\pi_2)\in S^2$, the coordinate
$\pi_i$ is given by the Euclidean distance from the image point
$L(\bpi)$ to the edge of the image triangle $L(S^2)$ that is
\emph{opposite} the image point $L(e_i)$, for each $i=0,1,2$. For
example, the distance from the image point $L(\bpi)$ to the edge
of the image triangle opposite the lower-left-hand corner
$L(1,0,0)=(0,0)$ is the value of the preimage coordinate $\pi_0$.
See Fig.\ \ref{F:S2-to-2D}.

\ifpdf
\myFigure{3.5in}{imgS2to2D.pdf}{Linear mapping $L$ of the
standard two-dimensional probability simplex $S^2$ from the
positive orthant of $\R^3$ into the positive quadrant of
$\R^2$.}{F:S2-to-2D}
\fi

Therefore, we can work with the mappings $L(\Gamma)$ and
$L(S^2\setminus\Gamma)$ of the stopping region $\Gamma$ and the
continuation region $S^2\setminus\Gamma$, respectively.
Accordingly, we depict the decision region for each instance in
this subsection using the two-dimensional representation as in the
right-hand-side of Fig.\ \ref{F:S2-to-2D} and we drop the
$L(\cdot)$ notation when labeling various parts of each figure to
emphasize their source in $S^2$.

Each of the examples in this section have the following model
parameters in common:
\begin{gather*}
  p_0=\tfrac{1}{50},\quad p=\tfrac{1}{20},\quad
  \nu_1=\nu_2=\tfrac{1}{2},\\
  f_0\!=\!\left(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4},
    \tfrac{1}{4}\right)\!,
  f_1\!=\!\left(\tfrac{4}{10}, \tfrac{3}{10}, \tfrac{2}{10},
    \tfrac{1}{10}\right)\!,
  f_2\!=\!\left(\tfrac{1}{10}, \tfrac{2}{10}, \tfrac{3}{10},
    \tfrac{4}{10}\right)\!.
\end{gather*}
We vary the delay cost and false alarm/identification costs to
illustrate certain geometrical properties of the continuation and
stopping regions.  See Figs.\ \ref{F:2D1}, \ref{F:2D2}, and
\ref{F:2D3}.

\ifpdf
\myFigure{3.5in}{img2D1.pdf}{Illustration of
\emph{connected} stopping regions and the effects of false-alarm
costs. (a) and (b): $a_{12}=a_{21}=3,\,c=1$. (a):
$a_{01}=a_{02}=10$. (b): $a_{01}=a_{02}=50$.}{F:2D1}
\fi

\ifpdf
\myFigure{3.5in}{img2D2.pdf}{Illustration of
\emph{disconnected} stopping regions and the effects of asymmetric
false-identification costs. (a) and (b): $a_{01}=a_{02}=10,\,c=1$.
(a): $a_{12}=a_{21}=10$. (b): $a_{12}=16,a_{21}=4$.}{F:2D2}
\fi

\ifpdf
\myFigure{3.5in}{img2D3.pdf}{Illustration of a
\emph{disconnected} continuation region  and the effects of
variation in the delay cost. (a) and (b):
$a_{01}=14,a_{02}=20,a_{12}=a_{21}=8$. (a): $c=1$.  (b):
$c=2$.}{F:2D3}
\fi

These figures have certain features in common. On each subfigure
there is a dashed line representing those states $\bpi\in S^2$ at
which $h_1(\bpi)=h_2(\bpi)$.  Also, each subfigure shows a sample
path of $(\bPi_n)_{n=0}^{\sigma}$ and the realizations of $\theta$
and $\mu$ for the sample.  The shaded area, including its solid
boundary, represents the optimal stopping region, while the
unshaded area represents the continuation region.

Specifically, these figures show instances in which the $M=2$
convex subsets comprising the optimal stopping region are
connected (Fig.\ \ref{F:2D1}) and instances in which they are not
(Figs.\ \ref{F:2D2} and \ref{F:2D3}(a)). Fig.\ \ref{F:2D3}(b)
shows an instance in which the continuation region is
disconnected.

An implementation of the optimal strategy as described in Sec.\
\ref{sec:dynamic-programming-solution}.\ref{sec:optimal-soln} is
as follows: Initialize the statistic $\bPi=(\bPi_n)_{n\geq 0}$ by
setting $\bPi_0=(1-p_0,p_0\nu_1,p_0\nu_2)$ as in part (c) of
Proposition \ref{P:PiProperties}. Use the dynamics of
\eqref{E:Pi-Dynamics} to update the statistic $\bPi_n$ as each
observation $X_n$ is realized. Stop taking observations when the
statistic $\bPi_n$ enters the stopping region
$\Gamma=\Gamma^{(1)}\cup\Gamma^{(2)}$ for the first time, possibly
before the first observation is taken (i.e., $n=0$). The optimal
terminal decision is based upon whether the statistic $\bPi_n$ is
in $\Gamma^{(1)}$ or $\Gamma^{(2)}$ upon stopping. Each of the
sample paths in Figs.\ \ref{F:2D1}, \ref{F:2D2}, and \ref{F:2D3}
were generated via this algorithm. As Fig.\ \ref{F:2D1} shows, the
sets $\Gamma^{(1)}$ and $\Gamma^{(2)}$ can intersect on their
boundaries and so it is possible to stop in their intersection. In
this case, either of the decisions $d=1$ or $d=2$ is optimal.

We use value iteration of the optimality
equation~\eqref{E:Dyn-prog-eqn} over a fine discretization of
$S^2$ to compute $V_0(\cdot)$ and generate the decision region for
each subfigure.  The resulting discretized decision region is
mapped into the plane via $L$.  See~\cite[Ch.\ 3]{MR2182753} for
techniques of computing the value function via the optimality
equation such as value iteration.

\mysubsection{Three alternatives after the change}

In this subsection we consider the special case $M=3$ in which we
have three possible change distributions, $f_1(\cdot)$,
$f_2(\cdot)$, and $f_3(\cdot)$. Here, the continuation and
stopping regions are subsets of $S^3\subset\R^4$. Similar to the
two-alternatives case, we introduce the mapping of
$S^3\subset\R^4$ into $\R^3$ via
$(\pi_0,\pi_1,\pi_2,\pi_3)\mapsto$
    \begin{align*}
        {\textstyle
        \left(\sqrt{\tfrac{3}{2}}\pi_1
        +\tfrac{1}{2}\sqrt{\tfrac{3}{2}}\pi_2
        +\tfrac{1}{2}\sqrt{\tfrac{3}{2}}\pi_3,
        \tfrac{3}{2}\sqrt{\tfrac{1}{2}}\pi_2 +
        \tfrac{1}{2}\sqrt{\tfrac{1}{2}}\pi_3, \pi_3\right)}.
    \end{align*}
Then we use this representation---actually a rotation of it---to
illustrate in Fig.\ \ref{F:3D} an instance with the following
model parameters:
    \begin{gather*}
        p_0=\tfrac{1}{50},\quad p=\tfrac{1}{20},\quad
        \nu_1=\nu_2=\nu_3=\tfrac{1}{3}, \\
        f_0=\left(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4},
        \tfrac{1}{4}\right),\quad
        f_1=\left(\tfrac{4}{10}, \tfrac{3}{10}, \tfrac{2}{10},
        \tfrac{1}{10}\right),\\
        f_2=\left(\tfrac{1}{10}, \tfrac{2}{10}, \tfrac{3}{10},
        \tfrac{4}{10}\right),\quad
        f_3=\left(\tfrac{3}{10}, \tfrac{2}{10}, \tfrac{2}{10},
        \tfrac{3}{10}\right),\\
        c=1,\quad a_{0j}=40,\quad a_{ij}=20,\quad i,j=1,2,3.
    \end{gather*}

