text:
\section{Introduction}

Anderson localization refers to the exponential localization in configuration space of the eigenstates of noninteracting systems with onsite disorder. It is caused by quantum interferences and was first studied in the context of electron conductance~\cite{Anderson1958,Lee1985,Kramer1993,Lagendijk2009,Abrahams2001}. The model employed to explain this phenomenon is a tight-binding model with random diagonal elements. In three dimensions (3D), the model may be found in the delocalized (metallic) or localized (insulating) phase, depending on the disorder strength, and a mobility edge is present.

The metal-insulator transition in the 3D model was also studied in terms of energy-level statistics~\cite{Efetov1983,Altshuler1988,Shklovskii1993,Mirlin2000}. The focus in~\cite{Shklovskii1993} was on the distribution $P(s)$ of the spacings $s$ between unfolded neighboring levels. The claim was that in the thermodynamic limit, there should be only three possible forms for $P(s)$: the Wigner-Dyson distribution [$P_{\text{WD}}(s)$] in the metallic phase, a Poissonian distribution [$P_{\text{P}}(s)$] in the localized phase, and an intermediate distribution between the two shapes at the critical point.

The Poissonian distribution appears when the eigenvalues are uncorrelated and degeneracies are not prohibited. The Wigner-Dyson distribution, first studied in random matrix theory (RMT)~\cite{MehtaBook,Guhr1998}, emerges when the eigenvalues are highly correlated in the sense they strongly repel each other. In the increasing order of the degree of correlations, we find first the Gaussian orthogonal ensemble (GOE), when the full random matrices are real and symmetric, then the Gaussian unitary ensemble (GUE), when the full random matrices are Hermitian, and finally the Gaussian symplectic ensemble (GSE), when the full random matrices are written in terms of quaternions. In the case of the 3D Anderson model, the only $P_{WD}(s)$ observed is that for the GOE.

For the 1D Anderson model in the thermodynamic limit, all eigenstates are exponentially localized in configuration space for any infinitesimal value of the disorder strength, so there is no metal-insulator transition. However, if the 1D system is finite, we observe the crossover between different kinds of energy-level statistics. Instead of going only from $P_P(s)$ to $P_{WD}^{GOE}(s)$~\cite{Wischmann1990}, even stronger levels of correlations are seen. Indeed we have the following picture: The eigenstates are localized on the scale of the sample size if the disorder is strong, in which case the energy-level spacing distribution is Poissonian. In the other extreme, where disorder is absent, the eigenstates are extended and the eigenvalues are nearly equidistant. In this last case, the spectrum is said to be of the ``picket-fence'' type~\cite{Berry1977,Pandey1991}, which means that the eigenvalues are even more correlated than what one finds in full random matrices and the energy-level spacing distribution has an abrupt peak away from $s=0$. Between these two limits, as the disorder strength decreases from the point where the energy levels are uncorrelated [$P_{\text{P}}(s)$] to the point where they are picket-fence like, the spectrum experiences all degrees of energy-level repulsion typical of full random matrices, going first from Poisson to GOE-like, then from GOE-like to GUE-like, from GUE-like to GSE-like, and finally from GSE-like to picket-fence, passing through all intermediate distributions in between~\cite{Sorathia2012}. Notice, however, that these different energy-level spacing distributions do not reflect the symmetries of the system. From large disorder to zero disorder, the Hamiltonian matrix of the 1D Anderson model is throughout real and symmetric, just as GOE matrices. The different shapes of $P(s)$ are caused by the unavoidable change in the degree of correlations that the eigenvalues have to experience as going from being uncorrelated to picket-fence like.

Since experiments are done with finite systems~\cite{Katsumoto1987,Waffenschmidt1999,Billy2008,Roati2008,Manai2015,Volchkov2018}, we decided to look into the finite 1D Anderson model in more detail, focusing not only on energy-level statistics, but also on the structure of the eigenstates and on dynamics. We consider a chain with open boundary conditions. The analysis of the dynamics is particularly relevant to experiments with cold atoms and ion traps that have limited access to the spectrum, but routinely investigate time evolutions. Our studies are made with respect to the parameter $\xi \propto (W^2 L)^{-1}$, where $W$ is the disorder strength and $L$ is the size of the chain.

We find clear relationships between different energy-level repulsion parameters and measures of the degree of delocalization of the eigenstates. However, despite the similarities between the energy-level statistics of the finite Anderson model and RMT, not all eigenstates of the 1D system are random vectors as in full random matrices, since their structures depend on their energies.

To study the dynamics, we place the particle initially on a single site and compute the evolution of the probability to find it on a specific site of the chain. The survival probability, which is the
probability to find the particle on its initial state, receives special attention. Its initial decay cannot distinguish systems with or without level-repulsion, a claim that has been made in the past in the context of many-body quantum systems~\cite{Torres2014PRA,Torres2014NJP,Torres2014PRAb}. Manifestations of the correlations between the eigenvalues emerge only at long times, in the form of the correlation hole~\cite{Leviandier1986,Guhr1990,Wilkie1991,Alhassid1992,Gorin2002,Torres2017,Torres2017Philo,Torres2018,TorresHerrera2019,Sergio2019PRE}. The hole is not necessarily a signature of quantum chaos. As we show, it emerges also in the limit where $W=0$ and the model is integrable, but the eigenvalues are strongly correlated. The hole gets deeper as the correlations become stronger and the statistics moves from GOE-like to GUE-like, to GSE-like, and finally reaches the picket-fence spectrum.

Before the correlation hole, the evolution of the survival probability depends on the shape and bounds of the energy distribution of the initial state~\cite{Tavora2016,Tavora2017}. We find that when the particle is on the first site of the chain, the shape of this distribution is the same found for the dynamics under full random matrices, resulting in the same power-law decay $\propto t^{-3}$ observed for RMT~\cite{TorresKollmar2015,Tavora2016,Tavora2017}. This behavior is much faster than the $t^{-1}$ decay obtained for a particle initially placed far from the edges of the chain. The physical justification of this behavior lies in the spread of the initial state over other site-basis vectors, which is almost unidirectional for the particle on site 1, but more homogeneous for the particle in the middle of the chain. This is an interesting border effect that may find applications for fast and efficient transfers of particles in quantum systems.

The paper is organized as follows. In Sec.~II, we describe the model. Section~III is dedicated to the analysis of the spectrum and the structure of the eigenstates. The dynamics is the subject of Sec.~IV.
Discussions are presented in Sec.~V.

\section{Finite 1D Anderson Model}

The 1D Anderson model with onsite disorder and open boundaries is described by the one-particle tight-binding Hamiltonian,
\begin{equation}
H = \sum_{n=1}^{L} \epsilon_n c_n^{\dagger} c_n - J \sum_{n=1}^{L-1} (c_n^{\dagger} c_{n+1} + c_{n+1}^{\dagger} c_{n} ).
\label{Eq:Anderson}
\end{equation}
Above, $c_n^{\dagger}$ $(c_n)$ is the creation (annihilation) operator of a particle on site $n$,  $\epsilon_n$ are uniform random numbers in the interval $[-W/2,W/2]$, so their variance is $W^2/12$,   and $L$ is the finite number of sites. The energy scale is set by choosing the hopping amplitude $J=1$.
The density of states (DOS) is bounded by the spectrum edges $\pm (2+W/2)$.

