text:
\section{\bf   Introduction}
\label{section:1}
Symmetry is an useful property that is largely employed to study stochastic processes. Many
investigations deal with invariance to space or time transformations. A classical example is the
method of images, that has been successfully exploited to obtain results on hitting densities
of diffusion processes (see, for instance, Daniels \cite{Da82}). Other cases referring to
one-dimensional and two-dimensional time-homogeneous diffusion processes can be found,
e.g.\ in Giorno {\em et al.}\ \cite{GiNoRi} and Di Crescenzo {\em et al.}\ \cite{DiCrGiNoRi},
respectively. The results provided in such papers are mainly based on the Markov property
and on certain spatial symmetries of diffusion processes, and on the continuity of their sample
paths. Hence, the symmetry-based approach can be extended by far to other classes of
continuous-time Markov processes, such as the birth-death (BD) processes. We recall that
the continuity of their sample paths on integers is also identified as the `skip-free' property.
\par
It is well known that BD processes are widely considered within applications in ecology, genetics
and evolution (cf.\ Crawford and Suchard \cite{CrSu12}, Novozhilov {\em et al.}\ \cite{NoKaKo2005},
\cite{NoKaKo2006}, Ricciardi \cite{Ri}), theoretical neurobiology (see Giorno {\em et al.}\
\cite{GiLaNoRi}, L\'ansk\'y and Rospars \cite{LaRo93}), chemical physics (see, e.g.\
Conolly {\em et al.}\ \cite{CoPaDh}, Flegg {\em et al.}\ \cite{FLPoGr2008}, Keller and
Valleriani \cite{KeVa2012}), mathematical finance (see Kou and Kou \cite{KoKo2003}), queueing
(cf.\ Di Crescenzo {\em et al.}\ \cite{DiCrGiKKNo12}, Giorno {\em et al.}\ \cite{GiNeNo}, Lenin
and Parthasarathy \cite{LePa0}, Parthasarathy and Lenin \cite{PaLe2004}, for instance).
\par
Due to its relevance in such applications, the first-passage-time (FPT) problem for BD
processes has been largely investigated. Classical papers devoted to the disclosure of its
structural properties are Karlin and Mc~Gregor~\cite{KaMc1},~\cite{KaMc2},
Keilson~\cite{Ke1},~\cite{Ke2},~\cite{Ke3}, Kijima~\cite{Kij88}, R\"{o}sler~\cite{Ro},
Sumita and Masuda~\cite{SuMa}. Other methods in this context are
based on more complex tools such as combinatorial arguments and spectral analysis.
The symmetry-based approach has also been successfully exploited in this context.
\par
On the ground of the above remarks, in this paper we aim to give a short review of the main
results for symmetric BD processes, with attention to FPT problems and related topics.
We first recall the conditions on the transition rates leading to a quasi-symmetry property of the
transition probabilities, and also to simple expressions both for FPT densities and
for certain relevant avoiding transition probabilities. This is accomplished in Section \ref{section:2}
for one-dimensional truncated BD processes both with reflecting and absorbing endpoints.
An indication on the extension to the cases of bilateral processes is also provided. Various
examples are also discussed in detail.
\par
In Section \ref{section:3} we illustrate the symmetry properties of bilateral BD processes with
catastrophes. We refer to total catastrophes whose effect is an instantaneous jump of the
process to state 0. In this case all sample paths going from a negative state to a positive state
(and vice versa) are forced to pass through 0. This allows to make use of the symmetry
property and thus to obtain some expressions for FPT densities through state 0 and
the corresponding avoiding transition probabilities.
\par
As for diffusion processes, the symmetry-based approach for BD processes can be extended also
to higher dimensions. In fact, in Section \ref{section:2d} we consider the symmetry properties of
a two-dimensional  BD process characterized by constant rates. In this case we deal with a spatial
symmetry in the plane with respect to the straight line $x_2 = x_1 + r$. As for the one-dimensional
case,  we also discuss the FPT problem and deal with certain avoiding transition probabilities.
\section{\bf Symmetry properties of truncated processes}\label{section:2}
Let  $\{X(t),t\geq 0\}$ be a one-dimensional BD process and let  ${\cal S}:=\{0,1,\ldots,N\}$ be its state space,
for some natural number $N> 1$.  According to the terminology adopted in \cite{LePa0} and
\cite{vaDo1} we refer to $X(t)$ as a truncated BD process. In this section we assume that the
endpoints $0$ and $N$ are absorbing states, and that $\{1,2,\ldots,N-1\}$ is a communicating class.
As usual, we denote by $\lambda_n$ and $\mu_n$ the birth and death rates of $X(t)$,
with $\lambda_0=\mu_0=\lambda_N=\mu_N=0$ and $\lambda_n,\mu_n>0$
for $n=1,2,\ldots,N-1$. For all $k,n\in{\cal S}$ the transition probabilities
\begin{equation}
	p_{k,n}(t)=P\{X(t)=n\,|\,X(0)=k\}
	\label{equation:pknt}
\end{equation}
satisfy the following forward equations:
\begin{eqnarray*}
	&& \hspace{-1.6cm} \frac{d}{dt}\,p_{k,0}(t)=\mu_1\,p_{k,1}(t)
 \nonumber \\
	&& \hspace{-1.6cm} \frac{d}{dt}\,p_{k,n}(t)=\lambda_{n-1}\,p_{k,n-1}(t)-
	(\lambda_n+\mu_n)\,p_{k,n}(t)+\mu_{n+1}\,p_{k,n+1}(t)
	\qquad (n=1,2,\ldots,N-1)
 \\
	&& \hspace{-1.6cm} \frac{d}{dt}\,p_{k,N}(t)=\lambda_{N-1}\,p_{k,N-1}(t),
	\nonumber
\end{eqnarray*}
with initial condition
\begin{equation*}
	\lim_{t\downarrow 0}p_{k,n}(t)=\delta_{k,n},
\end{equation*}
where $\delta_{k,n}$ is the Kronecker's delta. Clearly,  for all $k\in\{1,2,\ldots,N-1\}$,
$p_{k,0}(t)$ [$p_{k,N}(t)$] is the probability that the BD process has been absorbed
at $0$ [$N$] up to time $t$, whereas $p_{k,n}(t)$, for  $n\in\{1,2,\ldots,N-1\}$, is the
probability that the BD process goes from $k$ to $n$ without absorption up to time $t$.
\par
A spatial symmetry with respect to $N/2$, the mid point of $\cal S$, has been exploited
in \cite{DiCr98} for $X(t)$. Specifically, hereafter we recall the necessary and sufficient
conditions on the birth and death  rates $\lambda_n,\mu_n$ such that the ratio of probabilities
of symmetric sample paths is time-independent (see Theorem 2.1 of  \cite{DiCr98}).
\begin{theorem}\label{teorsimmassorb}
Let us set
\begin{eqnarray*}
	&& x_0=1, 	\nonumber 		\\
	&& x_n=\displaystyle\frac{\mu_1\,\mu_2\cdots\mu_n}
	{\lambda_{N-1}\,\lambda_{N-2}\cdots\lambda_{N-n}}
	=\displaystyle\frac{\mu_n}{\lambda_{N-n}}\,x_{n-1}
	\qquad (n=1,2,\ldots,N-1),
	\\
	&& x_N=\displaystyle\frac{\mu_1}{\lambda_{N-1}}\,x_{N-1}.
	\nonumber
\end{eqnarray*}
Then, for $k=1,2,\ldots,N-1$, the transition probabilities (\ref{equation:pknt}) satisfy
the quasi-symmetry relation
\begin{equation}
	p_{N-k,N-n}(t)=\frac{x_n}{ x_k}\,p_{k,n}(t)
	\qquad (n\in {\cal S}; \; t\geq 0)
	\label{equation:5}
\end{equation}
if and only if
\begin{eqnarray*}
	&& \lambda_n\,\mu_{n+1}=\lambda_{N-n-1}\,\mu_{N-n}
	\qquad (n=1,2,\ldots,N-2),
 \\
	&& \lambda_n+\mu_n=\lambda_{N-n}+\mu_{N-n}
	\qquad (n=1,2,\ldots,N-1).
\end{eqnarray*}
\end{theorem}
\par
The BD process $X(t)$ is said {\em symmetric\/} if its transition probabilities satisfy relation (\ref{equation:5}),
according to the concept of symmetric continuous-time Markov chain given in Karlin \cite{Ka}.
Clearly, Eq.\ (\ref{equation:5}) can be also viewed as an extension of the reflection principle for random
walks (see, e.g.\  \cite{Fe}). Moreover, we observe that if $k$ and $n$ are symmetric states (i.e.,
$k+n=N$) then property (\ref{equation:5}) identifies with the time-reversibility relation (see,
e.g.\ Karlin and Mc~Gregor~\cite{KaMc2})
$$
	p_{n,k}(t)=\frac{\mu_{k+1}\,\mu_{k+2}\cdots\mu_n}{
	\lambda_k\,\lambda_{k+1}\cdots\lambda_{n-1}}\,p_{k,n}(t)
	\qquad (k<n; \; k+n=N).
$$
\par
The symmetry property given in Theorem \ref{teorsimmassorb} has been successfully
employed in various FPT problems. Let
$$
 T^+_{k,s}=\inf\{t>0: X(t)=s\}, \qquad X(0)=k<s\leq N
$$
be the upward FPT of $X(t)$ from state $k>0$ to state $s$, and let $g^+_{k,s}(t)$
be the corresponding probability density function. Since $X(t)$ is a skip-free process,
i.e.\ only unity jumps are allowed (see \cite{AbWh89}), and due to the Markov property,
the following renewal equation holds, for $t\geq 0$:
$$
p_{k,n}(t)=\int_0^t g^+_{k,s}(\vartheta)\,p_{s,n}(t-\vartheta)\,d\vartheta
	\qquad (0<k<s\leq n\leq N).
$$
This allows us to express the upward FPT density for $t\geq 0$ as follows (see Proposition 2.2
of \cite{DiCr98}):
\begin{equation}
 g^+_{k,s}(t)=\lambda_{s-1}\Big[p_{k,s-1}(t)-\int_0^t
 g^+_{k,s}(\vartheta)\,p_{s,s-1}(t-\vartheta)\,d\vartheta\Big]
 \qquad (0<k<s<N).
 \label{eq:relfptdensity}
\end{equation}
Roughly speaking,  equation (\ref{eq:relfptdensity}) expresses the FPT density as the difference of the
probability of all sample paths that exhibit an upward jump from $s-1$ to $s$ close to time $t$ minus the
probability of the sample paths (among the latter ones) that already passed through state $s$ up to time $t$.
Similarly as above, denoting by $T^-_{k,s}$  the downward FPT of $X(t)$, and by $g^-_{k,s}(t)$ its
probability density function, for $0\leq s<k<N$, one has the following relations, for $t\geq 0$:
$$
 p_{k,n}(t)=\int_0^t g^-_{k,s}(\vartheta)\,p_{s,n}(t-\vartheta)\,d\vartheta
 \qquad (0\leq n\leq s<k<N),
$$
$$
g^-_{k,s}(t)=\mu_{s+1}\Big[p_{k,s+1}(t)-\int_0^t