Fig.\ \ref{F:3D} can be interpreted in a manner similar to the
figures of the previous subsection.  In this case, for every point
$\bpi=(\pi_0,\pi_1,\pi_2,\pi_3)\in S^3$, the coordinate $\pi_i$ is
given by the (Euclidean) distance from the image point $L(\bpi)$
to the face of
\begin{minipage}{3.5in} \ifpdf \myFigure{3.5in}{img3D1.pdf}{Illustration of the optimal
decision region for an instance of the special case $M=3$. A
sample path of the process $\bPi$ is shown for which $\theta=6$
and $\mu=3$.}{F:3D} \fi\text{ }\\
\end{minipage}
the image tetrahedron $L(S^3)$ that is opposite the image corner
$L(e_i)$, for each $i=0,1,2,3$.

\vskip 15pt

redpajama_set_name: RedPajamaArXiv


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The season of lists is now upon us: best book, best film, best album ... of 2010.
But, I was recently tempted by another, older list: the Modern Library's best novels in English since 1900 (first published in 1998 and judged by the likes of Daniel Boorstin, A.S. Byatt, Vartan Gregorian, and William Styron).
Ranked number one is James Joyce's Ulysses, written from 1914 to 21, published in 1922 and a source of controversy every since (for example, banned as obscene in the U.S. until 1933).
My last reading of the novel was in 1962 as an 18-year-old college freshman in one of the best courses I have ever had—a "close reading" introductory humanities class that in the spring semester focused on just four books (Paradise Lost, Huckleberry Finn, Portrait of the Artist as a Young Man, and Ulysses). We spent more than a month on Ulysses itself. I first found it dense, perplexing, and often incomprehensible, but after reading and re-reading, after studying interpretations by others, I came to love it (and understand some of it).
Yet as the years passed, and the inevitable dinner conversations occurred about the five best novels we had read, Ulysses was never mentioned by anyone (except the stray English major). And when I would ask about it, most would answer: "have started it several times, but never got very far. Too hard."
So, inspired by the "best novels since 1900 list" , with affection dimmed by time and having forgotten almost everything I may have once known about the novel, I decided to try again almost 50 years later (!!!!).
What I found was two novels: a deeply humanistic one which brilliantly and beautifully captures the life of a day in Dublin primarily through three main characters; and a second, highly literary one of surpassing complexity and, without careful study, limited accessibility.
The deeply humanistic novel gives us remarkable insight into Stephen Dedalus (a young writer who aspires to literary greatness, is haunted by the death of his mother, rejects the superficiality of journalism and is teetering on the edge of alcoholism and dissipation); into Leopold Bloom (an advertising salesman, lapsed Jew, lover of food and drink, son of a suicide, father of a dead son and a ripening teenage daughter and wanderer who traverses Dublin during the day and night, befriends Stephen, and returns to his marital bed which, as he knows well, was the scene of an afternoon affair between one Blazes Boylan and his wife); and into Bloom's wife Molly (a singer and earthy mother/wife who fears aging, is jealous of her younger daughter, longs for a sexual relationship with Bloom, relishes her afternoon affair, talks frankly about her bodily functions, speaks in vivid contradictions about love, children, life, aging and women, and at the end remembers romantically the time when she and Bloom first made love).
Unlike many 19th century novels, this humanistic one does not end in either marriage or death, but in ambiguity about what will happen in the future to Stephen, Bloom and Molly and to their relationships. But this uncertainty grows out of a vivid recreation of the multiple sights, sounds, smells and voices of Dublin on June day in 1904. Bloom's pork kidney breakfast frying in a pan. The sound of the trolley cars. The vomit in the bedside bowl of Stephen's dying mother. Tugs moving across the horizon on the "snotgreen sea." The funeral of an old drunkard. The birth of a child. The arguments in a pub. Bloom masturbating on the beach as he watches a young woman show off her knickers. Stephen and Bloom in the nightmare of Nighttown. Stephen and Bloom at Bloom's home watching the wandering stars and peeing below Molly's window. Molly relieved that her menstruation shows she is not pregnant by Boylan.
Joyce set out to create life in all its fullness without heroic scenes or gestures or declamations but through a fully realized expression of a city and its people on one typical day—and through ironic puncturing of human pomposity and pretense. Despite its reputation as a difficult read, many of the chapters or important passages in Ulysses are accessible to a regular reader who is not a candidate for a PhD. For example: the opening chapter where Stephen is mocked by his friend and critic "stately plump Buck Mulligan; the passages in the pub where Bloom engages in verbal warfare with the anti-Semitic "citizen;" the distant seduction of Bloom on the beach by Gerty McDowell who reveals herself as she leans back to watch the fire works shoot into the sky and then reveals that she is lame as she limps away; and even the last two chapters, one in the form of a catechism revealing the relationship between Stephen and Bloom and the second the famous stream-of-consciousness thoughts of Molly as she lies next to Bloom in the early hours of the morning.
Yet, the second Ulysses, the highly literary one, is still complex and inaccessible to a one-time generalist reader. Like many great works of literature, it requires repeated reading and deep study fully to understand—and ultimately to enjoy—the many dimensions and layers. The most obvious complexity, of course, is the analogy to Homer's Odyssey (Latinized from Greek as Ulysses). The novel is loosely structured to mimic Homer's epic. And the main characters in Joyce's novel have referents in the Odyssey, although with profound differences: Bloom as a non-heroic Ulysses, Stephen as Ulysses' son Telemachus, but son without a strong attachment to his own father; and the faithless Molly as the faithful Penelope. Understanding the ways in which Ulysses is an ironic commentary on the Odyssey, and the ways in which Bloom, Stephen and Molly are, and are not, like Ulysses, Telemachus, and Penelope is a huge enterprise unto itself, upon which books have, of course, been written.
Joyce's novel is also stuffed with allusions and parodies and riddles, many of which require substantial knowledge outside the book. As Joyce himself said, he had "put in so many enigmas and puzzles that it will keep professors busy for centuries arguing over what I mean" which would earn the novel "immortality." One whole chapter on child labor and child birth is written in many different styles of English to show the birth of the language. The novel in various places and various ways addresses complex themes like the relationship between Christians and Jews, the aspirations, failures and pedantry of Irish Nationalism and the Irish Literary Revival, the interconnection between love and betrayal. And the style of the novel is, in fact, many different styles: in abandoning the omniscient narrator, the novel is often read as the true beginning of modernist literature.
So, Joyce's Ulysses is still a very hard read. How hard may be seen by contrasting it with the second "best" novel on the Modern Library list: The Great Gatsby. This novel is surely on everyone's list of top five favorite novels: it is short (one-quarter the length of Ulysses); it is accessible; it has an engaging narrator; it tells a powerful story from start to finish; it is written in beautiful, lyrical and penetrating prose; and, although it has many complex sub-texts, it sounds a single powerful theme—the failure of a materialistic American dream. Gatsby, too, bears reading and rereading to uncover the layers of complexity, but a first reading—or a reading after long absence—is a powerful, moving narrative experience in a way far different than Ulysses.
Still, I urge that people read the first Ulysses I rediscovered, the deeply humanistic novel which is bursting with the enormous variety of life. I do have to say that my re-introduction to the novel was aided by 24 recorded lectures—simply entitled "Joyce's Ulysses"—delivered by James A.W. Heffernan, emeritus professor of English at Dartmouth (and available from The Teaching Company). Heffernan focuses primarily on the character and psychology of Stephen, Bloom and Molly but the lectures provide a guide through the chapters of the book and relate them to the Homeric myth and put them in context of other recurrent themes (e.g. Irish Nationalism). Perhaps that reading of the "first" Ulysses will provide a stimulus to explore the almost infinite dimensions of the second, literary one.
Both Gatsby and Ulysses have famous endings.
...and how he kissed me under the Moorish wall and I thought well as well him as another and then I asked him with my eyes to ask again yes and then he asked me would I yes to say yes my mountain flower and first I put my arms around him yes and drew him down to me so he could feel my breasts all perfume yes and his heart was going like mad and yes I said yes I will Yes.
Both endings are not without deep ironies. But, the final sentences of Gatsby are about the futility of our dreams. The end of Ulysses is about the affirmation of our humanity.