We use as basis vectors $|\phi_n \rangle$, states that have the single particle placed on a single site $n$. In this basis, the Hamiltonian matrix for Eq.~(\ref{Eq:Anderson}) is tridiagonal.
If the disorder strength $W$ is very strong, $\langle |\epsilon_n - \epsilon_m|\rangle \gg 1$, the diagonal part dominates and the eigenstates are localized in configuration space. This happens when the localization length is smaller than the chain size $L$.

In the absence of disorder, $W=0$, we have a Toeplitz tridiagonal matrix. The eigenvalues are
\begin{equation}
E_\alpha = 2\cos \left( \frac{\alpha \pi}{L+1} \right), \quad \alpha=1,2,\ldots L
\label{Eq:E}
\end{equation}
and the eigenstates are
\begin{equation}
|\psi_{\alpha} \rangle = \sum_{n=1}^L \sqrt{ \frac{2}{L+1} } \sin \left( \frac{\alpha n \pi}{L+1} \right) |\phi_n \rangle.
\label{Eq:psiAlpha}
\end{equation}

We use $\alpha$ as the index for the eigenstates of $H$ and $n$ for the basis vectors $|\phi_n \rangle$. We denote the overlaps between $|\psi_{\alpha} \rangle $ and $|\phi_n \rangle$ as $C^{(n)}_{\alpha} = \langle \psi_{\alpha}  | \phi_n \rangle  = \left(C^{(n)}_{\alpha}\right)^* = \langle \phi_n |\psi_{\alpha} \rangle $.

\subsection{Parameter for the Analysis}

The localization length $l_{\infty} (E)$ for the 1D Anderson model was obtained by Thouless via perturbation theory to second order in the disorder strength~\cite{Thouless1974,ThoulessBook,Heinrichs2003},
\begin{equation}
l_{\infty} (E) = \frac{96(1-E^2/4) }{W^2}.
\label{Eq:loc}
\end{equation}
This expression is valid inside the energy band, not at the edges. It also has a correction right at the center of the spectrum~\cite{Czycholl1981,Kappus1981,Izrailev1998,Tessieri2012}, where it becomes
\begin{equation}
l_{\infty} (0) =  \left[ \frac{\Gamma(1/4)}{\Gamma(3/4)} \right]^2 \frac{12  }{W^2} \approx \frac{105.045 }{W^2} .
\end{equation}

Motivated by the above equations, our studies are carried out as a function of the parameter
\begin{equation}
\xi=\frac{l_\infty(0)}{ L} \approx \frac{105.045 }{W^2 L},
\label{Eq:xi}
\end{equation}
which is varied from very small values, where for a fixed $L$, the disorder is very strong and the eigenstates are localized, to very large values, where the chain is almost clean and the eigenstates are maximally extended. Even though we vary $\xi$ over a large range of values, it is important to reiterate that Eq.~(\ref{Eq:loc}) holds only for weak disorder.

\section{Properties of the Eigenvalues and Eigenstates}
We analyze the properties of the eigenvalues and eigenstates as a function of the parameter $\xi$. Let us start with the DOS. In the clean limit, $W=0$, the DOS is obtained from~\cite{Schiulaz2018}
\begin{eqnarray}
\rho_{\text{DOS}}(E) &=& \frac{1}{L} \sum_{\alpha} \delta(E - E_\alpha) \nonumber \\
&=& \frac{1}{L}\sum_{\alpha} \int_{-\infty}^{\infty} \frac{d\tau}{2\pi} e^{i \tau [E-2 \cos \left( {\alpha \pi}/{(L+1)} \right)]}.
\label{Eq:preDOS}
\end{eqnarray}
In the continuum,
\begin{eqnarray}
\rho_{\text{DOS}}(E)
&=&  \int_{-\infty}^{\infty} \frac{d\tau}{2\pi} e^{i \tau E} {\cal J}_0(2\tau)  \nonumber \\
&=&  \frac{1}{\pi \sqrt{4 - E^2}},
\label{Eq:DOS}
\end{eqnarray}
where ${\cal J}_0$ is the Bessel function of first kind. The DOS has a U-shape, where the distribution peaks at the edges of the spectrum. This form persists even when the disorder is strong and the energy-level spacing distribution is Poissonian, provided $L$ is sufficiently large (for example, when $\xi \sim 0.002$ and $L\sim 16\, 000$, as we use below).

\subsection{Energy-Level Statistics}

To study the degree of short-range correlations between the eigenvalues, we use the energy-level spacing distribution $P(s)$ and also the ratio $\tilde{r}_\alpha$ between neighboring spacings. To get $P(s)$ we need to unfold the spectrum, but to compute $\tilde{r}_\alpha$ unfolding is not needed.

The Poissonian distribution is  $P_{\text{P}}(s) = \exp(-s)$. In the form of the Wigner surmise, the Wigner-Dyson distribution is written as $P_{\text{WD}}(s)=a_{\beta} s^{\beta} \exp(- b_{\beta} s^2)$. The constants $a_{\beta}$ and $b_{\beta}$ can be found in Ref.~\cite{MehtaBook,Guhr1998}, while $\beta$ is shown in Table~\ref{Tab:1} for the different energy-level statistics obtained for ensembles of full random matrices.

The ratio $\tilde{r}_\alpha$ is defined as~\cite{Oganesyan2007,Atas2012},
\begin{equation}
\tilde{r}_\alpha = \min \left( r_\alpha, \frac{1}{r_\alpha}\right), \quad \text{where} \quad r_\alpha=\frac{s_\alpha}{s_{\alpha-1}}
\end{equation}
and $s_\alpha = E_{\alpha+1} - E_\alpha$ is the spacing between neighboring levels. The values of $\langle \tilde{r}\rangle $ obtained by averaging $\tilde{r}_\alpha$ are also shown in Table~\ref{Tab:1} for different energy-level statistics.
\begin{table}[h]
  \centering
  \begin{tabular}{ | c | c | c| c|}
    \hline
     & $\beta$ & $\langle \tilde{r}\rangle $ &$P_R$ \\ \hline
    Poisson & 0 & $2\ln2 -1 \approx 0.39$ & $\approx 1$ \\ \hline
    GOE & 1 & $4-2\sqrt{3} \approx 0.54$ & $(L+2)/3$ \\ \hline
    GUE & 2 & $2\sqrt{3}/\pi -1/2 \approx 0.60$ & $(L+1)/2$ \\ \hline
    GSE & 4 & $(32/15) \sqrt{3}/\pi -1/2 \approx 0.68$ & $(2L+1)/3$ \\  \hline
    clean &$\gg4$  & $\approx 1$ & $2(L+1)/3$ \\
    \hline
  \end{tabular}
 \caption{Values of $\beta$ and $\langle \tilde{r}\rangle $ for different level statistics and the corresponding  values of the participation ratio. The results for the GOE, GUE and GSE rows refer to those for ensembles of full random matrices. The result for $P_R$ for the clean model is obtained using Eq.(3).}
  \label{Tab:1}
\end{table}

To study $\beta$ as a function of $\xi$, we extract $\beta$ from the expression suggested in~\cite{Izrailev1990},
\begin{equation}
P_\beta (s) = A \left( \frac{\pi s}{2} \right)^\beta
\exp \left[ -\frac{1}{4} \beta \left( \frac{\pi s}{2} \right)^2 -
\left( B s -  \frac{\beta}{4} \pi s \right)  \right] .
\label{Eq:beta}
\end{equation}
The parameters $A$ and $B$ are obtained from the normalization conditions
\begin{equation}
\int_{0}^{\infty} P_\beta (s)  ds =1\,\,\text{and}\,\, \int_{0}^{\infty} s P_\beta (s)  ds =1.
\end{equation}
The phenomenological expression (\ref{Eq:beta}) was written so that it reproduces for $\beta=1,2$, and $4$ the energy-level spacing distributions obtained from RMT for GOE, GUE, and GSE, respectively.
Variations of Eq.~(\ref{Eq:beta}) can be found in~\cite{Casati1991,Kottos1999,Sorathia2012}.