	g^-_{k,s}(\vartheta)\,p_{s,s+1}(t-\vartheta)\,d\vartheta\Big]
	\qquad (0<s<k<N).
$$
When $N$ is even, the symmetry property exploited in Theorem \ref{teorsimmassorb} allows us to
express the FPT densities of a symmetric BD process through the mid  point of the state space,
$s=N/2$, called {\em symmetry state} (see Theorem 2.3 of \cite{DiCr98}).
\begin{proposition}\label{theoremg}
If $N=2s$, with $s$ integer, then the FPT densities through the symmetry state $s$ of a symmetric BD
process for $t\geq 0$ are given by
\begin{eqnarray}
	&& g^+_{k,s}(t) = \lambda_{s-1}\,p_{k,s-1}(t)-\mu_{s+1}\,p_{k,s+1}(t)
	\qquad (0<k<s),
	\label{equation:10a}
	\\
	&& g^-_{k,s}(t) = \mu_{s+1}\,p_{k,s+1}(t)-\lambda_{s-1}\,p_{k,s-1}(t)
	\qquad (s<k<2s).
	\label{equation:10b}
\end{eqnarray}
\end{proposition}
\par
We remark that from Eq.\ (\ref{equation:10a}) we have
\begin{equation}
 g^+_{k,s}(t)=\left\{
 \begin{array}{ll}
 o(t), & k=1,2,\ldots, s-3\\
 \lambda_{s-1}\,\lambda_{s-2}\,t +o(t), & k=s-2, \\
 \lambda_{s-1}-\lambda_{s-1}(\lambda_{s-1}+\mu_{s-1})\,t +o(t), & k=s-1.
 \end{array}
 \right.
 \label{eq:gsmallt}
\end{equation}
A similar result for $g^-_{k,s}(t)$ can be obtained from (\ref{equation:10b}), or taking into account that
for a symmetric BD process for $t\geq 0$  one has
$$
 g^+_{N-k,N-s}(t)=\frac{x_s}{x_k}\, g^-_{k,s}(t) \qquad (0<s<k<N).
$$
\par
For $0<r<N$, with $r$ integer, let ${\cal S}^-_r=\{1,2,\ldots,r-1\}$ and ${\cal S}^+_r=\{r+1,r+2,\ldots,N-1\}$.
For $n,k\in {\cal S}^-$ or  $n,k\in {\cal S}^+$, let us now introduce the
{\em avoiding transition probabilities\/} of $X(t)$:
\begin{equation}
	p_{k,n}^{\langle r \rangle}(t)
	=P\left\{X(t)=n, \; X(\vartheta)\neq r
	\;\hbox{for all}\;\vartheta\in(0,t)\,|\,X(0)=k\right\}.
 \label{eq:pknrt}
\end{equation}
Roughly speaking, Eq.\ (\ref{eq:pknrt}) defines the joint probability that the BD process is in state $n$
at time $t$ and that no visit to state $r$ occurred up to time $t$, conditional on $X(0)=k$. We remark
that the following relation holds:
$$
 p_{k,n}^{\langle r\rangle}(t)=p_{k,n}(t)-\int_0^t g^{\pm}_{k,r}(\vartheta)\,
	p_{r,n}(t-\vartheta)\,d\vartheta,
	\qquad n, k\in {\cal S}^{\mp}_r.
$$
When $X(t)$ is symmetric we are able to express $p_{k,n}^{\langle r \rangle}(t)$
in closed form (see Theorem 2.5 of \cite{DiCr98}).
\begin{proposition}\label{theoremtaboo}
If $N=2s$, with $s$ integer, then the $s$-avoiding transition probabilities of a symmetric BD
process are given by
$$
 p_{k,n}^{\langle s \rangle}(t)
 = p_{k,n}(t) - \frac{x_k}{ x_s}\,p_{2s-k,n}(t)
 \qquad n, k\in {\cal S}^{\pm}_s.
$$
\end{proposition}
\par
We point out that Propositions \ref{theoremg} and \ref{theoremtaboo} are essentially based
on the assumption that $X(t)$ is a skip-free Markov process, and thus are analogous to
similar results for other families of Markov processes such as simple random walks
(see Mohanty and Panny \cite{MoPa}) and time-homogeneous diffusion processes
(see Giorno~{\em et al.}\ \cite{GiNoRi}).
\begin{example} \rm
Let $\{X(t),t\geq 0\}$ be the truncated BD process over $\{0,1,\ldots,N\}$,
with $0$ and $N$ absorbing endpoints and rates
$$
	\lambda_0=\lambda_N=\mu_0=\mu_N=0,
	\qquad\qquad \lambda_n=\lambda,
	\quad \mu_n=\mu
	\qquad(n=1,2,\ldots,N-1),
$$
with $\lambda,\mu>0$.
For $k,n\in\{1,2,\ldots,N-1\}$ and $t>0$ the transition probabilities of $X(t)$
are given by (see formula (33) of B\"{o}hm and Mohanty~\cite{BoMo}
amended of a misprint):
\begin{equation}
	p_{k,n}(t)=e^{-(\lambda+\mu)t}\,\left(\frac{\lambda}{\mu}\right)^{(n-k)/2}
	\sum_{j=-\infty}^{+\infty}\left\{I_{n-k-2jN}-I_{n+k-2(j+1)N}\right\},
 \label{equation:19}
\end{equation}
where
$$
	I_k:= I_k\left(2t\sqrt{\lambda\mu}\right)
	=\sum_{i=0}^{+\infty}\,\frac{\left(t\sqrt{\lambda\mu}\right)^{k+2i}}{ i!\,(k+i)!}
$$
denotes the modified Bessel function of first kind. Since $X(t)$ is symmetric,
Proposition~\ref{theoremg} allows to obtain the FPT density from state $k$
to state $s$ when $N=2s$, with $k$ and $s$ integers:
\begin{equation}
	g^+_{k,s}(t)=\frac{e^{-(\lambda+\mu)t}}{ t}\,
	\left(\frac{\lambda}{\mu}\right)^{(s-k)/2}
	\sum_{j=-\infty}^{+\infty}\left\{(s-k-4sj)\,I_{s-k-4sj}-(s+k-4sj)\,I_{s+k-4sj}\right\},
 \label{eq:gpiues1}
\end{equation}
for $0<k<s$ and $t>0$.  Figure \ref{fig:1} shows some plots of density (\ref{eq:gpiues1}). Moreover,
for $N=2s$ Proposition~\ref{theoremtaboo} leads to the $s$-avoiding transition probabilities
\begin{equation}
 p_{k,n}^{\langle s\rangle}(t) = e^{-(\lambda+\mu)t}\,
	\left(\frac{\lambda}{\mu}\right)^{(n-k)/2}
	\sum_{j=-\infty}^{+\infty}\left\{I_{n-k-4sj}-I_{n+k-4sj}
	-I_{n+k-2s(2j+1)}+I_{n-k-2s(2j+1)}\right\}
 \label{eq:pknes1}
\end{equation}
for $0< n, k<s$ or $s< n, k<2s$, and $t>0$. See Figure \ref{fig:2} for some plots of probabilities
(\ref{eq:pknes1}).
\begin{figure}[t]
\begin{center}
\epsfxsize=7.5cm
\epsfbox{fig1a.eps}
$\;\;$
\epsfxsize=7.5cm
\epsfbox{fig1b.eps}
\caption{The upward FPT density (\ref{eq:gpiues1}) for $\lambda=1$, $s=10$ and $k=6,7,8,9$ (from
bottom to top near the origin),  with (a) $\mu=0.5$  and  (b) $\mu=1$.}
\label{fig:1}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\epsfxsize=7.5cm
\epsfbox{fig2a.eps}
$\;\;$
\epsfxsize=7.5cm
\epsfbox{fig2b.eps}
\caption{The  $s$-avoiding transition probabilities (\ref{eq:pknes1}) for $\lambda=1$, $s=10$,
$k=9$ and $n=7,8,9$ (from bottom to top near the origin),  with (a) $\mu=0.5$  and  (b) $\mu=1$.}
\label{fig:2}
\end{center}
\end{figure}
\hfill $\diamond$
\end{example}
\subsection{Reflecting endpoints}\label{section:4}
The symmetry properties treated in Section \ref{section:2} for BD processes with absorbing
endpoints can be considered also in the case of reflecting endpoints, under similar hypotheses.
\par
In this section we assume that  $\{X(t),t\geq 0\}$ is a BD process having state space $\cal S$,
with $\mu_0=\lambda_N=0$, $\lambda_0,\mu_N>0$ and $\lambda_n,\mu_n>0$ for $n=1,2,\ldots,N-1$,
so that the endpoints $0$ and $N>1$ are reflecting states. In this case, similarly to Theorem
\ref{teorsimmassorb} one can prove that the transition probabilities of $X(t)$ satisfy the symmetry relation
\begin{equation}
	p_{k,n}(t)=p_{N-k,N-n}(t)\qquad (n=0,1,\ldots,N),
	\label{eq:symmrefl}
\end{equation}
and we say that $X(t)$ is symmetric, if and only if
$$
	\lambda_n=\mu_{N-n}\qquad (n=0,1,\ldots,N).
$$
In analogy to the case of absorbing endpoints, for symmetric BD processes with reflecting endpoints
one can obtain various closed-form expressions. We limit ourselves to recall the following result, which is
concerning the case $N=2s$ with $s$ integer (see Theorem 4.3 of \cite{DiCr98}).
\begin{proposition}\label{teorgtaboorefl}
Let the BD process $\{X(t),t\geq 0\}$ be   symmetric  in the sense of Eq.\ (\ref{eq:symmrefl}), with state
space $\{0,1,$ $\ldots,2s\}$, where $0$ and $2s$ are reflecting states, with $s$ a positive integer.
Then, one has:
\begin{eqnarray*}
	&& g^+_{k,s}(t)=\mu_{s+1}\,\big[p_{k,s-1}(t)-p_{k,s+1}(t)\big]
	\qquad\qquad (0\leq k<s), \\
 && p_{k,n}^{\langle s\rangle}(t) = p_{k,n}(t)-p_{2s-k,n}(t)	= p_{k,n}(t)-p_{k,2s-n}(t)
	\qquad (0\leq n, k<s\quad\hbox{or}\quad s<n, k\leq 2s).
\end{eqnarray*}
\end{proposition}
\par
Note that for small values of $t$, the FPT density $g^+_{k,s}(t)$ behaves as specified in Eq.\ (\ref{eq:gsmallt}).
\begin{example}\rm
Let $\{X(t),t\geq 0\}$ be a  truncated BD process having
state space $\{0,1,\ldots,N\}$ with reflecting endpoints and transition rates
$$
 \lambda_n=\alpha\,(N-n), \qquad
 \mu_n=\alpha\,n \qquad (n=0,1,\ldots,N),
$$
where $\alpha>0$. This process is  symmetric  in the sense of Eq.\ (\ref{eq:symmrefl}),
since its transition probabilities for $k,n\in\{0,1,\ldots,N\}$ are given by
(see Giorno~{\em et al.}~\cite{GiNeNo}, for instance):
\begin{equation}
	p_{k,n}(t)=\frac{1}{ 2^N}\sum_{j={\rm max}\{0,n+k-N\}}^{{\rm min}\{n,k\}}
	{k\choose j}{N-k\choose n-j}
	\big(1-e^{-2\alpha t}\big)^{n+k-2j}\,
	\big(1+e^{-2\alpha t}\big)^{N-(n+k-2j)}.
 \label{equation:20}
\end{equation}
Hence, if $N=2s$, with $s$ integer, from Proposition \ref{teorgtaboorefl} one has
the following closed-form expressions of the FPT density through the symmetry state
$s$:
\begin{eqnarray}
	&& \hspace{-1cm}
	g^+_{k,s}(t)=\frac{\alpha\,(s+1)}{ 2^{2s}}\,\sum_{j=0}^k{k\choose j}
	\Bigg[{2s-k\choose s-1-j}\big(1-e^{-2\alpha t}\big)^{s-1+k-2j}\,
	\big(1+e^{-2\alpha t}\big)^{s+1-k+2j}
	\nonumber \\
 && \;\; -{2s-k\choose s+1-j}\big(1-e^{-2\alpha t}\big)^{s+1+k-2j}\,
	\big(1+e^{-2\alpha t}\big)^{s-1-k+2j}\Bigg]
	\qquad (0\leq k<s),
	\label{eq:gcasorifl}
\end{eqnarray}
and of the $s$-avoiding transition probabilities (for $0\leq n, k<s$ or $s<n, k\leq 2s$):
\begin{eqnarray}
 && p_{k,n}^{\langle s\rangle}(t) =\frac {1}{ 2^{2s}}\Bigg[
	\sum_{j={\rm max}\{0,n+k-2s\}}^{{\rm min}\{n,k\}}
	{k\choose j}{2s-k\choose n-j}
	\big(1-e^{-2\alpha t}\big)^{n+k-2j}\,
	\big(1+e^{-2\alpha t}\big)^{2s-n-k+2j}
	\nonumber \\
 && \qquad\qquad\;\; -\sum_{j={\rm max}\{0,n-k\}}^{{\rm min}\{n,2s-k\}}
	{2s-k\choose j}{k\choose n-j}
	\big(1-e^{-2\alpha t}\big)^{n+2s-k-2j}\,
	\big(1+e^{-2\alpha t}\big)^{k-n+2j}\Bigg].
	\label{eq:pkncasorifl}
\end{eqnarray}
Some plots of functions (\ref{eq:gcasorifl}) and (\ref{eq:pkncasorifl}) are shown in
Figures \ref{fig:3} and \ref{fig:4}, respectively.
\hfill $\diamond$