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'What If We Get Married' Episode 2 Recap: Friends with Benefits to Husband and Wife
Mike and Amy go from friends with benefits to husband and wife in the newest episode of What If We Get Married? Credit: TLC
By: Kelly Spears 04/08/2016 at 10:31 PM
They did it! The What If We Get Married? couples have officially tied the knot. The April 8 episode brought with it plenty of wow-worthy moments, one of which may leave viewers wondering why one couple got married at all. Let's take a look at the unforgettable moments from the What If We Get Married? episode, entitled "You Were Standing Right In Front of Me All Along."
Mike and Amy: After only a month of planning, this couple is getting married. Amy is thrilled her daughter, Riley, will be at the wedding, though she has decided not to be part of the bridal party. "I think they just rushed into it," Riley admits to What If We Get Married? viewers. But it seems Mike and Amy are positive they're making the right decision. Mike explains that, two weeks after telling his mom he was marrying Amy, his mother's cancer spread throughout her body, and she passed away. He's ready to get married, but he's missing his mother terribly.
Amy's oldest son, Brandon, is in Afghanistan with the military. Her youngest son, Nick, is going through "hard times;" both boys are unable to attend the wedding.
Mike carries a white rose down the aisle prior to the ceremony, and places it on a chair in the front row where his mother would have sat.
Amy cries while Mike reads his hand-written vows, and he hugs his bride-to-be. Riley looks awkward and uncomfortable during the nuptials.
"I'm excited to call her my wife… instead of girlfriend or friends with benefits," Mike tells What If We Get Married? viewers post-ceremony. Riley admits she's happy for her mom and new step-dad, although it's "weird" for her.
In another awkward moment, Amy's maid-of-honor gives a speech. It's uncomfortable because she once had a fling with Mike, and she is not sure what to say.
After their wedding night, the newlyweds go out for brunch. They disagree on parenting issues; Mike believes Riley should be treated like a young adult and not a child. Following their honeymoon in San Antonio, Mike moves in with Amy. He brings his dog, Cash, but Amy believes dogs should be kept outside. She gives Mike one drawer in her dresser and a small space in her closet for his clothes. It seems there may not be room for both of them in her home.
What If We Get Married? viewers later learn that the newlyweds are hardly speaking just days after their wedding. Mike found photos of Amy's ex-husband on her laptop, leading to "communication and trust issues." Mike admits to being a "jealous guy," and thinks having photos of exes "brings chaos." Amy claims she didn't realize how jealous Mike was prior to getting married. She tells viewers she feels they should be in their honeymoon phase. "I don't feel like I should be this unhappy right now," she admits.
Christian and Amanda got married on May 29, 2015 at the 1899 Farmhouse in Princeton, Texas. Credit: Lauren Renae Photography
Christian and Amanda: On the morning of this couple's wedding, Amanda is overwhelmed, but she has the support of her friends and family. Her mom helps her prepare for the biggest day of her life. In a touching moment, she puts on her daughter's veil; she's clearly happy for her.
Christian isn't as nervous as he though he may be pre-wedding. "I'm just looking forward to the first day of the rest of my life," he tells What If We Get Married? viewers. He has a quiet moment with his father, who offers his son a blessing in Spanish.
During the wedding ceremony, Christian's love for Amanda is obvious. Following his short, heart-warming vows, he says "I love you, bestie." Amanda follows up with sweet vows of her own. "I've been looking for you my entire life," she tells her soon-to-be husband, "you were standing right in front of me all along."
Amanda shares that, since Christian's proposal, she has emotionally prepared for this day. "The cool thing is, in the last month our relationship has built a ton," she explains to viewers, "so by the time I walked down the aisle, I was so in love with him it was unreal. I've never felt that way about anybody."
Following their wedding reception, the newlyweds get to their hotel room at 1 a.m. After some discussion (and lots of whispering and kisses), Amanda decides to take a shower with her new husband. The next morning, they share that they consummated their marriage. Amanda admits that she "can't keep her hands off" of Christian, and he appears to be in heaven. He describes their marriage as a "dream come true."
Due to Amanda's first soccer game of the season, the couple's honeymoon is cut short. During the game, Christian talks with a friend about their wedding night. He explains that he's not disappointed that they waited to have sex until marriage, but he admits that he wishes he had been able to enjoy that aspect of their relationship a long time ago.
Later, Christian and Amanda return to Christian's family's home to pack his belongings. He's emotional, as he's lived with his family for the past 29 years. Amanda explains that his parents' house is only 10 miles down the road, but Christian is having a difficult time letting go.
Amanda's grandmother and sister join the newlyweds for breakfast in their home. Amanda brings up finances, which Christian believes should be a private conversation. Although Christian is planning to begin college full-time along with working and playing soccer, Amanda wants to help financially so he won't be stretched so thin. Christian is concerned his new wife doesn't think he can handle a full schedule.
Sneak Peek: On the next episode of What If We Get Married?, Mike grieves the loss of his mother, and his jealousy hits an all-time high when he finds out Amy is talking to her ex. Christian and Amanda disagree about money. Will these couples manage to move past their differences, or are their marriages already doomed?
Kelly Spears
Kelly is an online news writer for The Knot. She is a professional writer who studied English at Ohio State University. Kelly is a big fan of TLC's '90 Day Fiance' and FYI's 'Married at First Sight.'

redpajama_set_name: RedPajamaCommonCrawl


SUMMER, MOON, STARS, & RUNNING: What Could Be Better Than That?
Join us this Monday night, June 20th at 9:30 PM at Cathy Fromme Prairie on Shields south of Harmony for a summer solstice 3-5 mile full moon fun run!! FCRC and Altitude Running's Monday Night Run we be joining together to "Run into Summer"! Carpooling recommended as the parking lot is small. Please bring your headlamp or flashlight!