The main panel of Fig.~\ref{fig:beta}~(a) shows that $\beta$ is a linear function of $\xi$ for a broad range of values, $0.5 \lesssim  \xi \lesssim 10^2$. The linear fit becomes even better for the values of $\xi$ where we have the standard RMT distributions, that is for $1\leq \beta \leq 4$, as seen in the inset of Fig.~\ref{fig:beta}~(a).
This inset suggests that the energy-level repulsion parameter $\beta$ equals the rescaled localization length obtained at the middle of the spectrum~\cite{footnoteSorathia},
\begin{equation}
\beta \approx \xi.
\label{Eq:linear}
\end{equation}
Following discussions in Ref.~\cite{Sorathia2012}, the linear relation $\beta \propto \xi$ appears to be quite general and has been observed also for the kicked rotor~\cite{Izrailev1989,Izrailev1990} and Wigner banded random matrices~\cite{Casati1993}; see also~\cite{Angel2019} where a linear relation between the repulsion parameter $\beta$ and the degree of disorder of the 1D Anderson model $\xi$ was obtained through scattering quantities.

\begin{figure}[htb]
\includegraphics[width=\columnwidth]{Fig01}
\caption{Level repulsion parameter $\beta$ (a) and average ratio of spacings between consecutive levels $\langle \tilde{r}\rangle$ as a function of $\xi$. Four system sizes are considered, $L=512, 1024, 2048, 4096$, which are only distinguishable for very lager $\xi$ (hard to see for the chosen scales) (the size increases from bottom to top). Crosses indicate fitting curves: the linear functions are written in the main panel and inset of Fig.~\ref{fig:beta}~(a), while the function for Fig.~\ref{fig:beta}~(b) is given by Eq.~(\ref{Eq:sigmoid}). Averages over $100$ disorder realizations.}
\label{fig:beta}
\end{figure}

Figure~\ref{fig:beta}~(b) shows $\langle \tilde{r}\rangle $ as a function of $\xi$. The average ratio of level spacings goes from $\langle \tilde{r}\rangle \approx 0.39$ (Poisson) for very small $\xi$ to $\langle \tilde{r}\rangle \approx 1$ (picket-fence) for very large $\xi$. Four system sizes are shown and the curves fall on top of each other for most values of $\xi$, except for very larger $\xi$ (hard to see for the chosen scales), already in the picket-fence region. In this case, $\langle \tilde{r}\rangle \rightarrow 1$ as $L$ increases, as it should be, since the contributions from the edges of the spectrum, where the neighboring eigenvalues are not equidistant, become negligible. At the edges of the spectrum, according to Eq.~(\ref{Eq:E}), we have $r_{2} =5/3$ and $r_{L-1} =3/5$, while in the middle, $r_{(L+1)/2} =1$.

Contrary to $\beta$, we see in Fig.~\ref{fig:beta}~(b) that $\langle \tilde{r}\rangle$ is not a linear function of $\xi$. Its behavior is well fitted with the following function,
\begin{equation}
\langle \tilde{r}\rangle \approx \frac{\xi^{0.4} }{1 + \xi^{0.4}}
 \quad \text{for} \quad \xi > 1.
 \label{Eq:sigmoid}
\end{equation}
Taking Eq.~(\ref{Eq:linear}) into account, an expression very similar to Eq.~(\ref{Eq:sigmoid}) describes $\langle \tilde{r}\rangle$ {\em vs} $\beta$ for $1\leq \beta \leq 4$.

Notice that the values of $\xi$ that we obtain for each of the RMT level statistics using Eq.~(\ref{Eq:sigmoid}) are in the vicinity of what we get using Eq.~(\ref{Eq:linear}), but they do not agree exactly. For the GOE case, $\langle \tilde{r} \rangle_{GOE} = 0.54$ leads to $\xi \sim 1.5$, instead of $\xi \approx \beta =1$. The discrepancy increases as the correlations get stronger. For $\langle \tilde{r} \rangle_{GSE} = 0.68$, we reach $\xi \sim 6.6$, instead of $\xi \approx \beta =4$. The disagreement may in part be due to numerical factors, such as the unfolding procedure and the relatively small chain sizes, but also to the fact that we are not dealing with full random matrices and level repulsion caused by symmetries, but instead with a realistic model with short-range couplings described by real and symmetric matrices.

\subsection{Structure of the States}
The degree of localization of the eigenstates has been studied with measures, such as the localization length~\cite{Casati1992} and entropic quantities like the so-called structural entropy~\cite{Varga1994}. Here, we provide a detailed analysis of the structure of eigenstates and contrast our results with those for full random matrices.

To analyze the eigenstates, we use the participation ratio,
\begin{equation}
P_{R}^{\alpha} = \frac{1}{\sum_{n=1}^L|C^{(n)}_{\alpha}|^4 },
\end{equation}
which quantifies the number of basis vectors $|\phi_n \rangle $ that contribute to the eigenstate $|\psi_\alpha \rangle$. According to Table~\ref{Tab:1}, for full random matrices, in the region $1\leq \beta \leq 4$, we have approximately
\begin{equation}
\frac{\langle {P_{R}^\alpha}^{RMT}\rangle}{L}  \sim \frac{\sqrt{\beta} }{3}.
\label{Eq:prRMT}
\end{equation}

In Fig.~\ref{fig:PR}~(a), we report the average value of $P_{R}^{\alpha}$ for the eigenstates $|\psi_{\alpha}\rangle$ of the Anderson model at the middle of  the spectrum as a function of $\xi$.
The results make it evident that a relationship exists between the degree of delocalization of the eigenstates and the degree of level repulsion (recall that $\xi \approx \beta$), but it also exposes the limits of the analogy between the properties of full random matrices and of the finite 1D Anderson model. For the Anderson model, in the region $1\leq \beta \leq 4$, we actually find
\begin{equation}
\frac{\langle P_{R}^{\alpha} \rangle}{L}  \sim \frac{0.67 \beta}{0.56 + \beta},
\label{eq:PRvsLRM}
\end{equation}
instead of something closer to Eq.~(\ref{Eq:prRMT}). Indeed, the circles in the figure, which indicate the values from RMT,  do not match exactly the values of $\langle P_{R}^{\alpha} \rangle/L$ for $\beta=1$, 2, and 4.