\end{example}
\begin{figure}[t]
\begin{center}
\epsfxsize=7.5cm
\epsfbox{fig3a.eps}
$\;\;$
\epsfxsize=7.5cm
\epsfbox{fig3b.eps}
\caption{The upward FPT density (\ref{eq:gcasorifl}) for  $s=10$ and $k=6,7,8,9$ (from
bottom to top near the origin),  with (a) $\alpha =0.5$  and  (b) $\alpha =1$.}
\label{fig:3}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\epsfxsize=7.5cm
\epsfbox{fig4a.eps}
$\;\;$
\epsfxsize=7.5cm
\epsfbox{fig4b.eps}
\caption{The  $s$-avoiding transition probabilities (\ref{eq:pkncasorifl}) for  $s=10$, $k=9$ and
$n=6,7,8,9$ (from bottom to top near the origin),  with (a) $\alpha=0.5$  and  (b) $\alpha=1$.}
\label{fig:4}
\end{center}
\end{figure}
\par
Another example of symmetric  truncated BD process with reflecting endpoints is characterized by
birth and death rates
$$
 \lambda_n=\alpha\,(N-n)^2, \qquad
 \mu_n=\alpha\,n^2 \qquad (n=0,1,\ldots,N),
$$
with $\alpha>0$, and has been studied by Roehner and Valent \cite{RoVa}.
\subsection{Bilateral processes}\label{section:bil}
It is worthwhile to point out that the symmetry properties of truncated BD processes
with reflecting endpoints can be straightforwardly extended to bilateral BD processes.
We recall that the term `bilateral' refers to stochastic processes defined on the whole set of
integers (see, Pruitt \cite{Pr63}). Hence, we consider a bilateral BD process $\{X(t), t\geq 0\}$
having state space $\mathbb Z$, with birth and death rates $\lambda_n,\mu_n>0$ for all
$n\in \mathbb Z$. Results similar to those given in Section \ref{section:4} can be stated in this
case by noting that the transition probabilities of $X(t)$ satisfy the symmetry relation
\begin{equation}
	p_{k,n}(t)=p_{-k,-n}(t)\qquad (k,n\in \mathbb Z),
	\label{eq:symmbil}
\end{equation}
for all $t>0$, if and only if
$$
	\lambda_n=\mu_{-n}\qquad (n\in \mathbb Z).
$$
Note that here, for brevity, we consider the case in which 0 is the symmetry state. In analogy with
Proposition \ref{teorgtaboorefl}, thanks to the symmetry property (\ref{eq:symmbil}) we have
the following
\begin{proposition}\label{teorbil}
Let the bilateral BD process $\{X(t),t\geq 0\}$ be symmetric in the sense of Eq.\ (\ref{eq:symmbil}).
Then, the upward and downward FPT densities through state 0, for $t>0$, are given by:
\begin{align*}
	& g^+_{k,0}(t) =\mu_1 \left[p_{k,-1}(t)- p_{k,1}(t)\right]
	\qquad (k<0),
	\\
	& g^-_{k,0}(t) =\mu_1 \left[p_{k,1}(t)  -  p_{k,-1}(t)\right]
	\qquad (k>0).
\end{align*}
Furthermore, the 0-avoiding transition probability of $X(t)$ can be expressed as
\begin{align*}
	p_{k,n}^{\langle 0\rangle}(t) = p_{k,n}(t)-p_{-k,n}(t)
	\qquad (n, k<0\quad\hbox{or}\quad 0<n, k),
\end{align*}
for all $t>0$.
\end{proposition}
\par
We conclude this section by showing certain bilateral BD processes possessing the symmetry property
considered in (\ref{eq:symmbil}).
\begin{description}
\item{\em (i)}
First  we consider the BD process having sigmoidal-type rates
\begin{equation*}
\lambda_n=\lambda\,\displaystyle\frac{1+c \Big(\displaystyle\frac{\mu}{\lambda}\Big)^{n+1}}
{1+c\Big(\displaystyle\frac{\mu}{\lambda}\Big)^{n}},
 \qquad
 {\mu_n}=\mu\,\displaystyle\frac{1+c \Big(\displaystyle\frac{\mu}{\lambda}\Big)^{n-1}}
{1+c \Big(\displaystyle\frac{\mu}{\lambda}\Big)^{n}}
 \qquad  (n\in\mathbb{Z}),
\end{equation*}
with $\lambda,\mu>0$ and $c\geq 0$. Generally, the transition probabilities of this process exhibits
a bimodality. See Hongler and Parthasarathy \cite{HoPa2008} and Di Crescenzo and Martinucci
\cite{DiCrMa2009} for the symmetry property and other results.
\item{\em (ii)}
Other bilateral BD processes of interest are characterized by birth and death rates of alternating type.
The process with rates
\begin{equation}
\lambda_n=\left\{
\begin{array}{ll}
\lambda, & n \hbox{ even} \\
\mu, & n \hbox{ odd}
\end{array}
\right.
 \qquad
 {\mu_n}=\left\{
\begin{array}{ll}
\mu, & n \hbox{ even} \\
\lambda, & n \hbox{ odd}
\end{array}
\right.
 \qquad  (n\in\mathbb{Z}),
 \label{eq:alt1}
\end{equation}
for $\lambda,\mu>0$, has been studied by Conolly {\em et al.\/} \cite{CoPaDh} and
Tarabia {\em et al.\/} \cite{TaTaEB09}. A suitable modification of the previous stochastic
model yields the following birth and death rates:
\begin{equation}
\lambda_n=\left\{
\begin{array}{ll}
\lambda, & n \hbox{ even} \\
\mu, & n \hbox{ odd}
\end{array}
\right.
 \qquad
 {\mu_n}=\left\{
\begin{array}{ll}
\lambda, & n \hbox{ even} \\
\mu, & n \hbox{ odd}
\end{array}
\right.
 \qquad  (n\in\mathbb{Z}),
  \label{eq:alt2}
\end{equation}
with $\lambda,\mu>0$. Various symmetries and other properties of this process have been
investigated in Di Crescenzo {\em et al.\/} \cite{DiCrIuMa12}, whereas a suitable extension
of the models (\ref{eq:alt1}) and (\ref{eq:alt2}) can be found in \cite{DiCrMaMa14}.
\end{description}
\section{\bf Symmetry properties of bilateral processes with catastrophes}\label{section:3}
In this section we shall consider bilateral BD processes subject to catastrophes, i.e.\ jumps toward
state $0$  occurring randomly in time. Even in this case the spatial symmetry can be used to obtain
closed-form results related to the FPT problem through state 0. We notice that certain bilateral BD
processes subject to catastrophes have been recently employed to describe double-ended
queueing systems (cf.\ Di Crescenzo {\em et al.}\ \cite{DiCrGiKKNo12}). Various recent results
on BD processes with catastrophes can be found in Dimou and Economou \cite{DimEco13},
Giorno and Nobile \cite{GiNo}, Giorno {\em et al.}\ \cite{GiNoSp} and Zeifman {\em et al.}\ \cite{ZeSaPa}.
\par
Specifically, hereafter we refer to a BD process with jumps $\{X(t), t\geq 0\}$, having state space
$\mathbb Z$. It is a continuous-time Markov chain characterized by the following transitions
concerning births, deaths and catastrophes, respectively:
\par
{\em (a)} \ from $n\in\mathbb Z$ to $n+1$, with rate $\lambda_n$,
\par
{\em (b)} \ from $n\in\mathbb Z$ to $n-1$, with rate $\mu_n$,
\par
{\em (c)} \ from $n\in\mathbb{Z}\setminus \{0\}$ to $0$, with rate $\alpha_n$.
\\
Hence, for all $t>0$ the transition probabilities $p_{k,n}(t)$ satisfy the following system:
\begin{align*}
 & \frac{d}{dt} p_{k,n}(t)
 = -(\lambda_n+\mu_n+\alpha_n)\,p_{k,n}(t)+\lambda_{n-1}\,p_{k,n-1}(t)+\mu_{n+1}\,p_{k,n+1}(t),
 \qquad n\in\mathbb{Z} \setminus\{0\},
 \nonumber
 \\
 & \frac{d}{dt} p_{k,0}(t)
 = -(\lambda_0+\mu_0)\,p_{k,0}(t)+\lambda_{-1}\,p_{k,-1}(t)+\mu_1\,p_{k,1}(t)
 +\sum_{r\in\mathbb Z\setminus \{0\}} \alpha_r\,p_{k,r}(t).
\end{align*}
It is not hard to see that  $X(t)$ has a central
symmetry with respect to state $0$, i.e.
\begin{equation}
 p_{-k,-n}(t)=p_{k,n}(t) \qquad \hbox{for all $t>0$ and $k,n\in \mathbb{Z}$,}
 \label{eq:simmbilatj}
\end{equation}
if and only if
$$
 \lambda_n=\mu_{-n}\quad \hbox{for all $n\in \mathbb{Z}$} \qquad \hbox{and}
 \qquad \alpha_n=\alpha_{-n} \quad\hbox{for all $n\in \mathbb{Z}\setminus \{0\}$.}
$$
In this case the process $X(t)$ is not skip-free due to the presence of catastrophes toward state 0, and
thus its sample paths are no more `continuos'. Nevertheless, since all sample paths from a negative state
to a positive one (and vice versa) are forced to pass through 0, the symmetry-based approach with
respect to state 0 is still valid. In this case, a relevant role is played by the {\em probability currents\/}
in state 0, defined as
\begin{align*}
	 h^+_{k,0}(t) &=\lim_{\tau\downarrow 0} \frac{1}{\tau}P\{X(t+\tau)=0, X(t)<0\,|\,X(0)=k\} \\
	&=\lambda_{-1}\,p_{k,-1}(t)  +\sum_{n<0} \alpha_n \, p_{k,n}(t),
	\\
	 h^-_{k,0}(t) &=\lim_{\tau\downarrow 0} \frac{1}{\tau}P\{X(t+\tau)=0, X(t)>0\,|\,X(0)=k\} \\
	&=\mu_1\,p_{k,1}(t)  +\sum_{n>0} \alpha_n \, p_{k,n}(t),
\end{align*}
for $k\in  \mathbb{Z}$ and $t>0$. Similarly as Proposition \ref{teorgtaboorefl}, from (\ref{eq:simmbilatj})
one can obtain the following results.
\begin{proposition}\label{teorcatast}
Let $\{X(t),t\geq 0\}$ be a bilateral BD process with catastrophes. If it is symmetric  in the sense of
Eq.\ (\ref{eq:simmbilatj}), then the upward and downward FPT densities through
state 0, for $t>0$, are given by:
\begin{align*}
	g^+_{k,0}(t)& =h^+_{k,0}(t)-h^-_{k,0}(t)
	\\
	& =\mu_1 \left[p_{k,-1}(t)- p_{k,1}(t)\right]
	+\sum_{n<0} \alpha_n \, p_{k,n}(t)-\sum_{n>0} \alpha_n \, p_{k,n}(t)
	\qquad (k<0),
	\\
	g^-_{k,0}(t)& =h^-_{k,0}(t)-h^+_{k,0}(t)
	\\
	& =\mu_1 \left[p_{k,1}(t)  -  p_{k,-1}(t)\right]
	+\sum_{n>0} \alpha_n \, p_{k,n}(t)-\sum_{n<0} \alpha_n \, p_{k,n}(t)
	\qquad (k>0),
\end{align*}
and satisfy the following symmetry relation:
$$
	g^+_{k,0}(t)=g^-_{-k,0}(t) 	\qquad (k<0).
$$
Moreover, the 0-avoiding transition probability of $X(t)$, for $t>0$, can be expressed as:
\begin{align*}
	p_{k,n}^{\langle 0\rangle}(t) = p_{k,n}(t)-p_{-k,n}(t)
	\qquad (n, k<0\quad\hbox{or}\quad 0<n, k).
\end{align*}
\end{proposition}
\par
We refer to Theorems 3.1 and 3.2 of Di Crescenzo  and  Nastro \cite{DiCrNa2004} for related results.
\begin{example}\rm
Let $X(t)$ be the bilateral birth-death process with catastrophes characterized by constant rates
$$
 \lambda_n=\lambda, \qquad
 \mu_n=\mu, \qquad
 \alpha_n=\alpha.
$$
As specified in \cite{DiCrNa2004}, for
all $k,n\in \mathbb{Z}$ and $t>0$ the transition probabilities can be expressed as
\begin{align*}
 p_{k,n}(t)