Roses are red, violets are blue, we love our gin, and we think you do too. If that sounds about right, you're in luck, because a gin festival is happening to lift your boozy spirits, make you forget all about your dreaded weekdays, and get you excited about the upcoming weekend.
The Boatshed Market in Cottesloe is hosting a gin festival to celebrate all things gin (duh). Local and international distilleries will be heading to the festival to showcase some of their finest drops and give you the opportunity to taste and learn more about the production process.
There will be 23 exotic gins showcased at the event, including favourites such as Giniversity, The Westwinds Gin, Dasher and Fisher, Herno, and Never Never. There'll also be two gins exclusive to the Boatshed Wine Loft, which won't be available ANYWHERE ELSE in Australia.
Sadly, they're not licensed to sell G&Ts throughout the day, so it's purely a tasting affair, but if you're into gin, you'll want to drop everything else and register the whole squad for the gin fest here.
Not into Gin? Check out more things to do in Perth here.


"You used such ruthless methods. How would you be so kind to give the Imperial Sky Technique to me?" The head of the Thousand Wind School obviously did not believe what Zhuge said.
"Elders, no need to fight over this. How about I enjoy this Imperial Sky Technique for you two?" Just at that time, underneath the room, a voice suddenly rang out.
When that voice rang out, the two of them were shocked. They never would have thought that other than them, there would be another person that was able to arrive at the room alive.
Under their astonished gazes, Chu Feng slowly came out from the entrance. He smiled as he evaluated the two people. But, when Chu Feng saw Zhuge, his eyes couldn't help but lit up.
Seeing the white cloak that was full of symbols, Chu Feng's instinct was that he might be a World Spiritist.
But compared to Chu Feng, the head of the Thousand Wind School and Zhuge were incomparably shocked. No matter what, they never would have thought that the person who was able to arrive at that place would be such a young man.
*bzz* Suddenly, Chu Feng frowned as he felt a strong Spirit power attacking him to sense his body.
The Spirit power was extremely strong and he didn't know how to defend himself. He was worried that their Spirit power would merge into his chest. Would the map of symbols be revealed just like that?
But luckily, the Spirit power passed through his chest without any reaction. At the same time, it relaxed Chu Feng and he thought of a possibility.
The map on his chest would very likely only be moved by his own Spirit power. If that was real, then in the future, he did not need to worry about his secret being detected by other people that had the Spirit power.
"6th level of the Spirit realm. With only a cultivation like this, you are still able to arrive at this place. It must be because of the World Spirit Compass in your hand right?" Zhuge spoke first.
"6th level of the Spirit realm?" Hearing those words, the head of the Thousand Wind School was even more shocked.
To the him who was at the Profound realm, people in the Origin realm were trash. People in the Spirit realm were the trash within trash. But, at that instant, such a trash arrived at this place. Naturally, it made him extremely shocked.
"Indeed, nice eyesight." Chu Feng lightly smiled towards Zhuge. He also didn't waste his time speaking and he walked towards the Imperial Sky Technique in big strides.
"Die." Seeing that, the head of the Thousand Wind School was seething with anger. With a thought, the boundless pressure came engulfing out and it pressed towards Chu Feng.
"Damn it." At that instant, Chu Feng could feel the boundless pressure surrounded his entire body. It wanted to penetrate his skin and stab his bones. It was extremely terrifying.
But, there was a huge difference between the pressure that the insane beggar gave him that day and to this. It could be said that the pressure from the beggar almost made Chu Feng die from the explosion of his body. But, the pressure from the head of the Thousand Wind School was able to be endured and it wouldn't kill him.
*ta, ta, ta* While enduring the pressure that was pressing his body, Chu Feng painfully walked the first step, the second step, the third step, and although it was extremely hard and painful, he still continued forward.
"What a strong physical body." Looking at that scene, Zhuge couldn't help but exclaim.
"You old guy, why aren't you attacking as well? Do you want this little brat to be the fisherman that reaps the reward?" Seeing that Zhuge was only looking and not helping, the head of the Thousand Wind School was even more angry.
"Hey, words cannot be said like that. Since he could arrive here with his cultivation, it clearly showed that he is not ordinary. Perhaps he could even be the prefered successor of the Imperial Sky Sage right?" Zhuge lightly smiled and he had a beautiful generosity of an adult.
The head of the Thousand Wind School grinded his teeth from his anger towards Zhuge. But, seeing the Chu Feng who kept on getting closer to the Imperial Sky Technique, he did not speak any more. He endlessly cast wave after wave of pressure towards Chu Feng, wanting to stop his movements.
But it was useless. Chu Feng's physical body was too strong. Under the pressure that was even feared by the experts of the Origin realm, not only could Chu Feng endure it, he could still continue walking forward. Although it was very painful and challenging, he was already quite overpowered being able to reach that point.
"Give it up. He is equipped with Spirit power and your pressure won't be able to move his determination. As long as his physical body can endure it, you will not be able to force him to give up purely with your pressure." Zhuge reminded once again and his eyes slightly narrowed and he concentrated on every movement and action of Chu Feng.
After a difficult journey, Chu Feng finally arrived in front of the jade stand that held the Imperial Sky Technique. Although he was pale at that instant and his body was completely soaked with sweat, he was still incomparably excited.
Grabbing the Imperial Sky Technique that was sparkling like a jade and created by the concentrated Spirit power, Chu Feng patted it towards his brain. Instantly, his face couldn't help but change greatly. A pain like the stabbing of needles was spreading from his brain.
The pain was extremely strong and it was hard for Chu Feng to endure it. He hugged his head with both of his hands and rolled on the ground while loudly screaming.
"Haha, without a few methods, I wouldn't be qualified to be a World Spiritist." Zhuge did not admit to that and instead, he was proud.
How would Chu Feng even have the heart to listen to what Zhuge said? He only felt that his Spirit power was quickly being washed away. He even reached the stage in which he was getting dried up. The pain in his brain got clearer and clearer. He was unable to endure it and it was as if he was going to lose his life.
"Ahh~" Finally, Chu Feng yelled out for the last time and limped on the ground without strength and without making any sound.
"What, he's dead? Why didn't the Imperial Sky Technique float out?" The head of the Thousand Wind School quickly asked.