The differences between the realistic model and RMT are not only restricted to the average values of the participation ratio, but extends also to their energy dependence. Contrary to what we find in RMT, where all eigenstates are random vectors, so $\langle P_{R}^{\alpha} \rangle$ {\em vs} $\langle E_{\alpha} \rangle$ is a flat curve, for the Anderson model, the structure of the eigenstates depends on the energy. As it is known, they are more extended close to the middle of the spectrum than at the edges.

\begin{figure}[htb]
\includegraphics[width=\columnwidth]{Fig02}
\caption{Scaled average participation ratio for eigenstates in the middle of the spectrum (a) and for basis vectors (b) as a function of $\xi$. System sizes are (from top to bottom) $L=256,\,512,\,1024,\,2048$. Circles in (a) are the RMT predictions for $\beta=1,\,2,\,4$, while crosses (orange) represent Eq.~\eqref{eq:PRvsLRM}.  Circles in (b) are obtained from Eq.~\eqref{eq:PRvsLAM} with $\kappa=1/3$. For the averages we collected $10^4$ statistical data obtained from disorder realizations and states around to the middle of the spectrum.}
\label{fig:PR}
\end{figure}

In Fig.~\ref{fig:PR}~(b), we depict the degree of delocalization of the basis vectors $|\phi_n\rangle$ projected into the energy eigenbasis, that is $\langle P_{R}^{n} \rangle/L$ {\em vs} $\xi$. As we will see in the next section,  $|\phi_n\rangle$ are the initial states that we consider for the analysis of the dynamics. Interestingly, the dependence of this participation ratio on $\beta$ (recall that $\xi \approx \beta$) is similar to that for RMT, apart from a factor $\kappa$ that varies with system size. We find that in the region where $1\leq \beta \leq 4$ (but not beyond that),
\begin{equation}
\frac{\langle P_{R}^{n}\rangle}{L}  \sim \kappa \frac{\sqrt{\beta} }{3}.
\label{eq:PRvsLAM}
\end{equation}
For $L=1024$, we have $\kappa \sim 1/3$, and this factor decreases as $L$ increases, suggesting that the basis vectors written in the energy eigenbasis must be multifractal~\cite{Schreiber1991}. This separation between the curves in Fig.~\ref{fig:PR}~(b) contrasts with Fig.~\ref{fig:PR}~(a), where the curves for $\langle P_{R}^{\alpha} \rangle/L$ for different chain sizes and $\xi > 10^{-2}$ fall on top of each other. In the case of $\langle P_{R}^{\alpha} \rangle$, the curves do not overlap only in the insulating phase, where the eigenstates become localized.

In Fig.~\ref{fig:PR} the fluctuations in the values of $P_{R}/L$ are small, the largest dispersions being $3\%$ of the average $\langle P_{R}\rangle/L$.

After the characterization of the model in terms of level statistics and of the structure of the eigenstates, we now proceed with the analysis of the dynamics.

\section{Time Evolution}

To study the time evolution of the model, we assume that the particle is initially on the site $n_0$, so $ | \Psi(0) \rangle=|\phi_{n_0}\rangle$. This state, of energy $\epsilon_{n_0}$, evolves as
$|\Psi(t)\rangle = e^{-iHt} | \Psi(0) \rangle$.

\subsection{Survival Probability}

We start by analyzing the survival probability in time, which is the probability to find the particle in its initial position at time $t$. It is given by
\begin{eqnarray}
\label{Eq:SP}
S_{P_{n_0}} (t)&=&\left| \langle \Psi(0) | e^{-iHt} | \Psi(0) \rangle \right|^2 \nonumber \\
&=& \left| \sum_{\alpha}  \left|C_\alpha^{(n_0)}\right|^2 e^{-iE_\alpha t} \right|^2  \nonumber \\
&=& \left| \int_{E_{low}}^{E_{up}} \rho_{n_0}(E) e^{-iEt} dE \right|^2  .
\end{eqnarray}
Above,
\begin{equation}
\rho_{n_0}(E)=\sum_{\alpha} \left| C_\alpha^{(n_0)} \right|^2 \delta(E-E_{\alpha})
\label{Eq:LDOS}
\end{equation}
is the energy distribution of the initial state, which is also known as local density of states (LDOS) or strength function, and
$E_{low}$ and $E_{up}$ are the energy bounds of the LDOS (in our case, $E_{low}=-2-W/2$ and $E_{up}=2+W/2$). The variance $\sigma_{n_0}^2$ of the LDOS depends on the number of states directly coupled with the initial state,
\begin{eqnarray}
\sigma_{n_0}^2 &=&  \langle \Psi(0) |H^2| \Psi(0) \rangle  -   \langle \Psi(0) |H| \Psi(0) \rangle ^2 \nonumber \\
&=&\sum_{n \neq n_0} |  \langle \phi_n |H| \phi_{n_0} \rangle |^2 .
\label{eq:Gamma}
\end{eqnarray}
Since the chain has open boundaries, $| \Psi(0) \rangle$ is directly coupled with $2$ states when $n_0\neq 1, L$ and with $1$ state when $n_0= 1, L$.
By knowing the LDOS in detail, we should be able to determine the behavior of the survival probability, since the later is the square of the Fourier transform of the former.

\subsubsection{Local Density of States and the Survival Probability for $W=0$}

To get insight on the behavior of the survival probability, we study first the LDOS  for the clean model ($W=0$). The shape of the LDOS depends on the site $n_0$, where we initially place the particle, as
\begin{equation}
\rho_{n_0}(E)= \frac{2}{L+1}  \sum_{\alpha=1}^L \sin^2 \left( \frac{\alpha n_0 \pi}{L+1} \right) \delta(E-E_{\alpha}) .
\label{Eq:rhono}
\end{equation}
The sum above can be solved exactly (see Appendix~\ref{sec:derivation}), which gives the LDOS in the following closed form,
\begin{equation}
\rho_{n_0}(E)=\frac{1-T_{2n_0}\left(\frac{E}{2}\right)}{\pi\sqrt{4-E^2}} ,
\label{eq:LDOSAn}
\end{equation}
where $T_{2n_0}$ are the Chebyshev polynomials of the first kind.

For $| \Psi(0) \rangle = |\phi_1\rangle $, in particular, the shape of the LDOS is semicircular,
\begin{equation}
\rho_{1}(E)=\frac{1-T_{2}\left(\frac{E}{2}\right)}{\pi\sqrt{4-E^2}}=\frac{\sqrt{4 - E^2}}{2 \pi } .
\end{equation}
which is the same shape of the LDOS that we obtain for states that evolve under full random matrices~\cite{Torres2014PRA,Torres2014NJP,Torres2014PRAb}. Some examples for other sites $n_0$, also derived from Eq.~\eqref{eq:LDOSAn}, are listed below,
\begin{eqnarray}
\rho_{2}(E)&=& E^2 \rho_1(E) ,\nonumber \\
\rho_{3}(E)&=& (E^2-1)^2 \rho_1(E) ,\nonumber \\
\rho_{4}(E)&=& E^2(E^2-2)^2 \rho_1(E) .\nonumber
\end{eqnarray}
As we gradually move the particle from $n_0=1$ to the center of the chain, the number of peaks in the LDOS increases, this number actually coincides with $n_0$. The peaks get more prominent and concentrated at the edges of the spectrum. Once the particle is in the middle of the chain, $n_0 = (L+1)/2$, all overlaps between the initial state and the energy eigenbasis are equal,
\begin{eqnarray}
&&\rho_{(L+1)/2}(E) = \frac{2}{L+1}  \sum_{\alpha=1}^L \sin^2 \left( \frac{\alpha  \pi}{2} \right) \delta(E-E_{\alpha}) \nonumber \\
&&= \frac{1}{L+1}  \sum_{\alpha=1}^L  \delta(E-E_{\alpha})
=\frac{1}{\pi \sqrt{4 - E^2}},
\label{Eq:recover}
\end{eqnarray}
so we recover the U-shape obtained for the DOS, as given by  Eq.~(\ref{Eq:preDOS}) and Eq.~(\ref{Eq:DOS}). This result also follows from Eq.~\eqref{eq:LDOSAn} for $L\gg1$.