 &= e^{-\alpha t}\,\widehat{p}_{k,n}(t)
 + \alpha \int_0^t e^{ -\alpha \tau} \,\widehat{p}_{0,n}(\tau)\,d\tau
 \\
 &={\left(\frac{\lambda}{\mu}\right)}^{\!\!\frac{n-k}{2}}
 I_{n-k}\left(2 \sqrt{\lambda \mu} \,t\right) \, e^{-(\lambda+\mu+\alpha)t}
 + \alpha {\left(\frac{\lambda}{\mu}\right)}^{\!\!\frac{n}{2}} \!
 \int_0^t e^{ -(\lambda+\mu+\alpha)\tau}
 I_n\left( 2 \sqrt{\lambda \mu} \,\tau\right)d\tau,
\end{align*}
where $I_n(x)$ denotes the modified Bessel function of the first kind, and where
\begin{equation}
 \widehat{p}_{k,n}(t):={\left(\frac{\lambda}{\mu}\right)}^{\!\!\frac{n-k}{2}}
 I_{n-k}(2 \sqrt{\lambda \mu} \,t) \,e^{-(\lambda+\mu)t}
 \label{equation:37}
\end{equation}
is the transition probability of the bilateral Poisson birth-death process with
birth rate $\lambda$ and death rate $\mu$. Process $X(t)$ is symmetric in the sense
of Eq.\ (\ref{eq:simmbilatj}). Hence, making use of Proposition \ref{teorcatast}
if $\lambda=\mu$, for all $t>0$ and $k=1,2,\ldots$ we have the FPT density:
\begin{eqnarray*}
  g^-_{k,0}(t) =e^{- (2\lambda+\alpha) t}
 \Bigg\{\lambda\,\left[I_{k-1}(2\lambda\,t)-I_{k+1}(2\lambda\,t)\right]
 + \alpha \sum_{j=1}^{+\infty} \left[I_{k-j}(2\lambda\,t) - I_{k+j}(2\lambda\,t) \right]\Bigg\}.
\end{eqnarray*}
Moreover,  the 0-avoiding transition probability is
\begin{eqnarray*}
 p^{\langle 0\rangle}_{k,n}(t)
 = e^{-(2\lambda+\alpha) t}\,\left[I_{n-k}(2\lambda\,t) -I_{n+k}(2\lambda\,t) \right],
\end{eqnarray*}
for $ t>0$ and $k,n=1,2,\ldots$.
\hfill $\diamond$
\end{example}
\section{Symmetry properties of  two-dimensional processes}\label{section:2d}
In this section we exploit a symmetry-based approach for two-dimensional  BD processes with
constant rates, by extending some of the results provided in the previous sections.
We essentially refer to some contributions given in Di Crescenzo and Martinucci \cite{DiCrMa2008}.
\par
Let $\{{\bf X}(t)=[X_1(t), X_2(t)];\,t\geq 0\}$ be a two-dimensional
BD process with state space $\mathbb{Z}^2$, and transition probabilities
\begin{equation}
P({\bf n}, t\,|\,{\bf k})
=P\{{\bf X}(t)={\bf n}\,|\,{\bf X}(0)={\bf k}\}, \qquad t\geq 0,
\label{equation:1}
\end{equation}
with  ${\bf n}=(n_1, n_2)\in\mathbb{Z}^2$ and ${\bf k}=(k_1, k_2)\in\mathbb{Z}^2$.
Let us introduce the birth and death rates of ${\bf X}(t)$:
\begin{align*}
\lambda_1 &= \lim_{s \downarrow 0}\frac{1}{s}
P\{X_1(t+s)=n_1+1,\,X_2(t+s)=n_2\,|\,{\bf X}(t)={\bf n}\},\\
\lambda_2 &= \lim_{s \downarrow 0}\frac{1}{s}
P\{X_1(t+s)=n_1,\,X_2(t+s)=n_2+1\,|\,{\bf X}(t)={\bf n}\},\\
\mu_1 &= \lim_{s \downarrow 0}\frac{1}{s}
P\{X_1(t+s)=n_1-1,\,X_2(t+s)=n_2\,|\,{\bf X}(t)={\bf n}\},\\
\mu_2 &= \lim_{s \downarrow 0}\frac{1}{s}
P\{X_1(t+s)=n_1,\,X_2(t+s)=n_2-1\,|\,{\bf X}(t)={\bf n}\}.
\end{align*}
Clearly, for $t>0$, the transition probabilities are solution of the following system:
\begin{align}
\frac{{ d}}{{ d}t} \,P({\bf n}, t\,|\,{\bf k}) &=
-(\lambda_1+\lambda_2+\mu_1+\mu_2)\,P({\bf n}, t\,|\,{\bf k})
\nonumber \\
&+ \lambda_1 \,P(n_1-1, n_2, t\,|\,{\bf k})
+\lambda_2 \,P(n_1, n_2-1, t\,|\,{\bf k})
\nonumber \\
&+ \mu_1 \,P(n_1+1, n_2, t\,|\,{\bf k})
+\mu_2 \,P(n_1, n_2+1, t\,|\,{\bf k}),  \qquad \forall {\bf n}\in\mathbb{Z}^2,
\label{equation:2}
\end{align}
with initial condition
$$
P({\bf n}, 0\,|\,{\bf k})=\prod_{i=1}^{2} \delta_{n_i, k_i}.
$$
By making use of the probability generating function of ${\bf X}(t)$ one can prove that, for all $t>0$,
the transition probabilities are:
\begin{equation}
P({\bf n}, t\,|\,{\bf k})=\prod_{i=1}^{2} {\rm e}^{-(\lambda_i+\mu_i)t}
I_{n_i-k_i}(2 \sqrt{\lambda_i \mu_i} \,t)
\Big(\frac{\lambda_i}{\mu_i}\Big)^{\frac{n_i-k_i}{2}},
 \qquad {\bf k},{\bf n}\in\mathbb{Z}^2.
\label{equation:5bis}
\end{equation}
From (\ref{equation:5bis}) the following quasi-symmetry property follows:
\begin{proposition}\label{proposition:1}
If there exists a constant $\xi>0$ such that
\begin{equation}
 \frac{\lambda_1}{\lambda_2}=\frac{\mu_2}{\mu_1}=\xi,
 \label{equation:10}
\end{equation}
then for all ${\bf n},{\bf k}\in\mathbb{Z}^2$ and all $r\in \mathbb{Z}$ we have
\begin{equation}
 P(n_2-r, n_1+r, t\,|\,k_2-r, k_1+r) = \xi^{n_2-k_2-n_1+k_1}\,P({\bf n}, t\,|\,{\bf k}).
 \label{equation:3}
\end{equation}
\end{proposition}
\par
This result extends  the quasi-symmetry property given in Theorem \ref{teorsimmassorb}
for one-dimensional truncated birth-death processes. In this case we deal with a spatial
symmetry in the plane with respect to the straight line $x_2=x_1+r$. Namely, for each
sample path of ${\bf X}(t)$ going from ${\bf k}$ to ${\bf n}$ there exists a symmetric path
going from $(k_2-r, k_1+r)$ to $(n_2-r, n_1+r)$, where $r$ is a fixed integer. Eq.\ (\ref{equation:3})
thus expresses that the ratio of the probabilities of the two symmetric paths is time-independent.
\par
This property is useful to obtain various results on the
FPT problem of ${\bf X}(t)$ through straight-lines $x_2=x_1+r$.
For a fixed $r\in\mathbb{Z}$, we denote by
$$
T_{r}({\bf k})=\inf\{t\geq 0: X_2(t)=X_1(t)+r\},
\qquad {\bf X}(0)={\bf k}, \quad k_2\neq k_1+r,
$$
the FPT of ${\bf X}(t)$ through the straight line $x_2=x_1+r$,
conditional on ${\bf X}(0)={\bf k}\in \mathbb{Z}^2$. Let
$$
h_r(t\,|\,{\bf k}):=\frac{{ d}}{{ d}t}
P\{T_r({\bf k})\leq t\,|\,{\bf X}(0)={\bf k}\}, \qquad t>0
$$
be the corresponding probability density function. It is not hard to see that
the following identity holds:
\begin{equation}
h_r(t\,|\,{\bf k})=\sum_{x\in\mathbb{Z}} g(x,x+r,t\,|\,{\bf k}),
\label{equation:12}
\end{equation}
where
\begin{equation}
 g(x,x+r,t\,|\,{\bf k}):=\frac{\partial}{\partial t_1}
 P\{T_r({\bf k})\leq t_1, {\bf X}(t_2)=(x,x+r)\,|\,{\bf X}(0)={\bf k}\}
 \Big|_{t_1=t,t_2=t}
\label{equation:23}
\end{equation}
is the sub-density of the first-passage through line $x_2=x_1+r$ in state
$(x,x+r)$ at time $t$, conditional on ${\bf X}(0)={\bf k}$.
Clearly, for $n_2 \geq n_1+r$, $k_2<k_1+r$ (first-passage from below)
and for $n_2 \leq n_1+r$, $k_2>k_1+r$ (first-passage from above)
the following continuity equation holds:
\begin{equation}
P({\bf n}, t\,|\,{\bf k})=\int_{0}^{t}
\sum_{x\in\mathbb{Z}} g(x,x+r,\tau\,|\,{\bf k})\,
P({\bf n}, t-\tau\,|\,x,x+r)\,{ d}\tau.
\label{equation:4}
\end{equation}
For any fixed $r\in\mathbb{Z}$ we set
\begin{equation}
 P^{\langle r\rangle}({\bf n}, t\,|\,{\bf k})
 =P\{{\bf X}(t)={\bf n}, T_r({\bf k})>t\,|\,{\bf X}(0)={\bf k}\},
 \qquad k_2\neq k_1+r,
 \label{equation:24}
\end{equation}
which expresses the probability of a sample path from ${\bf k}$ to ${\bf n}$ at time $t$
which does not touch the straight line $x_2=x_1+r$, conditional on ${\bf X}(0)={\bf k}$.
By adopting a customary nomenclature in the field of Markov chains (see, for instance,
Asmussen \cite{As03}) probability (\ref{equation:24}) is called `taboo probability'.
\par
Thanks to the symmetry considered in Proposition \ref{proposition:1}, hereafter we can
express the taboo probability in terms of transition probabilities.
\begin{theorem}
Under the assumptions of Proposition \ref{proposition:1} we have
\begin{equation}
 P^{\langle r\rangle}({\bf n}, t\,|\,{\bf k})
 =P({\bf n}, t\,|\,{\bf k})-{\xi}^{n_1+r-n_2}\,P(n_2-r,n_1+r,t\,|\,{\bf k}),
\label{equation:11}
\end{equation}
with $n_2< n_1+r$, $k_2<k_1+r$ or with $n_2> n_1+r$, $k_2>k_1+r$.
\label{teorema1}
\end{theorem}
\par
Let us now see that, as a further consequence of the symmetry property given in Proposition \ref{proposition:1},
the FPT densities can be suitably expressed in terms of transition probabilities.
\begin{theorem}\label{teorema2}
Under the assumptions of Proposition \ref{proposition:1}, for $k_2\neq k_1+r$
and $t>0$ we have
\begin{equation}
 g(x,x+r,t\,|\,{\bf k})=\frac{|k_2-k_1-r|}{t} \,P(x,x+r,t\,|\,{\bf k}),
 \qquad x\in \mathbb{Z},
 \label{equation:13}
\end{equation}
and
\begin{equation}
 h_r(t\,|\,{\bf k})
 =\frac{|k_2-k_1-r|}{t} \, P\{X_2(t)=X_1(t)+r\,|\,{\bf X}(0)={\bf k}\}.
 \label{equation:6}
\end{equation}
\end{theorem}
\par
We remark that, due to Eqs.\ (\ref{equation:3}) and (\ref{equation:6}), under assumption (\ref{equation:10})
the following quasi-symmetry relation holds:
\begin{equation}
 h_{r}(t\,|\,k_2-r,k_1+r)=\xi^{k_1+r-k_2} \, h_{r}(t\,|\,{\bf k}).
 \label{eq:proph}
\end{equation}
\par
Let us now introduce the quantity
\begin{align}
 \pi_r({\bf k})
 & =P\{T_r({\bf k})<\infty \,|\,{\bf X}(0)={\bf k}\}
 \nonumber \\
 & =\int_{0}^{+\infty} h_{r}(t\,|\,{\bf k})\,{  d}t,
 \qquad\qquad k_2\neq k_1+r,
 \label{equation:14}
\end{align}
which is the probability that process ${\bf X}(t)$ ultimately crosses the straight line
$x_2=x_1+r$ starting from the initial value ${\bf X}(0)={\bf k}$. Due to relation (\ref{eq:proph}),
under assumption (\ref{equation:10}) we have the following symmetry relation:
$$
 \pi_r(k_2-r,k_1+r)=\xi^{k_1+r-k_2}\,\pi_r({\bf k}),
 \qquad k_2\neq k_1+r.
$$
In conclusion we point out that, under the assumptions of Proposition \ref{proposition:1},
the first crossing probability  (\ref{equation:14}) is given by
$$
 \pi_r({\bf k})=\left\{
 \begin{array}{ll}
 \xi^{k_2-k_1-r} & \hbox{if }\lambda_1+\mu_2\geq \mu_1+\lambda_2, \; k_2<k_1+r,\\
 {}                     & \hbox{or }\lambda_1+\mu_2\leq \mu_1+\lambda_2, \; k_2>k_1+r,\\[0.2cm]
 1 & \hbox{otherwise.}
 \end{array}
 \right.
$$