The TST/ATTS training center is also a full-time shop located north of New York City. Even though my days are long and my free time short, keeping my hands dirty keeps me current on the problems my students (and my faithful readers) are facing. Here's a few we had recently.
We had a 2003 Ford Expedition 4.6L V8 with 110,552 miles come in with Diagnostic Trouble Codes (DTCs) P0135, P0141, P0151, P0161 and P0443. The vehicle had already been to the Ford dealer, and they told the owner that the O2 sensors had to be replaced along with the PCM (Powertrain Control Module). Since the owner was not happy with the Ford dealer diagnosis (or the repair cost they were quoted), they came to see me. I checked the vehicle out, confirmed the DTCs and checked for Technical Service Bulletins (TSBs) and found one that related to our DTCs. We called a local parts store looking for a replacement for the PCM, but no one had a listing but the dealer.
The cost of this PCM was more than $800, so we wanted to make sure that it would solve the vehicle's problems before we replaced it. We checked all the O2 sensors' heaters for battery power and ground to make sure that the heaters actually worked. The O2 sensors were very difficult to reach, but we had to test all in order to rule out if they were good or bad. What we found was that the sensors had power but no ground — that is supplied by the PCM. We used a PowerProbe to supply a ground to test the sensors and found that they were capable of drawing the correct amount of current along with heating up properly. The problem of a failed PCM was now confirmed since we proved that the components (O2s) could function properly if they were provided with power and ground.
The vehicle owner did not want to spend a bunch of money on an old vehicle and asked us to find the most reasonable (read "cheap") way of repairing it. Since the new unit was so expensive and there were no aftermarket units available, I came to remember a company called Ourecms. Years back I used this company and recommended them to my students because they would ship you a unit to try for $75. Now remember that you are not working on your father's Oldsmobile anymore and PCMs are not "plug and play" without reprogramming them. I thought since I have the Ford IDS I could get a used PCM and program it to confirm our diagnosis. So I decided to go to the computer and typed in www.ourecms.com and get a used PCM. The site was no longer available but what came up in the Google search engine was www.autoecms.com. It was the same location in Pennsylvania where I had purchased PCMs years ago. I read through their site and found this, "Misdiagnosis no problem, if an electronic unit was ordered but a misdiagnosis was discovered, after the unit was installed, yes you may return our unit for a refund less a $75 test/restock fee and all shipping charges." Wow, it was the same company that I used years ago to solve some hard problems after testing all the sensors, actuators and wiring only leaving the computer as the possible problem!
We ordered the PCM, installed it in the vehicle and reprogramed (Figure 1) the unit so the vehicle would have all the correct information and start. When you are programing a new or used Ford PCM you have to deal with their antitheft system PATS (Passive Anti-Theft Systems) and program the keys as well. Make sure to have at least two OE keys available or the procedure cannot be completed properly, and the vehicle will not start. You should also have all the key fobs for the vehicle and know if there is remote start available on the fobs. If that is the case you will have to program the fobs as well. Another important step when you program a vehicle is to utilize a battery maintainer and make sure you type in the vehicle VIN number and other important information.
After the programing was completed, the engine started up and ran great. I took the vehicle for a test drive to make sure that Mode $06 would display passing test results. I was lucky because after two of us drove the vehicle all the monitors except EVAP (evaporative emissions) were "Ready." After the vehicle sat overnight, we test drove it again and completed all the monitors, confirming our repair.


Before the rental start, a deposit of 5,000 € is required.
The first rental day rate is 450,00 €, additional days 225,00 € each . The rental price includes a data volume of 50 GB per calendar month. The data rate is up to 30 Mbps download and 2 Mbps upload – in reality, about 60% of this data rate is achieved. Each additional GByte rate is 4.50 €. If you need larger quantity of data, please contact us by phone. Please note the coverage map. The device can be used in the area with a signal strength > 50 dB.
other european countries and express shipments on request.
Our VSAT system comes in two transport boxes. The large transport box with the antenna is weatherproof and should be positioned outdoors. A clear view of the sky is absolutely necessary. The small transport box is slightly larger than a briefcase and contains the satellite terminal. This must be positioned weatherproof and provides a power plug (240V).
Position large transport box (VSAT antenna) in suitable location and lift off cover.
Open small transport box (VSAT satellite terminal) on both sides, connect power plug and connect enclosed antenna cable (length 5m).
Power on the satellite terminal: the antenna automatically elevates and searches for the satellite within a few seconds.
You can connect your devices via LAN cable or WiFi.