Since the shape of the LDOS determines the decay of the survival probability via the Fourier transform in Eq.~(\ref{Eq:SP}), using Eq.~\eqref{eq:LDOSAn} (see Appendix~\ref{sec:derivation}), we also obtain a closed form for the survival probability,
\begin{equation}
S_{P_{n_0}}(t)=\left[{\cal J}_0 (2 t)-(-1)^{n_0}{\cal J}_{2n_0} (2 t)\right]^2.
\label{eq:SPno}
\end{equation}
The equations for some particular values of $n_0$ are given below,
\begin{eqnarray}
&& S_{P_1} (t) =\frac{[{\cal J}_1 (2  t)]^2 }{ t^2}, \nonumber \\
&& S_{P_2} (t) =  \frac{4[{\cal J}_2 (2 t) - 2 t {\cal J}_3(2t)]^2 }{ t^4} , \nonumber \\
&& S_{P_3} (t) = \frac{9 \left[ \left(3 t^3-20 t\right) {\cal J}_1(2 t)-4 \left(3 t^2-10\right) {\cal J}_2(2 t)\right]^2}{t^8} , \nonumber \\
&&\vdots  \nonumber \\
&&S_{P_{(L+1)/2}} (t) = [{\cal J}_0 (2  t)]^2.
\label{Eq:Bessel}
\end{eqnarray}

\subsubsection{Power-law Decay of the Survival Probability for $W\neq 0$}

Remarkably, when $W\neq 0$, the shapes of the LDOS remain very similar to those obtained for $W=0$ for values of $\xi \approx \beta \geq 1$, that is, for cases where the eigenvalues are strongly correlated and the localization length is larger than the system size. This is confirmed in the left panels of Fig.~\ref{fig:SP}, where the shapes obtained numerically with $W\neq0$ closely agree with those obtained from Eq.~(\ref{eq:LDOSAn}). In this figure, from the top  to the bottom panel, the initial position of the particle goes from site $n_0=1$ to site $n_0=L/2$. The results are shown for eigenvalues that have statistics close to GSE, but similar results are found for GOE and GUE.

The right panels of Fig.~\ref{fig:SP} display numerical results for the survival probability for $W \neq 0$ (black lines). For comparison, we also show the analytical expressions in Eq.~(\ref{Eq:Bessel}), obtained for $W=0$ (orange lines). The analytical curves are shown only up to $t=300$, since they go to zero for larger times, while numerical results saturate (see discussion about the saturation in the next subsection). Because the survival probability is not self-averaging~\cite{SchiulazARXIV2019}, in order to clearly reveal all of its dynamical features, we perform averages over ensembles of $10^3$ disorder realizations. The agreement between the analytical and numerical results is excellent. This confirms that the number of disorder realizations considered in the time interval $[0,300]$ is enough.
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{Fig03a}\\
\includegraphics[width=\columnwidth]{Fig03b}\\
\includegraphics[width=\columnwidth]{Fig03c}
\caption{Local density of states (left) and survival probability (right) for GSE-like statistics;  $n_0=1,2,3,8,20,L/2$ from top to bottom. Right panels: Dashed (Blue) and dot-dashed (green) lines correspond to the asymptotic decays $t^{-3}$ and $t^{-1}$, respectively. Horizontal double dotted-dashed (red) lines correspond to the infinite-time average.  In addition to the numerical curves for $W\neq0$, we show with orange lines the corresponding analytical expressions (\ref{Eq:Bessel}) for $W=0$. All panels are for $L=16\,384$; averages over $10^3$ disorder realizations; and $\xi=6.31$, which implies $\langle \tilde{r}\rangle \sim \langle \tilde{r} \rangle_{GSE}$.}
\label{fig:SP}
\end{figure}

The behavior of $|\Psi(0)\rangle = |\phi_1 \rangle$ is rather counterintuitive. Only at very short times, the decay of $S_{P_1}(t)$ is slightly slower than for the other initial states, since the particle is attached to the border.
But for $t \gg 1$, the asymptotic behavior of $S_{P_1} (t)$ is $\propto t^{-3}$, as seen in Fig.~\ref{fig:SP}~(b). This power-law decay is what we have for evolutions under full random matrices~\cite{TorresKollmar2015,Tavora2016,Tavora2017}, where the LDOS is also semicircular. The evolution under full random matrices corresponds to the fastest dynamics that one can have for many-body systems with a unimodal LDOS~\cite{Torres2014PRAb}. It is  surprising to find the same behavior in a system with a single particle, and not only for $W=0$, but also in the presence of onsite disorder.

As we change the initial position of the particle, placing it further from the edge, the $t^{-3}$ behavior gets postponed to later times, while the initial decay is substituted by the power-law $t^{-1}$, as seen in the right panels of Fig.~\ref{fig:SP}, from top to bottom. When $n_0$ is closer to the middle of the chain [Fig.~\ref{fig:SP}~(l)], only the $t^{-1}$ decay is seen~\cite{Arias1998,Santos2016}, which is the asymptotic behavior of $[{\cal J}_0 (2  t)]^2$ in Eq.~(\ref{Eq:Bessel}).

The power-law exponent of the algebraic decay of $S_{P}(t)$ can be derived from the bounds of the LDOS as follows~\cite{Erdlyi1956,Urbanowski2009,Nowakowski2008,Tavora2016,Tavora2017}. If $\rho_{n_0}(E)$ decays abruptly close to one of the energy bounds, such that
$\rho_{n_0}(E)=(E-E_\text{low})^{\nu}\eta(E)$, for $0<\nu<1$ and $\lim_{E\to E_\text{low}}\eta(E)>0$, then $S_{P}(t)\propto t^{-2(\nu+1)}$. This explains the $t^{-3}$ decay, since for the semicircular LDOS, $\nu=1/2$. For the case where $n_0$ is away from the edges of the chain and the LDOS has a U-shape, we actually have $\nu = -1/2$, which is outside the range of values for $\nu$ above, but the relationship still holds, leading to the $t^{-1}$ decay.

\subsubsection{Correlation Hole}

The power-law decay is followed by the correlation hole, which is a dip below the saturation value of the survival probability. The saturation value is indicated with a horizontal double dotted-dashed line on the right panels of Fig.~\ref{fig:SP}. It corresponds to the infinite-time average of the survival probability,
\[
\overline{S_{P}}_{n_0} = \sum_{\alpha=1}^{L} |C_{\alpha}^{(n_0)}|^4 .
\]
The correlation hole gets below this point and develops only when the eigenvalues are correlated. It is nonexistent when the eigenvalues are uncorrelated and the $P(s)$ is Poissonian.

As it is evident from Fig.~\ref{fig:SP}, the hole is better visible when the decay is entirely $\propto t^{-3}$ than when the decay is $\propto t^{-1}$, because in the former case, $S_{P_1} (t)$ has enough time to reach very low values.
One sees from the top panels that the survival probability is already below $\overline{S_{P}}_{n_0} $ for $t\gtrsim10$, which is not a very long time. This suggests that the correlation hole might be detectable experimentally even in platforms without exceedingly long coherence times. The same features observed in the figure are present also for $L\sim 100$, which are sizes within the reach of experiments with ion traps.