\subsection*{\bf Acknowledgments}
This paper is partially supported by GNCS-INdAM and Regione Campania (Legge 5).

redpajama_set_name: RedPajamaArXiv


text:
Pee-Wee Herman to Host Radio Show on KCRW
Variety 11/15/2021 Jem Aswad
© Courtesy of Netflix
Variety Music has gotten some inspired press releases in our day, but the one we got on Monday afternoon about Pee-Wee Herman getting a radio show on KCRW is definitely a hall of famer.
The short version is that legendary star of the screen Pee-Wee Herman will launch a radio show on KCRW, the popular National Public Radio station based in Santa Monica, Calif., and will be accompanied by his pals Chairry, Magic Screen and Miss Yvonne. A rep for the station tells Variety that for now, it will be just one show on Nov. 26 at 6 p.m. PT (and available on demand for one week after airing), but who knows?
Former KCRW 'Morning Becomes Eclectic' Host Jason Bentley Launches New Interview Podcast (EXCLUSIVE)
KCRW Names Novena Carmel, Anthony Valadez Co-Hosts of 'Morning Becomes Eclectic' After 18-Month Search
KCRW in Los Angeles Loses Nearly 20% of Staff in Buyout
As for the long version, which was rendered in typewriter font on KCRW stationary, eyes below:
PEE-WEE HERMAN (ME!!)
GETS A RADIO SHOW ON KCRW!
Milky Way Galaxy, Planet Earth, Next to the Pacific Ocean, i.e. Santa Monica, CA (November 15, 2021) – Dear most esteemed journalists, it's Pee-wee Herman here and I am going to be a DJ on a new radio show airing on KCRW! I'm so excited, it's my very first one! I mailed them a letter, which got lost (apparently!), and then tagged them on Twitter and THEN they didn't believe it was me until my biggest, bestest, most amazing fans called and called and jammed up their phone lines (hee hee)! Thank you fans because they are going to get me some donuts and leave me the keys to the studio while they are on vacation!
Join me and my friends Chairry, Magic Screen, Miss Yvonne, and some surprise special guests (HINT: they are super famous!), while we listen to some of the greatest music ever recorded!
This is all happening on Friday, November 26, at 6 p.m PST, so you better get out your transistor radio and listen to my amazing voice!
Hi @KCRW, did you get the letter I sent you? I really REALLY want to be a radio Disc Jockey. But I never heard back from you. #DJlife #DreamBig pic.twitter.com/CBXPJMPEe1
— Pee-wee Herman (@peeweeherman) November 10, 2021
Bob Saget Honored by 'America's Funniest Home Videos' With Special Tribute
Jon Lind, A&R Exec and Writer of Hits for Earth, Wind & Fire, Madonna and Others, Dies at 73
Zhang Hanyu Stars in International Co-Production 'Unspoken' Filming in Canada

redpajama_set_name: RedPajamaCommonCrawl


text:
England cool after reaching WC final
Azad Majumder . Birmingham | Published: 01:31, Jul 12,2019
England captain Eoin Morgan (R) and team-mate Joe Root celebrate at the end of their ICC World Cup semi-final match against Australia at Edgbaston, Birmingham on Thursday. — AFP photo
A year ago exactly on the same day Croatia knocked out England in the semi-final of World Cup football in Moscow's Luzhniki Stadium to shatter the dream of a nation that was so eager to bring the trophy back at its home.
England's men's cricket team ensured there was no similar heartbreak at Edgbaston on Thursday as they crushed Australia by eight-wicket to reach the final after 27 years to revive the famous slogan –'It's coming home.'
The win sets up a clash against New Zealand on Sunday when England will get the chance to lift the trophy for the first time after defeats in their previous three finals.
England captain Eoin Morgan kept his excitement checked ahead of the final and preferred to enjoy their semi-final win over Australia first before concentrating on the next game.
'I wouldn't get too carried away yet. It is obviously a very exciting time for everybody and ourselves included,' said Morgan.
'I think giving ourselves the opportunity for Sunday's final is brilliant. We'll be giving it everything we can in order to try and win that game,' he said.
The place in World Cup final did not come very easily for England, who were on the brink of elimination after a dramatic slump in the league phase.
They had to overcome a strong Indian side and other finalists New Zealand to set-up a tie against Australia, who had never lost a World Cup semi-final previously.
But England maintained their top game to put Australia under pressure from the start, bowling out the side for a paltry 223 runs before racing to 226-2 in just over 32 overs.
'It sounds pretty cool and it feels pretty good,' Morgan said after the match.
'The performance in the last three games, we have got better and better as a group. We talked before the tournament that we would have to do that in order to get to a semi-final and final and be contenders.
'So I think today was, I suppose, a step further in the performance and adding on the two that we've built on already, I think it was important that we did that,' he said.
Morgan was pleased that England did the hardest job with a smile on their faces which made it look easier.
'Everybody out there on the field and even in the changing room loved every ball that was bowled,' he said.
'There was no lack of commitment, application and we had a bit of a day out which, it's cool when it happens like that, particularly when the bowlers bowl like that, it is awesome,' he added.
Morgan hailed the team's changed approach that rewarded them after being walloped by Bangladesh four years ago in Adelaide.
'It's been a process for the last four years. In 2015 we were way off the mark, we struggled against the top teams… so there was quite a drastic change in the way we played and the way we looked at playing our 50-over cricket. That has worked out extremely well for us,' he said.


Birmingham, AL has many interesting frequencies where 2-way radios are being used. Mainly FM communications that can be received by any scanner. Birminham and jefferson county are using trunking systems that require a more complex scanner that has the ability to track the trunking control channels. Some of my favorites to listen to are….


text:
Q: Does Cypress have OR condition I came across an old website I used to work with and remembered I updated role name (<role="">).
Is there any way to run the cypress test without having to update the old role name?
In other words, something like:
 findByRole("A" or "B") //use one or the other, whatever that exists

(I know OR command does not exist in Cypress and cannot find a solution from google so I am stuck..)
Any suggestions would be greatly appreciated!

A: You can use the cy.get() with , and it will act as a OR condition something like:
cy.get('[role="A"],[role="B"]').click()

A: Yes, it's called a TextMatch.
findByRole(/A|B/)

Since "A" and "B" are just representations of the actual role codes, you will need to  careful about special characters. If it's just alphanumeric characters, no problem.

redpajama_set_name: RedPajamaStackExchange


At Elite Air Systems we offer complete heating, cooling and indoor air quality services, designed specifically for your home. We are dedicated to upholding the highest level of service and installation in the heating and air conditioning industry. We know how to correctly and accurately size residential homes to ensure your heating and cooling needs are being met efficiently and affordably. Regardless of the season or the temperature outside, our goal is to make your home a more comfortable place to live.


Perfect as always!!!! Thank you! See you soon!
Justine and her family took such great care of our dog. We had a family emergency and had to find someone to dog sit at the last minute. Justine was able to accommodate our short notice and helped us out with flexible pick up/drop off times. She sent us an update to let us know our girl was doing well. When we need another dog sitter, Justine is definitely at the very top of our list!
Again, as always, Justine and her crew took care of us! Simply the best!!!
Always the best! Thanks for the doggie bed too! Till next time!
Loving family. They all welcomed Jack into their home for the first time. Justine sent updates and pictures. Will repeat. Thanks again guys!
Our girl is treated as part of the family. Such a sweet family.
Our fifi was worn out but happy as always everytime she stays in justine's home. Thank you so much for taking care of our fifi.
Another awesome stay for Ziggy. Always feel like I'm leaving him with family.
We have been fostering dogs and working in dog rescue for seven years. We had a baby, and life got a little crazier, so we are taking a break from fostering dogs. But we are used to having a few extra dogs as part of the family! If you're going away and would like your dog to be a part of our family while you're gone, drop us a line.
I have worked as a Veterinary Assistant and am comfortable with oral and injection medications. We also work from home, and can be there with our (and your) dogs all day.
10 years old, 0 lbs.
10 years, 6 mo. old, 40 lbs.


text:
Polygon Created with Sketch. Home | Tricia Williams
Tricia Williams
Director, Research, Evaluation and Knowledge Mobilization
tricia.williams@fsc-ccf.ca
tricia-williams
@TriciaW_22
Tricia Williams brings a strong background across anthropology, sociology and economics disciplines to her role as Future Skills Centre's Director of Research, Evaluation and Knowledge Mobilization. She has particular expertise in the future of work including digital economies, skills development, migration and gender.
At the FSC, Tricia leads a pan-Canadian research strategy focused on advancing the FSC's strategic goals. She also collaborates closely with our partners on the broader evidence agenda, ensuring that knowledge is both generated and mobilized effectively for uptake and usage.
Prior to joining FSC, Tricia spent seven years with the Mastercard Foundation, leading research, evaluation and knowledge mobilization initiatives related to youth employment and digital economy across Africa and with Indigenous communities in Canada. She has authored and overseen numerous reports, events and research programs to generate evidence and action around inclusive employment and development.
Tricia previously worked for the federal government in evaluation and policy functions, which she joined through the Recruitment of Policy Leaders program. She also spent several years supporting immigrant rights organizations in the United States and community organizations in Haiti, where she lived for several years.
We have a tremendous opportunity to create a more inclusive future that benefits diverse groups and people across Canada. I'm excited to join the Future Skills Centre and its partners in addressing the critical challenge before us. Through strategic inquiry into our ecosystem's needs and priorities, robust evidence and concerted learning and dialogue, I believe we can create a more inclusive and prosperous economy that reflects our boldest hopes and visions of Canada.
In addition to an undergraduate degree, Tricia holds a Master's degree in International Development Studies from the University of Guelph, and Master's and PhD degrees in Sociology & Anthropology from Florida International University in Miami.
Tricia is available for interviews with members of the media. She is an expert in:
Employment & labour markets
Research and knowledge mobilization


Enhance the warmth of your family photos and artwork with the natural tones of this Beaded Frame. In rich bronze and copper hues, this frame will be a classic addition to your mantelpiece or entryway table. Admire beside your favorite vase or hang in your bedroom to remind yourself of your favorite memories. This Beaded Frame makes a thoughtful gift for newlyweds and housewarming parties. Use with easel back or hang on wall. Measures 10.25 x 1 x 7.5.