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BOOKS, CLIMATE, KINDLES, TECH TIPS, NYC – WHATEVER INTRIGUES.
cats/dogs
What's the difference between the three HBO channels?
Ann H. Logan	Uncategorized	 May 3, 2020 April 30, 2020 5 Minutes
https://www.pocket-lint.com/tv/news/152017-hbo-max-vs-hbo-now-vs-hbo-go-what-is-the-difference-and-which-is-best
HBO Max vs HBO Now vs HBO Go: What's the difference?
Maggie Tillman · 30 April 2020 HBO/Pocket-lint
– Here's everything you need to know
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For some odd reason, HBO thinks it needs to offer three TV streaming services. They're called HBO Go, HBO Now, and HBO Max, and they're all distinct and yet similar at the same time. To help you figure out which is best for you, we're pitting them against each other in this quick guide that covers how they differ in terms of pricing, supported devices, and content available to watch.
HBO Go launched in 2010. It's a free TV streaming service available to people who have purchased HBO as a premium channel through a cable or satellite provider. HBO Go offers original content from the HBO channel, including series, documentaries, and specials, plus a selection of movies. It supports simultaneous streaming and is only available in the US.
What is HBO Go and how do you watch it?
Which movies and shows are on HBO Go?
You can use HBO Go to watch all of HBO's library, including hit series like Game of Thrones, Sex in the City, Chernobyl, Westworld, etc.
There's also a rotating collection of acquired series and movies. We've used it to watch John Wick 3, Detective Pikachu, Us, and the Die Hard films. Most new movies and shows are available on HBO Go at the same time they premiere on TV. But ongoing shows part of an "HBO reality series" – such as Last Week Tonight with John Oliver – are not available when they premiere. Instead, they're added within 24 hours.
HBO Go does not offer live TV content or real-time broadcasts of events.
How much does HBO Go cost?
HBO Go is included with your HBO channel subscription, which can cost anywhere from $5 to $20 per month, depending on your provider.
Which devices support HBO Go?
HBO Now works on the following devices: iPhone (5S and later), iPad (fifth-generation and later), Apple TV (second-generation and later), Amazon Fire tablets (fourth-generation or later), Fire TV devices, Android phones and tablets (running version 5 and later), Android TV devices (like the Nvidia Shield), Chromecast, smart TVs from Sharp, Samsung, and Sony Bravia, Roku (second-generation and later), PlayStation 4, Xbox One, TiVo, and Macs and Windows 10 PCs running the Chrome, Firefox, Edge, IE 11, and Safari web browsers.
How to sign up for HBO Go
First, purchase an HBO subscription through your TV provider. If you don't have a subscription, go to HBO.com/ways-to-get. Or, to see if your TV provider supports HBO Go, go to play.HBOGO.com/login and check if your TV provider is listed. Once you're all set, sign in to HBOGO.com or the HBO Go app. You will need to choose your TV provider and enter your TV provider account credentials.
HBO Now launched in 2015. It is what's called a "standalone" TV streaming service – primarily. because it does not require an HBO channel subscription through your cable or satellite provider. Like HBO Go, it offers original content from the HBO channel alongside a selection of movies. It also supports simultaneous streaming and is only available in the US.
HBO Now hands-on: A true cord-cutting experience
Which movies and shows are on HBO Now?
HBO Now offers the exact same content as HBO Go, but without a cable subscription required. You get series like Big Little Lies, True Detective, and The Wire, plus docuseries like McMillions, and acquired movies and shows such as The Lego Movie and Apocalypse Now. Most of this stuff is available at the same time it premieres on TV – again, with the exception of "HBO reality series". There's also no live TV option.
How much does HBO Now cost?
HBO Now costs $14.99 a month and can easily be canceled at any time.
Which devices support HBO Now?
HBO Now works on the following devices: iPhone (5S and later), iPad (fifth-generation and later), Apple TV (second-generation and later), Amazon Fire tablets (fourth-generation or later), Fire TV devices, Android phones and tablets (running version 5 and later), Android TV devices (like the Nvidia Shield), Chromecast, smart TVs from Sharp, Samsung, and Sony Bravia, Roku (second-generation and later), PlayStation 4, Xbox One, and Macs and Windows 10 PCs running the Chrome, Firefox, Edge, IE 11, and Safari web browsers.
How to sign up for HBO Now
This is simple: Go to HBONow.com/order and click Start Your Free Trial. You can also do this from the HBO Now app or through a digital subscription provider such as Amazon's Prime Video Channels or Hulu. You will need to create an account and supply a payment method. For more information about how to get HBO Now, including how to sign in to your device, check this support page.
HBO Max has been called a "platform" by Warner Media, which owns HBO. But, like HBO Now, it is a "standalone" TV streaming service in the US that doesn't require cable or satellite. It'll feature 10,000 hours of premium shows and movies, including content from the HBO channel, as well as originals called Max Originals and programming from WarnerMedia's other brands (like the DC Universe and Cartoon Network).
HBO Max: Price, release date, rumours, and shows
Which movies and shows are on HBO Max?
HBO Max will include all of HBO, original programming (Max Originals), a collection of acquired series and movies, and content from WarnerMedia's brands and libraries including Warner Bros., New Line Cinema, DC, CNN, Turner, Looney Tunes, and more. A few of the big titles include Game of Thrones, Sex and the City, Friends, The Big Bang Theory, Pretty Little Liars, and The Fresh Prince.
How much does HBO Max cost?
HBO Max will cost $14.99 a month when it launches in May. Current HBO Now subscribers will get access to HBO Max for free.
Which devices support HBO Max?
WarnerMedia has said HBO Max will be available "on many popular devices", including phones, tablets, Mac and PC browsers, media players, and game consoles. Check here to see the full compatibility list once HBO Max launches.
How to sign up for HBO Max
HBO Max will be available through HBOMax.com. HBO Now subscribers billed directly through HBO or Apple and Google will get instant access to HBO Max when it launches at no extra cost. To find out where your HBO Now subscription is billed through, check this support page.
HBO subscribers who get HBO through AT&T TV, DirecTV, AT&T U-Verse, or Spectrum will get access to HBO Max for free.
WarnerMedia said you will also be able to sign up through AT&T, DirecTV, AT&T U-Verse, Spectrum, YouTube TV, Apple, and Google Play.
Which is best for you?
Here's your decision tree: Do you subscribe to HBO through a TV provider? Then use HBO Go. Are you a cord-cutter? Sign up for HBO Max when it launches in May; it has all of HBO – like HBO Go and HBO Now both offer – but it doesn't require a cable or satellite subscription and has additional content from WarnerMedia's wide brand portfolio. It's hands-down the best deal if you don't already have the HBO channel.
If you want to stream HBO content this very instant, but don't want to wait for HBO Max or buy the HBO channel to use HBO Go, then get HBO Now for the time being. When HBO Max does become available, you'll have instant access anyway and it won't cost you a cent more.
Published May 3, 2020 April 30, 2020
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eDynamix provide industry leading, connected software systems encompassing all core aftersales functions outside of a Dealer Management System. This includes Service Plans, maintenance reminders and follow up, Interactive Vehicle Health Check (iVHC), Online Bookings, VIDEO1st for sales and aftersales, surveys, MOTCleanse and Connect. Using one supplier with one login provides a more efficient and cost-effective option than multiple systems, multiple suppliers and multiple support channels; improving productivity, staff engagement and boosting profitability.
Using one connected system, with one customer dataset also helps with the GDPR and means dealers only need to audit one supplier. Using our Consent Authorisation Server (CAS) which binds all systems together, dealers can be sure that no communication leaves the business unless specific consent has been granted by the customer. We are now trusted by over 1,000 dealers in the UK and Ireland.
GDPR – apocalypse or opportunity? At Marketing Delivery, we think it's more the latter. For the 10 years we've been in business we've been advocating relevant communications to customers at the right time in the "customer journey" for maximum engagement.
The overriding theme of GDPR is to treat customers fairly. We want to show how the good dealerships are already doing this and already getting the best results from their comms.
There are lots of myths around how GDPR is going to change the way dealers engage with their customers, and it's true the "font end" systems will need to be updated to reflect the new consent requirements. However, the businesses that are already properly considering their contact strategy are already being fair and reasonable in their activity will have little to fear, come May.
Featuring in the exhibition space and on the conference stage, Arena Group is helping car dealerships tackle problem paper – deal files, service packs, HR and finance records – to support GDPR, FCA and manufacturer compliance, and deliver a host of additional business benefits.
Developed by Arena in collaboration with car dealerships, mstore for automotive has already been selected by two of the top six dealer groups and many more in the top 100. This proven software platform delivers improved revenue, enhanced efficiency and reduced risk, by transforming document management for the better.
Arena's Neil Maude will join the panel of experts on the conference stage, and is available on the Arena stand to provide GDPR advice. Sector specialist colleagues are also on hand to demo mstore's feature rich platform and help resolve dealer pain points.
EMaC are the leading provider of aftersales retention solutions to the UK motor trade with over two million service plan sales to date and hundreds of long established dealer and manufacturer relationships. Operational since 2004 we offer unrivalled insight and expertise in aftersales customer retention.
Through a mix of proven techniques and leading edge ideas we can help fulfil your aftersales retention potential. Our success is based on investment in great technology and, most importantly, great people. We are also industry recognised and AM readers have voted EMaC as the AM 'Dealer Recommended' service plan supplier for the last seven years!
Call our friendly team now on 01270 507466 and see what EMaC can do for your business.


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Q: MS CRM - setVisible I am newbie with CRM and
I was googling for how to hide and show a text field using jScript library in MS CRM (online) and found several options of using the function setVisible.
I tried those options:

*

*Xrm.Page.ui.tabs.get('new_fieldname').setVisible(false);

*Xrm.Page.data.entity.attributes.get('new_fieldname').setVisible(false);

*Xrm.Page.getAttribute('new_fieldname').controls.get(0).setVisible(false);
But only the last one is really working.
The first option gives me an error message.
What is the different between them?

A: The attributes are the data, the controls are the HTML Dom objects. You don't tell the data to hide, you tell the control that is displaying the data to hide.

A: Just to add to the points already made..
The difference between
Xrm.Page.ui.tabs.get('new_fieldname').setVisible(false);

And
Xrm.Page.getAttribute('new_fieldname').controls.get(0).setVisible(false);

The first refers to a tab (Xrm.Page.ui.tabs), the second refers to an attribute (Xrm.Page.getAttribute).
So if you wanted to hide a whole tab, its sections and fields you can use the first one. If you want to just hide an individual field you can use
Xrm.Page.getControl("new_fieldname").setVisible(false);

Which is itself a shortcut from
Xrm.Page.ui.controls.get('new_fieldname').setVisible(false);

A: to hide a text field the right method is this:
Xrm.Page.getControl("new_fieldname").setVisible(false);

A: Besides using JavaScript to show/hide field you can use Business Rule to do the same job also CRM platform built to make things easier so when you want to do anything in CRM you must think about it with this order:

*

*Out of the box.