The correlation hole becomes deeper as the spectrum becomes more rigid (larger $\xi, \beta$), as seen in Fig.~\ref{fig:Hole}, where we go from Poisson and GOE-like [Fig.~\ref{fig:Hole}~(a)] to GUE-like [Fig.~\ref{fig:Hole}~(b)], GSE-like [Fig.~\ref{fig:Hole}~(c)], and picket-fence [Fig.~\ref{fig:Hole}~(d)]. In this figure, $n_0=L/2$. Notice that the hole is not exclusive to spectra that show statistics similar to those of RMT, but develops whenever correlations are present. The strongest correlations for the tight-binding model in Eq.~(\ref{Eq:Anderson}) happen for $W = 0$, when the eigenvalues away from the borders of the spectrum become nearly equidistant and the spectrum is of the picket-fence type. The hole is therefore deeper in Fig.~\ref{fig:Hole}~(d).

\begin{figure}[htb]
\includegraphics[width=\columnwidth]{Fig04}
\caption{Survival probability for initial states with $\epsilon_{n_0}\approx 0$ for Poissonian (turquoise) and GOE-like nearest-neighbor energy-level spacing distributions (a), GUE-like (b), GSE-like (c) and the picket-fence spectrum with $W\approx 0$ ($\xi=1000$)  (d).  Dotted-dashed lines indicate the $t^{-1}$ decay. Horizontal double dotted-dashed lines correspond to the infinite-time average. The insets in (a), (b) and (c) zoom in the correlation hole for the GOE-, GUE-, and GSE-like statistics. $L=16\,384$; averages over initial states and disorder realizations that give a total of $10^4$ data. An additional average is done over small windows of time to further reduce fluctuations.}
\label{fig:Hole}
\end{figure}

Figure~\ref{fig:Hole} also makes evident that the behavior of the survival probability before the correlation hole is independent of the level statistics. This observation has been made also in studies of nonequilibrium many-body quantum dynamics~\cite{Torres2014PRA,Torres2016Entropy}. In Fig.~\ref{fig:Hole}, since $n_0=L/2$ and the LDOS has the U-shape,  the same power-law decay $\propto t^{-1}$ emerges for all level spacing distributions, even for the Poissonian $P(s)$.  To capture correlations among the eigenvalues, times beyond the power-law decay need to be reached for the dynamics to resolve the discreteness of the spectrum~\cite{Schiulaz2019}.

After the correlation hole, the dynamics finally saturates. Notice that in this region of equilibration, where the survival probability fluctuates around $\overline{S_{P}}_{n_0}$, one sees isolated peaks. They are visible in Fig.~\ref{fig:SP}, in the insets of Fig.~\ref{fig:Hole}, and specially in Fig.~\ref{fig:Hole} (d). They happen at times that are integer multiples of the system size $L$, when the particle reaches the edges of the chain and partial revivals take place.  Recurrences of this kind have been discussed in the context of a 1D quantum Ising chain at criticality~\cite{CamposVenuti2010}.  Experimental observations of recurrences in small quantum systems were verified in~\cite{Rempe1987,Greiner2002}.

\subsection{Particle Spread on the Lattice}

To get a better picture of why the dynamics is faster when the particle is initially on the edge of the chain than when $n_0$ is away from the edges, we study how the probability to find the particle on any site spreads in time. This is shown with density plots in Fig.~\ref{fig:2D} for $n_0=1$ (a), $n_0=2$ (b), and $n_0=L/2$ (c). The checkerboard pattern of the panels are reminiscent of the Bessel functions that characterize the dynamics [see Eqs.~\eqref{Eq:Bessel}].

\begin{figure}[htb]
\includegraphics[scale=0.35]{Figure05}
\caption{Spread in time of the probability to find the particle on any site. Three initial states are shown, particle on $n_0=1$ (a), particle on $n_0=2$ (b), and particle on $n_0=L/2$ (c). Parameters: $\beta\approx 4$, $L=1000$, and one random realization.}
\label{fig:2D}
\end{figure}

The motion of the particle for $|\Psi(0)\rangle = |\phi_1\rangle$ is very concentrated [Fig.~\ref{fig:2D}~(a)]. The particle moves away from the edge, hopping successively from site $n$ to site $n+1$, with very little spreading onto sites in the vicinity. For the case  $|\Psi(0)\rangle = |\phi_2\rangle$, the particle hops from site 2 to site 1 at $t\sim1$, and from there, its motion is again well directed [Fig.~\ref{fig:2D}~(b)], although there is more leaking than in Fig.~\ref{fig:2D}~(a). These behaviors contrast with that in Fig.~\ref{fig:2D}~(c), where the particle very soon spreads simultaneously through several sites in the vicinity of $n_0$ and has a significant chance to be brought back to its initial position even late in time. This is clearly not the case for Fig.~\ref{fig:2D}~(a). There, we see that once the particle detaches from the border, it is more difficult to bring it back to $n_0=1$. As a result, the particle moves away from $n_0=1$ faster than from $n_0=L/2$.

\section{Discussion}

All degrees of energy-level repulsion typical of full random matrices and even stronger (picket-fence like) can be obtained by varying the disorder strength and the system size of the 1D Anderson model, specifically by increasing $\xi \propto (W^2 L)^{-1}$ from very small to very large values. These different degrees of level correlations can be detected directly from the analysis of the spectrum or by studying the dynamics of the system. Both approaches were explored in this work.

Our analyses of the spectrum and also of the degree of delocalization of both the eigenstates and the initial states were done as a function of $\xi$. The parameter $\beta$, obtained from the level spacing distribution, depends linearly on $\xi$, while a less trivial relationship emerges for $\xi$ and the average ratio $\langle \tilde{r}\rangle $ between neighboring level spacings. The latter is a recently introduced quantity that has been gaining popularity in analysis of level repulsion.

Correlated eigenvalues affect the dynamics of quantum systems by creating the so-called correlation hole. As we show, the correlation hole reflects all the different degrees of energy-level repulsion achieved by the 1D Anderson model, with stronger correlations leading to deeper holes. The correlation hole has been the subject of recent studies associated with many-body quantum systems in high energy physics~\cite{Cotler2017GUE,Gharibyan2018,Nosaka2018} and condensed matter physics~\cite{Torres2017Philo,Torres2018,Schiulaz2019,TorresHerrera2019}. However, for many-body quantum systems with local short-range couplings, the hole emerges at exceedingly long times~\cite{Schiulaz2019}, which prevents it from being observed experimentally. In this work, we showed that the correlation hole in the 1D Anderson model happens at times that are experimentally accessible.

To investigate the dynamics, the particle was initially placed on a single site. When this is the first site of the chain, or a site very close to the edges, the survival probability shows a power-law decay $\propto t^{-3}$, which is the same behavior one encounters when dealing with full random matrices. This fast decay allows for the emergence of the correlation hole at times that are not very large, being thus reachable by current experiments. In contrast, when the particle is initially close to the middle of the chain, the survival probability decays as $t^{-1}$ and the correlation hole takes longer to become visible.