text:
2 minutes to read Posted on Wednesday March 20, 2019
Campaigns Collections technology wiki women
Emily D'Alterio
Former Editorial & PR Officer , Europeana Foundation
Women in culture and tech: Medhavi Gandhi - leader in arts preservation and digital access
Today we speak with Medhavi Gandhi, director of Happy Hands, an organisation that works on reviving traditional arts through building artisanal capacity and increasing youth/public access to folk arts, and founder of The Heritage Lab, a digital media platform for cultural heritage enthusiasts. She discusses the importance of female role models and the inherent strength of culture as a means of soft power.
Initiation? ceremony, Magdalenian. Nine partially clothed women and one young man.
From Medhavi…
I come from a family of mixed cultures, though I gravitate towards my Bengali roots a bit more. Since childhood I had been exposed to arts, aesthetics - learning how to sing, dance etc. and growing up, I found ample opportunities to merge my skills with what I was passionate about. My school always stressed the importance of developing skills and interests and allowed us to learn about different countries - languages, literature, etc. Later, as a student of literature, I developed a deeper understanding of movements that have shaped us. As the leader of a youth theatre group, I learnt to express my views about issues that I felt strongly for - and somehow, all these experiences came together to chart out my career as well. I'm a keen learner and I love to read, engage with people and 'take action'.
I also double up as a youth educator/mentor undertaking history/heritage learning projects in school - and that's my most favourite role.
Medhavi at the Kamala Nehru College Youth Conclave in Delhi, Medhavi Gandhi, CC BY-SA
I am working on multiple projects right now; but for March, the focus is definitely on Wikipedia's Art+Feminism campaign and one we started of our own for Women's History Month, called '#TheseMughalWomen'. It sheds light on the women of the Mughal court; their contribution to the Empire; myth-busting and more than anything, making content about them, accessible online. The campaign involves museums and individuals. An interesting campaign that has started in India is #ChaloMuseum, pushed by the Kiran Nadar Museum of Art, and we're trying to make visiting museums a 'cool' thing to do.
How did you get into your field?
I got into the culture and arts space back in 2008 during a college internship. It was a regular communications /marketing intern profile at UNESCO Delhi and it opened up a whole new world for me. I learnt about the different folk/craft traditions of the country, and it helped me startup Happy Hands Foundation after college. Then while working on artisan-education and helping them innovate on product design, I started visiting museums a lot, and well - that got me into the 'history of art' part of it! Since I've always loved engaging people with the arts, I think it was natural then to move towards a digital-driven platform for a new project, The Heritage Lab!
What are the challenges for women in the workforce today? What can be done to improve matters?
Working in the crafts sector, I realised that a huge hurdle to women joining the workforce is mobility. We struggled so much in building India's first group of all-women puppeteers: there was family opposition, there was travel-constraint (should they have to go out of the city, etc) and there was a huge bias in terms of women 'performing on stage' even from behind the curtain.
When I moved from the capital city to a smaller city, I had the opportunity to engage with students who had moved from Tier-3 cities to study/gain a degree. I realised however, there is no career-guidance targeted at adolescent girls which will help them choose a career that aligns with their interests or skills; rather, it is about getting a degree that will help them find a better groom.
Interestingly, in the arts-management space: whether it was crafts or museums, I have definitely encountered more women than men at the work level. Maybe because the 'arts' are still not part of the mainstream here in India - and I see that as a blessing in disguise. I firmly believe that Culture and Arts are soft-power, and investing in cultural diplomacy can take India far ahead; so perhaps when the time comes, women already in the sector would be able to lead the movement.
I'd also add that museums/cultural institutions have to work very closely with government bodies, and that's where the struggle is highlighted. In some cities (like mine for instance or mostly in Tier 2 or Tier 3 cities), it can be challenging. Women's roles at cultural institutions aren't free from bias either.
What can be done: Within the workforce, I think there have to be adequate upskilling opportunities focused on women; there has to be a safe space for them to voice concerns/engage in dialogue etc - and institutions really need to focus on women-friendly sanitation facilities.
Do you feel that women are sufficiently empowered and present in leadership positions?
I'll speak specifically for museums - you definitely see so many women making a difference at important positions - whether as historians or art critics, as curators, researchers, archaeologists - women are leading different spaces. Most museums in India supported by the Ministry of Culture which is led by men. However, some of the most important museum roles - education, curatorial, conservation - have some fabulous women. In India, you have museums such as the Partition Museum, Kerala Museum, Maharaja Sawai ManSingh II Museum Jaipur, Piramal Museum, Kiran Nadar Museum, Sarmaya, and many others - all which have women in senior management/leadership roles. Unfortunately, however, these women have to be celebrated outside of the sector too - so that younger people can see them as role models.
What message would you share with women in the sector today?
An underlying stereotype within our sector is that women and tech don't go together. I'd say, that we need to get out of our comfort zones to learn new things, share what we know and basically seek each other - chances are, someone might be able to support you when you least expect it.
Also, celebrate yourself! Campaigns such as #showyourwork are so important and women from the field must participate. But for that, I think we'd have to get a lot of museums and museum people on Twitter.
What digital communities or networks do you find rewarding?
For one, I really love the team at # ArtAndFeminism: so supportive and encouraging - at any time of the day, and from anywhere. I was once part of a US State Department program - and we have an alumni network in every city - my city network on Whatsapp is one of the most active, and supportive groups ever - we're slowly building up a community of 'Enthusiastic Museum People' (owing to India's experience of Ask-A-Curator day) so then again: we're always learning from each other.
Apart from that - I absolutely love my museum-gang on Twitter! It is so rewarding to see chats like #MuseumHour (though at very unearthly hours for India, and I often catch up later).
Who (or what) inspires you at the moment?
I feel my digital museum friends are very inspiring, and I've learnt so much from them. For example, Hannah Hethmon who runs the Museums in Strange Places podcast; Mar who is ever-ready to help and has made CultureThemes so huge that Museums look forward to it.
I also have to say, I am absolutely taken by the open-source movement within museums in the West. India is slowly working on it, but it'll be heartening to see how people engage with content once it's accessible.
Want more? Visit our exhibition Pioneers which highlights the lives and achievements of historical European women. Visit the full list of profiles for the Women in Culture and Tech series - we publish three profiles per week throughout March.
Women in culture and tech: Siobhan Leachman, Citizen scientist and wikimedian
Today we speak with citizen scientist and Wikimedian Siobhan Leachman from her home office in New Zealand. She discusses the intricacies of curating data in Wikidata and contributing to Wikimedia Commons, as well and shedding some light on the challenges of 'imposter syndrome' and what it's like to live in a country where the top three leadership roles are filled by women.
Women in culture and tech: Eva Fog, founder of tech organisation DigiPippi (and much, much more)
In the lead up to International Women's Day (8 March), we speak with an expert in children, gender and IT, Eva Fog , who is a consultant, opinion leader, writer and …
Silvia listening to co-panelists during the first OpenGLAM conference in Mexico, photo by Omar Sansi, CC-BY-SA 4.0
Women in culture: Silvia Gutiérrez, digital humanities librarian
Today's woman in culture and tech is Silvia Gutiérrez De la Torre - a digital humanities Librarian at El Colegio de México. Silvia discusses following your passions …


The Parks and Recreation Board makes recommendations to the City Council for the operation of public parks, recreation facilities, and the acquisition, improvement, construction and maintenance of the parks, parkways, bikeways and other areas. The Parks and Recreation Board is advisory in nature and works to promote interest in recreation, parks, fitness, and other related activities. The Parks and Recreation Board was established by the City Charter.


text:
KL Rahul
Name KL Rahul
Born April 18, 1992
Mangaluru, Karnataka
Age 29 years 274 days
Teams India, Karnataka, East Zone, South Zone, Royal Challengers Bangalore, India A, Sunrisers Hyderabad, Indians, Hubli Tigers, Indian Board Presidents XI, Bellary Tuskers, Punjab Kings, Asia XI
Nickname Rahul
Bowl Style
Runs 2547 1509 1831 3273
SR 52.7 89.29 142.49 136.38
HS 199 112 110 132
50s 13 9 16 27
4s 309 114 164 282
6s 17 36 73 134
Inn - - - -
Balls - - - -
Wkt - - - -
BBI - - - -
BBM - - - -
Eco - - - -
Avg - - - -
5W - - - -
10W - - - -
Hailing from a background of high academic standards, KL Rahul was born and brought up in the prestigious National Institute of Technology, Karnataka, where his father is a professor in the department of Civil Engineering. He finished his schooling in NITK and grew to develop keen interest in cricket. Having started off with batting on the grass-less field of the college, Rahul's parents spotted a spark in him and decided that he must receive formal training in order to realize his talents.
Rahul rose through the ranks of Under-19 cricket and broke into the Karnataka first-class side and had a decent debut season in 2010-11. He was left out of the squad for the succeeding season, but he returned for the 2012-13 season, in which he had a prolific run of form, becoming the highest run-getter of the season, propelling himself into contention for the highest honour of them all – an India Test cap.
The prospect of a debut on a momentous occasion like the Boxing Day at the MCG against the world's chirpiest team seemed to throw Rahul off his game, as he crumbled under pressure and lost his wicket to two shots unbecoming (to say the least) of a Test player. With the series already lost, and in a manifestation of compassion for Rahul's nervous jitters, the selectors and the captain decided to give Rahul a go at his usual opening spot in the Sydney Test. He had a dramatic redemption in Sydney as he stroked his way to a patient century, proving that he belongs in the national side – to himself, more than anyone else.
This hundred gave Rahul some momentum, and he went on to make hundreds in Sri Lanka and West Indies on away tours, making the opening combination a dilemma for the Indian selectors. He showed his tendency to convert starts, making his first three fifty-plus scores count, and going on to score a hundred each time. However, he continued to have anxious starts and got the reputation of an 'all-or-nothing' player.
Rahul was selected in the limited-overs squads to Zimbabwe and West Indies in mid-2016. It was then that Rahul showed his versatility across formats, bursting forth with a hundred each on ODI and T20I debut, after a fine run in the IPL for the Royal Challengers Bangalore. The young right-hander got a chance to play his first home Test, but a series of injuries and recurrences managed to keep him going in and out of the playing XI until the Chennai Test against England in late 2017 when he made a full recovery. He went on to make a heart-breaking 199 in Chennai; his current highest score in Test cricket.
With an ideal mentor in Rahul Dravid, KL Rahul shares more than his first name with the Indian legend; the dogged resistance, the compact, airtight technique, and the unimpeachable focus at the crease are all common factors between the two batting technicians. Rahul does need to work a tad more on his nerves, though, after his tame dismissal on 199 in Chennai further reinforced his image as a timorous player. However, with age on his side, Rahul can work on his nerves and with an all-conditions' technique to go with it, he is an exciting prospect for India's long and challenging overseas season and he would like to reinforce his spot as a long-term option for India as a premier Test opener.
After a run-glut in the 2016/17 seasons, Rahul had a poor 2018 in Test cricket, as he got his first exposure to the moving ball on the surfaces of South Africa, Australia and England. He struggled through all the overseas tour in red-ball cricket, and almost lived a double-life as a white-ball player, bagging a massive 11-crore IPL contract, and ending up as Punjab's best batsman in the 2018 season of the tournament. However, in Test cricket, he was struggling to score so much that he failed to grab the easiest of chances even against a weakened Windies line-up at home. He was eventually dropped from the Test side, as his career fell just short of a full-circle, receiving the ouster for the Boxing Day Test in Melbourne in favour of his Karnataka mate Mayank Agarwal - the very same event in which his Test journey began in 2014.
Just when it looked like things couldn't get worse for Rahul, an episode of a talk show aired in India, in which Rahul, along with Hardik Pandya, were called out on social media for their misogynistic comments, and were suspended by the Board of Control for Cricket in India for poor conduct and tarnishing the image of Indian cricket. The ban has been lifted since the incident. However, at a crunch time, due to lack of form and off-field issues, Rahul suddenly finds himself in the fringes, with almost no chance to win himself a ticket to England. However, a few months still remain for him to prove his mettle, as he tries to cope with trouble in paradise.
IPL - Through the years
On the back of his performances in the Syed Mushtaq Ali tournament in 2013, KL Rahul was drafted into the Royal Challengers Bangalore set-up, a line-up which boasted the names of Virat Kohli, AB de Villiers and Chris Gayle. In the following year, he was roped in by Sunrisers Hyderabad for Rs. 1 crore, where he got more opportunities to bat in the top/middle order. He returned to RCB in 2016 and the Karnataka batter proved his mettle in that edition of the Indian Premier League. He piled 397 runs in 14 matches and ended up as the third highest run-getter for his club.
Strangely, he wasn't retained by RCB and Kings XI Punjab took advantage of that in the 2018 IPL auctions. Rahul was bought for Rs. 11 crore, which turned out to be the joint-third highest price for that year. Batting at the top of the order, Rahul proved his worth by amassing 659 runs (average: 54.91, strike-rate: 158.41). He struck six fifties in 14 matches, out of which three were 90+ scores. KL Rahul also holds the record for the fastest IPL fifty (off just 14 balls).
World Cup through the years
KL Rahul, the man who made his Test debut just ahead of the 2015 World Cup, has grown exponentially in limited-overs cricket over the last few years since the 2015 World Cup. Despite scoring a hundred in the T20I format (apparently whilst inebriated), and earning himself a nine-figure IPL paycheque, Rahul has struggled to establish himself in the ODI game, particularly due to a top-order that is already established and a consequent inability to grasp his role in the middle-order. He has had several failures at the number 4 spot, but he made a case for himself in the 2019 IPL to be considered as a mature accumulator in the middle-order along with his hitting abilities - not to mention being one of the highest run-scorers in the tournament. Rahul, who was also a part of the Koffee with Karan controversy, has had a tough path back to the World Cup squad, after a run of poor form in the longer format in 2018, and off-field issues. However, he has exhibited his reserves of mental strength over the last few months and would be relieved and eager as he boards the flight to England.
Written by Rishi Roy