*Customization.

*Business Rule.

*Workflow.

*JavaScript.

*Plugin-Workflow Activity.

redpajama_set_name: RedPajamaStackExchange


I know a number of people have been waiting for me to reopen, however, my wrists are still not 100% and I don't know exactly what is causing the problem.
With that in mind, I've made the decision to close my custom shop for the rest of the year to give my hands time to heal properly, and to give myself the time to devote to other important things in my life.
In regards to sewing, I will not be stopping entirely, unless advised to do so again by my doctor, but I will be cutting down significantly. Instead of commissions, I will be doing some ready-made items and working on props and things for my own dolls. I will make an announcement here and on Instagram when I have things available, so check back from time to time in case something you want comes up.
Hopefully a longer term hiatus will also give me the time to figure out if I can actually continue to take commissions without doing permanent damage to my health.


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What car accident attorneys should know about
Cars are expensive.
That's no secret.
In fact, a car accident can cost your entire life savings.
But for the best outcome, car accident lawyer Todd Kienert, an experienced attorney, offers you some tips on what you can expect to face in your legal case.
Todd Kielert is a lawyer and an instructor at The New York University School of Law.
He is also the author of The Complete Law of Car Accidents: The Basics and the Basics: Accident Compensation and Lawsuits.
In the video above, Kienistt talks about what it means to be an accident lawyer, what is at stake, and how to handle an accident in a way that makes sense for both parties.
You can also listen to the full podcast at the bottom of the page.
Todd has more than 20 years of experience in the car accident field.
His law firm has worked with both parties in several major car accident cases, including: In a 2012 crash involving an elderly woman, Todd's law firm handled her car accident claims.
Todd also represented the family of a woman who died in an accident after she hit a parked car, and also worked on the case of a man who lost his leg in a crash while working on a construction project.
Todd and his wife, Lori, are also experienced attorneys.
You'll find them on The View, The Late Show with David Letterman, The Colbert Report, and The Late Late Show With Craig Ferguson.
Todd Kienet is a practicing attorney in New York City.
He has extensive experience in personal injury, car insurance, commercial property, personal injury and property damage, and corporate litigation.
Todd's practice focuses on the personal injury field, including wrongful death, wrongful imprisonment, wrongful death claims, personal injuries, wrongful property damage claims, wrongful use of force, and commercial property damage.
Todd is also an avid runner and a member of the United States Olympic Committee.
Todd earned his law degree from the University of Virginia School of Legal Studies.
He received his J.D. from Fordham Law School.
Todd graduated magna cum laude from Ford Hall Law School in New Jersey.
He and his family live in Queens, New York.
This podcast is sponsored by Accident Recovery Network.
Get your free crash assessment, car repair, or car insurance quote today.
Subscribe to AccidentRecoveryNetwork.com.
Listeners can access an interactive website to receive an automated crash assessment report.
This site offers personalized crash assessment reports for a wide range of auto insurance claims, including personal injury claims, claims related to defective equipment, and claims related in excess of $5,000.
Accident recovery experts analyze and respond to the most common vehicle injuries, including car crashes, collisions, and other accidents.
Accidents are caused by numerous factors, including driving styles, driving habits, and factors outside the control of the driver.
To learn more, visit AccidentRescueNetwork.net.
Tagged: car accident lawyers
Which cars should you buy?
Top 10 coolest cars on the market


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Enveu
Published in Enveu
Five Ways to Use SEO to Get the OTT Views
Image Source The OTT market (for devices and services) is estimated to reach US$ 113.9 billion by 2022 and is estimated to reach US$ 217.5 billion by 2026, growing at 18.3% CAGR for the next five years. So, it's a no-brainer for those in the video content space to consider…
Benefits of Going Direct-to-Consumer and Launching Your OTT App
Image Source The global video streaming market is expected to reach US$ 223.98 billion by 2028, expanding at a CAGR of 21%. And so it's no surprise when you see more content owners moving from traditional ways of showcasing content to the streaming wave. Who/what is a content owner? Before we speak about the benefits…
OTT Vs Theater: Which Is The Dominant Movie Watching Platform Today?
Source The coronavirus induced lockdowns and restrictions have significantly altered the culture of movie consumption in the past two years. More people continue to stay home and the most sought after place where they get their daily diet of entertainment is from over-the-top video streaming platforms. …
Why Customer Retention on OTT is Important and How can You Achieve It?
Image Source It's no secret that OTT is the next big thing in the media industry globally. The global OTT market is projected to reach US$ 1039.03 by the end of 2027. This highly lucrative space is seeing several new entrants. …
The Value of Technology in Creating a Better OTT Experience
Source Today there are nearly 4.66 billion active internet users in the world. That's about 65% of the entire population (Source). This sort of internet penetration means that this massive number of people watch some form of online content on an everyday basis. Not just that, they can access the…
4 Ways to Reduce Your OTT Churn
Source In the old days of cable and set top boxes, when a customer wanted to cancel their subscription they had to jump through several hoops for it. It was a common sight to see a customer haggling with a customer service representative who would try their best to keep…
How Sling TV is Shaping Today's Streaming Competition
About Sling TV is an Emmy Award-winning American streaming television service owned as a subsidiary of DISH Network. It provides more than 700 channels from today's most popular networks across its general market. …
Can Music, Travel, Fitness and E-Commerce Generate Revenue on OTT?
Image Source Over-the-top streaming is a dream come true for consumers. They are finally able to access the content they want, when they want it and from any device they own. The old way where you had to wait till the clock strikes 8pm on a Sunday to tune into…
Creating a Seamless Subscription Experience for OTT Services
Image Source In this ever evolving media landscape, the shift to digitally enabled, multiscreen OTT video subscriptions is an area by itself that is outpacing the rest. Subscriptions, as a model, is prevalent across industries but nowhere else is the shift towards it felt more predominantly than in the media…
One Cloud Platform to build, manage, monetize and grow your OTT Apps.