The difference in the value of the power-law exponent for the survival probability decay is, of course, nonexistent in a chain with periodic boundaries, where the only power-law behavior observed is $\propto t^{-1}$. Yet, experimental setups usually deal with open chains. This border effect can be explored to better transfer particles and to detect dynamical features typical of full random matrices, such as the power-law decay $\propto t^{-3}$ and a deep correlation hole.

\begin{acknowledgments}
E.J.T.-H. and J.A.M.-B. acknowledge funding from VIEP-BUAP (Grant Nos.~MEBJ-EXC19-G, LUAG-EXC19-G, and CELU-EXC19-I),
Mexico. They are also grateful to LNS-BUAP for allowing the use of the supercomputing facility.
L.F.S. was supported by the NSF Grant No.~DMR-1603418.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Tag: motorcycle
Valentino Rossi does not rule out competing in the WEC in 2022
December 6, 2021 December 6, 2021 Lou Modestino
No one is surprised that after announcing his retirement from MotoGP at 42, the Italian Valentino Rossi continue competing in motorsports. In fact, whenever his schedule allowed, he has participated in different categories. The big difference is that in 2022 you will have enough time to do it consistently.
Lou Modestino's Track Talk, Motorcycle Racing, Road Racingracing, 2022, Lou, Road, motorcycle, track, modestino's, talk, the, out, does, not, competing, wec, rossi, rule, valentinoLeave a comment
Racing on TV Listings for October 7-10
October 7, 2021 October 7, 2021 Lou Modestino
Here's where to find NASCAR Cup, Xfinity, NHRA, F1 and more on TV this weekend.
NASCAR, Lou Modestino's Track Talk, Cup Series, XFINITY Series, Motorcycle Racing, Flat Track Racing, Formula One, Road Racing, IMSA/Sports Cars, NHRA Drag Racing, Stock Car Racingnascar, cup, xfinity, series, racing, Cars, Stock, car, Lou, Road, motorcycle, listings, track, modestino's, talk, for, imsa/sports, one, formula, nhra, drag, october, 7-10, flatLeave a comment
Racing on TV Listings for September 27-October 3
September 28, 2021 Lou Modestino
Here's where to find NASCAR Cup, Xfinity and Truck Series Playoffs at Talladega and more motorsports TV listings.. Listings compiled by Mike Pryson of Auto Week…
NASCAR, Lou Modestino's Track Talk, ARCA, Cup Series, XFINITY Series, Motorcycle Racing, Camping World Truck Series, Whelen Modified Tour, Road Racing, NHRA Drag Racing, Sebring, Talladega, Texas, Watkins Glennascar, truck, whelen, cup, arca, xfinity, series, racing, Talladega, Sebring, texas, Glen, Lou, Road, modified, tour, motorcycle, listings, camping, world, track, modestino's, talk, for, nhra, drag, september, watkins, 27-octoberLeave a comment
Racing on TV Listings for September 14-19
Here's where to find the major racing action on TV and online this week:
NASCAR, Lou Modestino's Track Talk, ARCA, Cup Series, XFINITY Series, Motorcycle Racing, Whelen Modified Tour, Road Racing, IMSA/Sports Cars, Indy Cars, NHRA Drag Racingnascar, whelen, cup, arca, xfinity, series, racing, Cars, indy, Lou, Road, modified, tour, motorcycle, listings, track, modestino's, talk, for, imsa/sports, nhra, drag, september, 14-19Leave a comment
Racing on TV Listings for August 25-29
August 25, 2021 Lou Modestino
Mike Pryson of Auto Week has collected TV listings for NASCAR, Extreme E, return of Formula 1 and more for you…
NASCAR, Lou Modestino's Track Talk, Cup Series, XFINITY Series, Motorcycle Racing, Camping World Truck Series, Formula One, Road Racing, IMSA/Sports Cars, NHRA Drag Racing, Stock Car Racingnascar, truck, cup, xfinity, series, racing, Cars, Stock, car, Lou, Road, motorcycle, listings, camping, world, track, modestino's, talk, for, imsa/sports, one, formula, nhra, drag, august, 25-29Leave a comment
AMA Motorcycle Hall of Fame Offers Condolences Following Death of Yvon Duhamel
PICKERINGTON, Ohio — The AMA Motorcycle Hall of Fame offers its condolences to the family, friends and fans of the outgoing and talented Yvon Duhamel, a member of the Hall of Fame Class of 1999 known for riding the temperamental and brutally fast Kawasaki H2R in AMA road race nationals in the 1970s.
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Racing on TV Listings for August 9-15
August 10, 2021 August 10, 2021 Lou Modestino
The latest schedule for TV motorsports, including NASCAR, NHRA, Indycar, ARCA, Whelen Modifieds, Formula E, motocross, superbike, and more can be found here…
NASCAR, Lou Modestino's Track Talk, ARCA, Cup Series, XFINITY Series, Motorcycle Racing, Camping World Truck Series, Whelen Modified Tour, Road Racing, IMSA/Sports Cars, Indy Cars, NHRA Drag Racing, Indianapolis, Kansas, Lime Rock Park, Nashville, New Hampshire, Stafford Speedwaynascar, truck, whelen, cup, arca, xfinity, series, racing, speedway, Cars, Nashville, motorsports, Indianapolis, indycar, indy, Stafford, new, Lou, Road, Motocross, modified, Formula E, tour, Rock, TV listings, motorcycle, listings, camping, world, track, modestino's, talk, for, hampshire, park, lime, imsa/sports, nhra, drag, august, 9-15, kansas, Whelen Modifieds, superbike, television listingsLeave a comment
SRX Teams Up with Progressive for Inaugural Season
May 11, 2021 Mike
MOORESVILLE, N.C. (May 11, 2021) – Superstar Racing Experience (SRX), the six-race short-track series airing this summer on Saturday nights in primetime on the CBS Television Network and streaming live on Paramount+, announced today that Progressive, the third-largest auto insurer in the country, has become the series' exclusive car and home insurance sponsor. Progressive is also a leading seller of motorcycle and commercial auto insurance, and one of the top homeowners insurance carriers.
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New Hampshire Motor Speedway Waves The Green Flag For A Season Full Of Racing
May 11, 2021 Lou Modestino
New Hampshire Motor Speedway (NHMS) is gearing up for an exciting and action-packed racing season that includes Motorcycle Week at NHMS featuring the 98th Annual Loudon Classic, a two-day NASCAR race weekend highlighted by the NASCAR Cup Series (NCS) Foxwoods Resort Casino 301, plus the Granite State Legends Cars Road Course Series (RCS) and many other racing clubs, driving experiences and more throughout the season.
NASCAR, Lou Modestino's Track Talk, Cup Series, Modifieds, XFINITY Series, Motorcycle Racing, Whelen Modified Tour, Road Racing, Stock Car Racing, New Hampshirenascar, website, cup series, win, whelen, cup, champion, xfinity, series, racing, international, speedway, Chase Elliott, Brad Keselowski, Ryan Preece, Cars, United States, WELCOME, Lakes, NASCAR Xfinity Series, Xfinity Series, Kevin Harvick, Events, motor, Keselowski, motorsports, free, winner, New Hampshire Motor Speedway, race, Harvick, 's, Weekend, NASCAR Cup Series, Twitter, Facebook, NASCAR Whelen Modified Tour, David, mobile app, top, 've, kevin, driver, NASCAR Cup, president, car, 're, Small, Ryan, Said, Chase, Elliott, american, 2021, nascar xfinity, championship, Champion Chase Elliott, Our Motorsports, Harley-Davidson, at, new, Road, 2021 schedule, modifieds, Loudon, road racing, legend cars, legends, Day, modified, New England, New Hampshire, schedule, fans, Brad, tour, Will, NHMS, April, Preece, road course, lineup, of, Laconia Motorcycle Week, Davidson, Laconia, David McGrath, motorcycle, harley, Foxwoods Resort Casino 301, ticketsLeave a comment
A History of Flat Track Racing
May 6, 2021 May 6, 2021 Mike
Flat track racing is an American motorsport that gained popularity shortly after the end of WWI. It was a craze during the 1920s and 1930s where it stole the majority of the audience that was already interested in horse racing. It was much similar to board track racing that has been covered extensively in our other entries.
Flat Track RacingHarley-Davidson, Honda, slide job, Kawasaki, motorcycle, motorcycles, suzuki, triumph, harley, flat track, flat track racing, ktm, indianLeave a comment

redpajama_set_name: RedPajamaCommonCrawl


SODA for Python is a Python API that implements Simple Oracle Document Access (SODA). It is part of the Oracle Python driver, cx_Oracle — no additional installation is needed.
You can use SODA for Python to perform create, read (retrieve), update, and delete (CRUD) operations on documents of any kind, and you can use it to query JSON documents.
SODA is a set of NoSQL-style APIs that let you create and store collections of documents (in particular JSON) in Oracle Database, retrieve them, and query them, without needing to know Structured Query Language (SQL) or how the documents are stored in the database.