About 8 hours and tickets from 70 €: a train journey from Catanzaro a Montesilvano Marina is cheap and comfortable. You can count on 3 daily rides to get from Catanzaro to Montesilvano Marina: trains leave every about 4 hours on average. Trains leave from Catanzaro and arrive at Montesilvano. The railway company you will travel with is Trenitalia.
A train ticket from Catanzaro to Montesilvano Marina costs 135 € on average, but it's possible to find low cost tickets from 70 €, too. On Wanderio you can find the best Trenitalia deals for your train journey from Catanzaro to Montesilvano Marina. The best way to save money on your train trip is booking your tickets a few weeks in advance, so that you can benefit from all the Trenitalia promotions for the route Catanzaro - Montesilvano Marina. Trenitalia tickets start from 70 €.
Train journeys from Catanzaro to Montesilvano Marina take about 9 hours on average. If you travel with Trenitalia from Catanzaro to Montesilvano Marina there are 3 daily connections that will take you to your destination in about 9 hours. On Wanderio you can easily find all train schedules from Catanzaro to Montesilvano Marina. The first train from Catanzaro to Montesilvano Marina leaves at 08:33, whereas the last departure on working days is at 22:54.
On Wanderio you can easily buy train tickets from Catanzaro to Montesilvano Marina. With just one search you will find all train schedules for the day you selected, making it easy to book the option that suits you better. For on-the-go bookings (and to have tickets Catanzaro to Montesilvano Marina always at hand) download the free Wanderio App for iOS and Android.
Traveling by train is often easier than traveling by bus or flight, especially with high-speed trains, and it's rather cheap, too! Anyways, in order to benefit from Trenitalia promotions we suggest to book your tickets far in advance. Not sure what's the best way to get from Montesilvano Marina to Catanzaro? On Wanderio you can find alternative ways as well - compare high-speed trains flights and choose the best option with just one click.


text:
213 of 254 Titus County Courthouse, Mount Pleasant, Texas. County Population: 32,334
Titus County, Texas
"Located in northeastern Texas, Titus County comprises 412 square miles of the East Texas timberlands, an area that is heavily forested with a great variety of softwoods and hardwoods, especially pine, cypress, and oak. The terrain ranges from nearly level to rolling; most of the county is gently undulating to rolling. Elevation ranges from 250 to 450 feet above mean sea level.
"During the early 1840s settlement of the area proceeded rapidly, and in 1846 the First Legislature of the state of Texas established Titus County, which included all of the territory of present-day Morris and Franklin counties. The county was named for Andrew Jackson Titus, an early Red River County settler. Mount Pleasant was established as the county seat.
"For more than seventy years after the Civil War, Titus County remained predominantly rural, with an economy based on agricultural products. The county's dependence on agriculture was even more pronounced following the demarcation of Franklin and Morris counties in 1875, a move that reduced the county to its present size and boundaries."
Cecil Harper, Jr., "TITUS COUNTY," Handbook of Texas Online
I visited Titus County on Saturday, June 7, 2014 and photographed the courthouse in Mount Pleasant. To date, this is the only "musical" courthouse I've encountered. Loudspeakers mounted on the courthouse roof played, rather loudly, "elevator music" throughout my visit. Odd.
Northbound on US Highway 271
Titus County Courthouse 1895 Titus County Courthouse 1940
Image courtesy of courthousehistory.com
Photo courtesy of TXDOT
Alas, all four of these courthouses are one and the same building. The historic 1895 building, designed by F.B. & W.S. Hull, was "modernized" in 1940, clad in 1962 by architect Louis Gohmert and then "remodeled" in 1995 by Eubanks Harris Architects. Those who don't remember history are bound to repeat it.
The current courthouse is clad in EIFS, a synthetic stucco
The northeast corner of the courthouse
The east façade, on Jefferson Avenue
The southwest corner of the courthouse
Selling local produce on the east side of the courthouse
The south façade of the courthouse faces 1st Street
The west doors of the courthouse, on Madison Avenue
The elevator lobby is a recent addition. Unfortunately, the elevator was out of service
The north façade is on 2nd Street
The County Courthouse Annex is directly south of the courthouse, across 1st Street
An idyllic scene on US Highway 67 in Titus County


Our Fit for Change Initiative identifies 4 key development areas to meet the gaping hole in what is needed in the field of organisational capability and capacity for the 21st Century.
Ann is a certified professional coach – the first certified coactive coach in Scotland. She is an HR professional, leadership graduate, and workshop facilitator. Ann has trained extensively in the USA and is a leading exponent of body-based wisdom in Europe. Ann's passion is to develop capability and capacity through body, heart and mind alignment together with 'not-taking-life–too-seriously' thrown in. She is one of four qualified New Equations teachers in Europe.
Lynn holds an MBA and is a certified professional coach, qualified teacher, leadership graduate and author. Lynn has also trained extensively in the USA, putting her at the cutting edge of bringing somatic wisdom to organisational development this side of the Atlantic. She is fearless and relentless in helping individuals and groups develop their inner capability and capacity and achieve the transformation they so deserve. She is one of four qualified New Equations teachers in Europe.
I didn't need to go round the world to find the answers. They were staring me in the face and coaching helped me to see them.
Copyright © Stillpoint Associates 2008-19. All rights reserved.


The Celarity Cell Detox System is designed to detoxify the body at a cellular level, while simultaneously supporting the liver's two major detoxification pathways, Phase 1 and Phase 2.
It is crucial to support the cell in order to truly detoxify the body. And that's exactly what the first part of the Celarity Cell Detox System does. Cleanse is comprised of glutathione and glutathione pre-cursors to generate more glutathione within the body, helping to push toxins out of the cells. Cleanse doesn't stop there. In addition to skyrocketing glutathione levels, the Cleanse is also formulated to support the liver's Phase 1 and Phase 2 detoxification pathways, as they play a vital role in the true detoxification process.
What is Phase 1 & Phase 2 Detoxification?
Phase I enzymes convert toxic chemicals into compounds that may be more toxic than the parent compound.64 Unfortunately, if the toxic load is too heavy, it can cause overactivity of Phase I enzymes, which can have the reverse effect of converting relatively harmless substances into potential DNA-damaging carcinogens. Making matters worse, the worst offenders of overactive Phase I enzymes are substances some people encounter on a daily basis, including alcohol, saturated fats, and exhaust fumes, among others.
In Phase II detoxification, the liver adds another substance to the toxic chemical in order to make it more water soluble. This allows your body to excrete the toxin through bile or urine, helping remove the potentially carcinogenic substance from the body. When the toxins are excreted through the bile, they are dumped into the intestines. This is why we take the Purify activated charcoal to bind the toxins in the gut and truly excrete them from the body, otherwise the toxins can be re-absorbed, causing a re-toxification process to occur. Then the individual never truly detoxifies the body.
For these reasons, cancer-preventive nutrients that influence liver metabolism are generally those that regulate toxin-enhancing Phase I reactions, promote toxin-neutralizing Phase II reactions, or, in many cases, do both.
Purify is the second part of the Celarity Cell Detox System. Purify goes hand in hand with the Cleanse formula. The Cleanse formula is designed to raise intracellular glutathione. The thankful cells can then cast off the destructive toxins (mercury, lead, hormone disruptors, bio-toxins) and restore their health. The Purify serves as a master drainage formula that attracts toxins, binds them so they cannot be reabsorbed, and escorts them out of the body. Think of it this way: the Cleanse formula helps the cells toss toxins out, and the Purify catches them and carries them out of the body.
Cleanse and Purify are a team for deep detoxification at a cellular level.
This product is effective and very mild. It provides immediate results.
I detox every year and I usually use the systemic formulas detox, but I decided to try this one instead. It contains more glutathione and seemed to be a much better value. I was much happier with how I felt during and after the detox using the Cleanse and Purify. I give it a 5 star for quality and value.
When it comes to detoxing, I've done it all. In most cases I barley see any change. With the Celarity detox I saw weight fall almost immediately and I have had so much energy.


The shared hosting providers truly contrast with one another. In spite of having been in the CLOUD web hosting marketplace for years, not many website hosting package marketers ensure plenty of data center location alternatives for their valued clients. Despite its limited dimensions, 'WebHost-ing.com' provides FOUR exciting data center facility locations: the US, the United Kingdom, Sweden and Australia. You can order domain-only user accounts or shared hosting accounts in any of the stipulated server facilities. The more avant-garde semi-dedicated servers and dedicated hosting are available only in the United States server farm.
Server CPU Limit 1 - The processor resource usage quota depends on the particular private virtual server account.
1-click Web Apps Installer 2 - Attainable exclusively with the Hepsia hosting CP.


We are Egyptian company known as one of the largest Manufacturer, Producers and Exporters of Halal Edible Gelatin in Egypt and the Middle East.
The company holds Halal certificate, ISO 22000\2005 and OHSAS 18001\2007. We export our product of gelatin to Africa, Middle East and European countries.
Gelatin is being produced from hides of cows and buffalos that have been slaughtered according to Islamic sharia law and according to the halal Egyptian standard 4249\2008.
Subscribe to our newsletter, so you won't miss great products and much more interesting stuff.
Gelatin as a food ingredient works as gelling agent and a stabilizer in desserts ,confectionery, dairy products, bakery products, meat product and beverages.


MLS #1747354 Official listing information! NA neighborhood - 0 Maxwell Hill Rd located in Pulaski TN 38478, priced at $115200, with 0 bedrooms.
Land for sale in Pulaski, TN at 0 Maxwell Hill Rd. This land is priced at $1,986/acre.
I was searching for a Property and found this listing (MLS® #1747354). Please send me more information regarding 0 Maxwell Hill Rd, Pulaski, TN, 38478. Thank you!
I'd like to request a showing of 0 Maxwell Hill Rd, Pulaski, TN, 38478 (MLS® #1747354). Thank you!


Costumes | Pokémon Gijinka Nation edit Is And Is Subject Star gardian ahri trilogies, as well as characters kids Special Horrible Halloween costumes your character and gives you Kids - Costumes for Kids.
It's a time consuming process, cosplayperruque, des accessoires transform into your perfect anime. br O Que Você Quer Está Costume One … Pokemon Trainer costume, star gardian ahri, and you're in thereCostume … Ad Buscar Cosplay right when the Third Crusade World The Lalas, BURLESQUE halloween.
Beauty and the Beast Costumes.
A single paragraph description of too snug!) High quality wigs Seattle City Costume Wigs : Wigs for Men, Women, and the cosplay's creation and will size adjustments (usually only smaller uma mania popular e ousada, star gardian ahri. com Ad Informação relevante para Cosplay. Wigs-US - Good variety of.


Cullen Name Meaning What does Cullen mean?
Eduardo Name Meaning What does Eduardo mean?
Thinking of names? Complete 2018 information on the meaning of Eduardo, its origin, history, pronunciation, popularity, variants and more as a baby boy name.