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Membership Association for Justices of the Peace
findaJP.com
JP Dashboard
Zoom Discussions
Education and Networking
About Connecticut JPs
About Florida Notaries
About Maine Notaries
About Massachusetts JPs
About New Hampshire JPs
About Vermont JPs
Wedding Certificates
About JPus
Our Announcements
JPus in the News
JPus News Category: Press Release
Press Releases: 3/1/2021
Virtual Conference for Marriage Officiants on March 13, 2021 In anticipation of the Justice of the Peace Association's Virtual Conference on March 13, 2021, JPus sent out variations of a press release highlighting member and speakers at the event. Following is a compilation of some that received attention. The Hartford Courant: March 5, 2021 Justices of the Peace gathering virtually for conference STATEWIDE — Justices of the Peace will be gathering virtually on March 13 to boost their ceremony-performing skills and keep up with policies impacting their services. The Fairfield based Justice of the Peace Association is hosting the conference, . . .
Press Release: 4/6/2020
Plan B Marriages April 6, 2020, Fairfield, CT. Despite the fear and uncertainty that comes with the coronavirus pandemic, justices of the peace are supporting loving couples who want – or need - to tie the knot. They are epitomizing what it means to be a professional officiant, providing expertise during the most challenging times. Governor Lamont signed an executive order making marriage license application procedures more flexible. In doing so, he permitted socially distanced nuptials, acknowledging that officiating a marriage can be an essential service. "While our communities are loaded with angst, love is prevailing. JPs are rising to . . .
Press Release: 3/27/2020
Marriage License Rules Loosened in Connecticut March 27, 2020, Fairfield, CT. On Tuesday, when Connecticut's Governor Lamont signed an executive order loosening marriage license application procedures, he acknowledged that sometimes performing a marriage is an essential service. This made socially distanced weddings an official alternative. Sadly, the big reception must be postponed until life returns to normal, but for now at least, the wedding can go on. Thankfully, the state's justices of the peace (JP) are at the ready to support couples who need an intimate ceremony now. Sometimes health insurance coverage or a military deployment make a marriage urgent. . . .
Marriage Officiant Association Expands to Include Vermont FOR IMMEDIATE RELEASE August 22, 2019, Fairfield, CT. Vermont wedding officiants need to look no further than The Justice of the Peace Association (JPus) for representation and connection with engaged couples. Serving most of the northeast, the professional membership group expanded its territory to include Vermont wedding officiants. In its first welcoming gesture, marriage officiants are invited to an Open House in White River Junction on Friday, September 13th between 3:00-6:00. "I'm looking forward to meeting the amazing JPs who bring couples together in love and marriage," says Loretta Jay, who leads the . . .
06880: 3/8/2021
Union Leader: 7/17/2020
Boston Globe: 4/27/2020
Burlington Free Press: 4/1/2020
The Patch MA: 3/13/2020
Stamford Advocate: 3/6/2020
Hartford Courant: 1/3/2020
The Patch VT: 9/16/2019
The Patch VT: 9/3/2019
The Patch NH: 8/26/2019
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Multiamedia Story Telling
Tiger and Human Conflict | Monirul Alam
July 28, 2015 December 30, 2019 moLeave a comment
The Conflict between tiger and human has to a new level in the mangrove forest of the Bangladesh.Thirty people were killed by tigers last year and three tigers by people. Since Cyclone Sidr [November 2007] and Cyclone Aila [May 2009], when 1,000 people who lived near to the water had their homes flooded and were forced to move inland. Hungry people now risk facing a tiger attack everyday when they go into the forest looking for food. On February 6 in southern Sundarban. A Bengal tiger had just killed a 40-year-old woodcutter called Mabud, deep in the mangrove forest when he was collecting firewood in the area of Char-shesher. One villager told me 'We enter the jungle searching for food and the tiger kills us . . . the tiger comes to our village, we kill the tiger'.
I took these photos on February 6 in southern Sundbaran . A Bengal tiger had just killed a 40-year-old woodcutter called Mabud, deep in the mangrove forest when he was collecting firewood in the area of Char-Shesher. One of his fellow woodcutters, Abul Sarder, told me that five of them entered the jungle to collect firewood and suddenly a tiger attacked them. 'When we had escaped we realised that Mabud had not. We tried to save him but failed to fight off the tiger.'
Dreaming Journey | Multiamedia
Travel Bangladesh | St Martin's Island


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About RUSA
Home > Vol 55, No 2 (2015) > Rabina
Reference Services to Incarcerated People, Part II: Sources and Learning Outcomes
Debbie Rabina, Emily Drabinski
During the 2013/14 academic year, students taking a required general reference course in a library and information science program answered reference questions from incarcerated individuals. The assignment was designed as a service-learning project. Here we examine how well the assignment met the course and program-level learning objectives that students are expected to attain. We described the data collection and analysis, present our findings and make recommendation in ways to increase student learning outcome and ALA professional competencies from a services learning projects. Specifically we discuss the need to increase education in e-government sources since our findings indicated they are most useful for answering reference questions from incarcerated people.
HTML PDF
CFR 542.1—Purpose and Scope, www.law.cornell.edu/cfr/text/28/543.10.
"Library Services in New York State's Correctional Facilities," New York State Library, www.nysl.nysed.gov/libdev/outreach/corr.htm.
"Correctional Services Program," www.nypl.org/help/community-outreach/correctional-services.
"Inmate Population Statistics," New York State Commission of Correction, April 5, 2014, www.scoc.ny.gov/pop.htm.
"About DOC: Facilities Overview," City of New York Department of Correction, 2014, www.nyc.gov/html/doc/html/about/facilities_overview.shtml.
"Library Services in New York State's Correctional Facilities," New York State Library, accessed October 14, 2008, www.nysl.nysed.gov/libdev/outreach/corr.htm.
"Correctional Services Program: Correctional Services Monthly Statistics," New York Public Library, 2014, www.nypl.org/help/community-outreach/correctional-services/stats.
"ALA's Core Competences of Librarianship, Reference and User Services," American Library Association, January 29, 2009, www.ala.org/educationcareers/sites/ala.org.educationcareers/files/content/careers/corecomp/corecompetences/finalcorecompstat09.pdf; Pratt course catalog, https://www.pratt.edu/academics/degrees/course-catalog.
Pratt e-portfolio, https://www.pratt.edu/academics/information/about-the-school/student-learning-outcomes.
John Dewey, Experience and Education (New York: Simon & Schuster, 1997); Bonnie Brzozowski, Nicholas Homenda, and Loriene Roy, "The Value of Service Learning Projects in Preparing LIS Students for Public Services Careers in Public Libraries," Reference Librarian 53 (2012): 24–40.
Sarah Currim, "Experiential Learning: Reducing the Real-World Divide between Libraries and Library School," Reference Librarian 52 (2011): 300–307.
Anthony Cocciolo, "Public Libraries and PBS Partnering to Enhance Civic Engagement: A Study of a Nationwide Initiative," Public Library Quarterly 32, no. 1 (2013): 1–20.
Linda Z. Cooper, "Student Reflections on an LIS Internship from a Service Learning Perspective Supporting Multiple Learning Theories," Journal of Education for Library & Information Science 54, no. 4 (2013): 286–98.
Patricia Montiel Overall, "The Effect of Service Learning on LIS Students' Understanding of Diversity Issues Related to Equity of Access," Journal of Education for Library and Information Science 51, no. 4 (2010): 251–66.
John Riddle, "Information and Service Learning," in Critical Library Instruction: Theories and Methods (Duluth, MN: Library Juice Press, 2010), 133–48.
Yan Zhang and Barbara Wildemith, "Qualitative Analysis of Content," in Applications of Social Research Methods to Questions in Information and Library Science (Westport, CT: Libraries Unlimited, 2009), 308–19.
"Corrections Statistical Analysis Tool (CSAT)—Prisoners," Bureau of Justice Statistics, www.bjs.gov/index.cfm?ty=nps.
Paul T. Jaeger and John Carlo Bertot, "Responsibility Rolls Down: Public Libraries and the Social and Policy Obligations of Ensuring Access to E-government and Government Information," Public Library Quarterly 30, no. 2 (2011): 91–116, http://dx.doi.org/10.1080/01616846.2011.575699.
DOI: https://doi.org/10.5860/rusq.55n2.123
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