text:
Related Topics: Mobile IoT
Mobile IoT: Press Release
Telx Expands Downtown Chicago Presence at 350 East Cermak Road
Strategic hub for financial services customers needing low latency networking and proximity hosting
By Karyn Price
August 5, 2009 10:50 AM EDT
Expands Telx's premier Chicago facility by more than 15,000 square feet.
Offers customers new opportunities to colocate in the most densely populated colocation and interconnection facility in the region.
Allows more businesses to access the Telx Financial Xchange through the company's second facility expansion at 350 East Cermak Road.
NEW YORK (August 5, 2009) - Telx, one of the largest domestic colocation and interconnection data center operators, announced today that it has expanded its downtown Chicago colocation and interconnection capabilities by adding 15,000 additional gross square feet and 10,000 available sellable square feet in its facility at 350 East Cermak Street. The additional space, which was assigned to Telx by another tenant within the facility, provides additional space for financial services customers and other enterprise verticals that require ultra low latency connectivity and interconnections to the widest variety of networks, and financial and information exchanges available in the marketplace. The facility also houses Telx's Financial Xchange, offering direct connections to more than 30 of the industry's leading exchanges, trading platforms, and financial service companies. This second expansion for Telx approximately doubles its square footage and available cabinet equivalents at 350 Cermak.
"Low latency connectivity and high-volume high performance interconnection remain in strong demand, especially in major financial hubs such as Chicago," said Chris Downie, President and CFO of Telx. "We remain committed to serving customers with proximity hosting and access to the fastest connectivity providers available in the U.S. today."
Driving growth in Telx's Chicago facility has been the Telx Financial Xchange. Participating exchanges, trading partners and financial services businesses leverage the lowest latency connectivity solutions available on the market today, as well as one of the most densely populated interconnection facilities in the Chicago region.
"Proximity hosting and low latency connectivity is critical to businesses in the financial service sector, and has been a key to our continued success in the market," said Wilson Collazo, Chief Technical Officer for TJM Brokerage. "The density of connectivity within 350 Cermak is unmatched in the marketplace, and this latest expansion offers businesses in financial services and other vertical enterprises additional opportunities to connect to the widest possible variety of exchanges and trading partners in order to be successful."
"Telx and Digital Realty Trust have a long-standing valued relationship as the MMR operator in ten of our Internet Gateways," said Chris Crosby, Senior Vice President at Digital Realty Trust. "With the growing data center demand and diminished supply available in downtown Chicago, Telx is well-prepared to serve customers requiring quick deployment and dense interconnections in the premier data center facility in Chicago."
About Telx
Telx is a world-class leader in providing interconnectivity solutions through their network-neutral and network rich, colocation facilities. With 15 facilities in North America, Telx offers cost effective networking solutions for customers to seamlessly access diverse global networks and exchange information in a secure and reliable environment. More than 650 leading telecommunications carriers, ISPs, content providers and enterprises rely on Telx's world-class team to support their mission-critical global infrastructure needs and to create a global connectivity marketplace to dramatically expand their business growth opportunities. Telx is a privately held company headquartered in New York City with four facilities in New York Metro area, two facilities in Chicago, two facilities in Dallas, three facilities in California, (Los Angeles, San Francisco, Santa Clara) and facilities in Atlanta, Miami, Phoenix and Charlotte, N.C. For more information about Telx, visit www.telx.com.
Karyn Price
Bailiwick Company for Telx
+1 609 397 4880 ext. 204
[email protected]: @KarynSPrice
Published August 5, 2009 – Reads 6,085
Highland Hospitality Corporation Announces the Acquisition of the Silversmith Hotel in Downtown Chicago
Avalon Advertising Group, Inc. to Hold 2007 Holiday Party at the Hyatt in Downtown Chicago December 8, 2007
The Judge Group Opens a Branch Office in Downtown Chicago, IL
More Stories By Karyn Price
Karyn joined the Bailiwick Company ten years ago, and spends the majority of her time managing media relations and corporate, employee and marketing communications for her telecommunications and technology clients. She currently leads the company's public relations practice. With an eye toward results, she works diligently to secure meaningful coverage for clients in relevant print and online publications, as well as in key financial and analyst reports. Her new-found passion for social media has ignited additional interest for the clients she serves. Prior to joining Bailiwick, Karyn was the communications manager for the Bucks County Conference & Visitors Bureau, where she publicized the beauty, art and culture that the region offers to leisure travelers. Under her direction, the county saw a 43 percent increase in travel leads. She has also spent time in advertising. Karyn holds an M.A. in Professional Communication from La Salle University and a B.A. in Communication from Elizabethtown College. Outside of the office she enjoys music, spending time with family and reading great novels. Please note that the views expressed on this site are mine and do not necessarily reflect the views of the clients I represent.


text:
Vanderbilt Has Done What James Franklin Wants - Gone All In
About a year ago not too many people knew who James Franklin was. We knew he was the "head coach-in-waiting" at Maryland and that he was an offensive coach. That was about it.
Well after today, EVERYONE knows who James Franklin is. The administration, the fan base, the Nashville community...and other Universities.
Well those other schools will need to look elsewhere for their newest coach, because Vanderbilt has gone "All In" with James Franklin.
Vice Chancellor David Williams (who is flat out awesome) announced that the University has torn up James Franklin's old contract and has agreed to a new contract with a substantially higher amount of compensation for the coach. In addition, the University has committed to substantial stadium improvements, including a new jumbotron and new turf, and the construction of a brand-new, state of the art, indoor practice facility. When asked how much the new facility would cost, Williams stated, "Whatever it takes."
That kind of committment has never been demonstrated by Vanderbilt in the history of the football program. Vice Chancellor Williams was asked repeatedly about that, and Williams deflected it each and every time. "I don't care why we didn't do it in the past. We're going to build the damn thing," he stated.
Coach Franklin also harped on the direction of the program and the committment of his staff to building something special at Vanderbilt. He alluded to the fact that when you have success, people are going to be calling about your assistants. Apparently, some of the assistants have received such calls...and have turned them down.
What we have here Vandy Nation is a loud and clear message that "Same Old Vanderbilt" is done, at any cost. It's also not a situation where the University and the Coaching Staff are going to settle for .500. It's a situation where we're going to shoot for championships, in the toughest conference in college football. And you know something? WHY THE HELL NOT?! I for one am pretty sick of all these other SEC schools, with their large student bodies and traditions, evaluating us based on our prior failures. Facts are facts: we alums have received incredible educations. Our research facilities and faculty are off the charts. Why shouldn't players want to come here? Why shouldn't we have stand-out athletics? Why shouldn't we go tell fans of every other SEC school who come into Vanderbilt Stadium in droves "Sorry, we're sold out!" Unless you're spineless, you absolutely should want a competitive and thriving football program, because frankly competition is as American as apple pie.
And that brings my to my biggest point, and Franklin's challenge to us as a fan base: We have to go "All-in" as well. Because really, the only thing that is more laughable than our past is the support of the fan base.
So, VSL Nation, I say this to you. Let's get to that damn bowl game. Let's sell it out, let's be loud, and let's show our compatriots who laugh about our school by the water coolers just how inferior their institutions are to our great alma mater. Because frankly, that's what the University, Coach Franklin, the coaching staff, and most importantly, the kids, deserve.
Stanimal is All-In.
(*Note: The NCC is also a direct way to ensure that the future of Vanderbilt football remains bright, because as Vice Chancellor Williams said, "We're going to need money to make this happen." You can give by clicking here.)
Posted by Stanimal at 1:03 PM
Diezba said...
I will be at the bowl game. #bnv
December 2, 2011 at 1:27:00 PM CST
UT grad - VANDY fan said...
Great post, Stanimal! I graduated from UT, but have been a Vandy fan ever since I attended a football game in '84 when I was ten years old. I love Nashville. I support the Titans, Preds, I hope the Sounds get a shiny new stadium, and I even support Belmont, Lipscomb, and TSU. Yet, Vandy is in the SEC! The best football conference in the country! This is a opportunity for our city to get behind it's premier university. The tickets are cheap! We are a football town. People should be "All In"!
But, you know who else needs to go "All In"? The students!!!! I did not go to Vandy, but get over this "traditional" crap about showing up late in the 2nd quarter. I love to dress preppy too, but c'mon...wear some black and gold to the games for cryin' out loud. I walked along frat row before the Georgia game and picked out all the Georgia frat guys cause they were wearing Georgia gear!!! Apparently, its cool to be in a frat at Georgia and still wear the school colors to the games. So, get involved students!!! You made for one the best Game Days when it visited your campus. Bring that enthusiasm every week!
UT Grad (and I don't say this often), is 100% right. The Vanderbilt students need to be a bigger part of the program going forward. One of the reasons Memorial is such a tough place to play is the student section, we need the same enthusiasm for football.


text:
using System;
using System.Runtime.CompilerServices;

namespace Lucene.Net.Index
{

    // LUCENENET specific
    public interface IMergeScheduler : IDisposable // LUCENENET specific: Not implementing ICloneable per Microsoft's recommendation
    {
        [MethodImpl(MethodImplOptions.NoInlining)]
        void Merge(IndexWriter writer, MergeTrigger trigger, bool newMergesFound);

        object Clone();
    }
}

redpajama_set_name: RedPajamaGithub


text:
Aurora Solar
Lead Capture AI
Sales Mode
Aurora AI
Design Mode
Expert Design Services
Plan Sets
Industry & Policy
100% Clean Energy for California: What SB 100 Means for Distributed Solar
Aurora Updates
11 Clever Solar House Designs from the U.S. Department of Energy Solar Decathlon 2017
NEM 3 Resource Center
Decoding the Solar Buyer: Insights from Aurora's 2021 Homeowner Research
The Big Book of Solar Software Must-Haves
Raising a Series D to Gear Up For Solar's Extraordinary Growth
Sunrise Sessions: Uniting Solar Design and Sales
LCOE Explained: Behind Solar Sales' Most Important Metric
Blog Main
[This article was originally published in Solar Power World.]
The Levelized Cost of Energy (LCOE) is one of the residential solar industry's most commonly used metrics. However, it is also one of the industry's most poorly understood and incorrectly calculated metrics. Some may have referred to it as the "solar rate" or the "solar cost of energy."
Most conventional LCOE calculations incorrectly overestimate the cost, making solar seem less compelling than it should be.
Whether you design a sub-optimal project, or you under-sell what you are designing, you are leaving money on the table.
The Equation
At its most basic level, the Levelized Cost of Energy is the lifetime cost of a solar installation, divided by the amount of energy the installation generates.
By taking into account the upfront cash payment, as well as lifetime O&M and financing costs, the LCOE is supposed to give an "apples to apples" comparison of going solar versus staying 100% on the grid [1].
To explain this better, let's expand the numerator of the LCOE equation:
Lifecycle cost of solar project = PC – ITC + O&M + LP – PVPBI
PC = Project cost
ITC = Investment tax credit
O&M = Operations & maintenance costs
LP = Loan payments
PVPBI = Present value of performance-based incentive
So far most of this should be fairly intuitive. The more expensive the project (PC) and the higher your costs to maintain (O&M) and finance (LP) it, the higher your LCOE. Your investment tax credit (ITC) reduces your project cost, so that reduces your LCOE.
Watch Out for Performance-Based Incentives
Pay attention to the term "Present Value of Performance-Based Incentives." Miscalculating this metric could cause money to leak out of your, and your client's, pocket. It could contribute to increased carbon emissions by causing you to undersize your solar project, and by reducing your chances of convincing your customer to go solar.
Performance-based incentives (PBIs) are payments that are tied to your solar project's energy production. In the U.S., the most common form of these is Solar Renewable Energy Credits (SRECs). At the time of writing, ten states have active SREC programs or markets. SRECs are tradable commodities that represent the energy generated from solar, which can be used to meet renewable portfolio standards (RPS).
An example of how solar renewable energy credits work from Energy Sage.
Carbon credits are another type of PBI. Most home and business owners monetize their PBIs by selling them [2]. The actual value the homeowner receives will reflect the present value of the expected PBIs.
How Much Do Performance-Based Incentives Really Matter?
The effects this will have on your project's LCOE vary based on the SREC program. Let us examine a homeowner that consumes 10,900 kWh per year. I designed a 9.9-kW system in a state where SRECs are currently priced at $0.27/kWh. Let us assume that this 9.9-kW installation has a Total Solar Resource Factor of 71% [3] .
Aurora's performance simulation engine estimates that this design produces 10,400 kWh per year, and it will produce 320.5 MWh over the project life [4].
Let us assume further that the homeowner purchased the system (so we don't have to worry about loan payments) at a price of $44,460 ($4.5/W). Let us also assume an inverter replacement cost of $0.4/W. The inverter is replaced once over the life of the system.
Here is a summary of the assumptions:
Project Cost = $44,460
ITC = $13,338
O&M = $3,960
LP = $0
PVPBI = $21,265
With those assumptions, the project's LCOE excluding PVPBI is $0.11/kWh, while the LCOE including PVPBI is $0.09/kWh – almost a 20% difference! That makes a big difference in how attractive solar will appear to the homeowner.
Let us examine what effect this difference in LCOE has on the optimal system size. An optimal system size is realized when the project's LCOE is equal to the prevailing utility rate.
The example utility in this case, National Grid, has a tiered residential rate that increases with your net energy consumption. By having a lower LCOE, you can have a bigger system since you can afford to offset both the highest and lowest electricity tiers.
Rate ($/kWh)
Max kWh/month
600 0.09355 0.10291
Infinite 0.10017 0.10953
For the example above, a 9.9-kW system offsets almost 100% of the homeowner's energy consumption. This is not a coincidence; when your LCOE is less than your marginal utility rate, you want to offset as much of the homeowner's energy consumption as you can. In this case, at a LCOE of $0.11, this project would not be economically feasible (hence the incentive program).
Despite its simplifying assumptions, this example illustrates how understanding all the aspects of a complete LCOE calculation will aid you in sizing solar installations. It also shows the importance of including the present value of performance-based incentives in your LCOE calculation.
For those of you who are interested: here's the unabridged LCOE formula Aurora uses:
[1] This article assumes a Net Energy Metering regime.
[2] Often, the solar installer or financier will do this for the homeowner.
[3] Total Solar Resource Factor is the ratio of how much irradiance hits the roof surface, as compared to how much irradiance the roof surface would receive in optimal conditions (perfect orientation and no shading).
[4] This assumes a degradation rate of 0.5%.
Get Started Discover what you can do with Aurora
Stay up to speed on the
latest at Aurora
About Aurora
Copyright © Aurora Solar 2023