Real New Yorkers Swipe it Forward!Wednesday morning, join Access-A-Ride Reform Group (AARRG!!!) and The People's MTA for a protest, sit-in and people's speakout at the MTA's monthly board meeting.Tell the MTA board you're sick of no elevators, too few buses and subway trains, racist policing and fare hikes!
Last week the NY state comptroller let the cat out of the bag: "unplanned" fare hikes are in our future, due to the increased money the MTA is spending.But where is that money going? It's not being used to make stations accessible. The hundred-million-dollar makeovers that are taking stations out of service don't even include new elevators. No, that money is going to line the pockets of banks for debt service, or others who are profiting off the MTA's expansion in gentrified areas.
As the MTA promotes gentrification, the racist police crackdown intensifies in the boroughs and in the subways. Eighty percent of 2017's 15,000 arrests for "fare evasion" are of African American or Latinx riders.
Somehow more money is being spent — all while buses have been cut, seats taken out of trains, service is deteriorating and more cuts of transit workers and station agents are planned. Wall Street is looting public transportation!
Come to the MTA Board meeting and tell the rich, unrepresentative board members that public transportation belongs to the people! If they are only capable of running the subway for Wall Street, we'll take it over and run it ourselves!


text:
package com.chinomars.bccAndroidViewer;

//import java.security.acl.AclNotFoundException;
//import com.chinomars.bccAndroidViewer.R;

import android.app.Activity;
import android.content.Intent;
import android.os.Bundle;
import android.view.Window;
/**
 * Created by Chino on 3/8/16.
 */
public class SplashScreen extends Activity {
    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        requestWindowFeature(Window.FEATURE_NO_TITLE);
        setContentView(R.layout.splash);

        Thread splashTread = new Thread() {
            @Override
            public void run() {
                try {
                    int waited = 0;
                    while(waited < 5000) {
                        sleep(100);
                        waited += 100;
                    }
                } catch(InterruptedException e) {
                    // do nothing
                } finally {
                    startActivity(new Intent("com.chinomars.bccAndroidViewer.BccController"));
                    finish();
                }
            }
        };
        splashTread.start();
    }

}

redpajama_set_name: RedPajamaGithub


The mission of Visit Albuquerque is to stimulate economic growth by marketing Albuquerque as a convention and visitor destination.
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Selling and marketing Albuquerque as a destination.
Focused on conventions, meetings, leisure travelers and travel professionals.
Funded by lodgers tax through a contract with the City of Albuquerque.
Supported by a partnership of nearly 900 organizations.
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Funded with ~$7M (~$6M from lodgers tax and ~$1M from private sector activities).
We are goal-oriented and results-directed.
We look for opportunities, and we build on them.
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Dr. D. W. McKee, General Electric Company (1987)
Dr. I. C. Lewis, UCAR Carbon Company (1986)
Dr. R. T. K. Baker, EXXON Corporation (1985)
Dr. J. L. White, The Aerospace Corporation (1984)
Dr. L. S. Singer, Union Carbide Corporation (1983)
Prof. M. S. Dresselhaus, Massachusetts Institute of Technology (1982)
Prof. D. B Fischbach, University of Washington (1981)
Prof. P. L. Walker, Jr., The Pennsylvania State University (1980)
Prof. S. Mrozowski, State University of New York at Buffalo (1979)
Prof. R. J. Diefendorf, Rensselaer Polytechnic Institute (1978)
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Emirates SkyCargo successfully transports Brazilian satellite
Wednesday, December 30th, 2020 5:23pm
Emirates SkyCargo, the freight division of Emirates airline has successfully executed a cargo charter to transport the Brazilian satellite Amazonia-1 from Sao Jose dos Campos, Brazil to Chennai, India.
This is the first time that Emirates SkyCargo has transported a space satellite from South America.
Amazonia-1 is the first satellite to have been developed completely in Brazil by the National Institute for Space Research (INPE), Brazil's apex body dedicated for space research and exploration.
The satellite took eight years to be developed and once launched into space, will help monitor the ecosystem of the Amazon rainforest, the world's largest tropical rainforest.
The satellite is due to be sent to space in February 2021 from the Satish Dhawan Space Centre on the East coast of India.
Emirates SkyCargo had to undertake extensive preparations for this charter flight to ensure the safe and secure transport of the satellite. The air cargo carrier had to apply for special permission to operate a Boeing 777 freighter from Sao Jose dos Campos airport to uplift the satellite.
Emirates SkyCargo together with the INPE, the airport and local partners conducted two comprehensive simulation sessions ahead of the transport date to be able to transport the sensitive cargo safely from the airport.
During the transportation process, the Amazonia-1 satellite was dismantled into multiple components to facilitate easy loading and unloading from the aircraft. The satellite components were packed inside large containers to avoid any damage during the transport.
Overall, the total cargo weighed around 22 tonnes. Four members of the team from INPE also travelled with the satellite to continuously monitor the status of the cargo during the flight from Sao Jose dos Campos to Dubai and then onwards to Chennai, India.
Emirates SkyCargo has expertise in transporting satellites and other cargo developed for space technology. In 2018, the carrier transported Khalifasat, the first satellite developed completely by Emirati engineers at the Mohammed Bin Rashid Space Centre from Dubai to Seoul.
In Brazil, Emirates SkyCargo operates scheduled cargo flights to Sao Paulo and Viracopos airports, allowing exporters and importers from the country to connect with their global customers.
Etihad to restart flights to Doha
Etihad Airways has announced plans to restart passenger services between Abu Dhabi and Doha.
Britain's Burberry says COVID-19 closures drag Q3 sales down 9%
British luxury brand Burberry said underlying sales fell 9% in the three months ended December as the COVID-19 pandemic closed shops and meant fewer tourists visiting its European stores.
Etihad Cargo records 50% year-on-year rise in pharma shipments
Etihad Cargo has reported a 50 per cent year-on-year rise in pharma shipments under its specialized PharmaLife product.
Abu Dhabi Executive Council approves AED 40 billion project framework
The Abu Dhabi Executive Council has approved a framework between Abu Dhabi Government and Aldar Properties for the development of capital projects in the emirate worth AED 40 billion.
Dubai Consumer Confidence Index at highest level since 2017
Dubai's Consumer Confidence Index during the last quarter of 2020 reached its highest level in more than three years.
Combatting COVID-19: Dubai Municipality shuts five outlets
Five establishments have been shut down by Dubai Municipality for violating the COVID-19 precautionary measures.
H.H. Sheikh Hamdan launches Al Fahidi Fort restoration project
His Highness Sheikh Hamdan bin Mohammed bin Rashid Al Maktoum, Crown Prince of Dubai and Chairman of the Dubai Executive Council, has announced the launch of a project to renovate the historical Al Fahidi Fort, which is home to the Dubai Museum.
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text:
Charles Burnett III
Flamboyant multimillionaire businessman and philanthropist who set a land-speed record in 2009 and died in a helicopter crash
January 31 2018, 5:00pm, The Times
Charles Burnett III in 2009 after breaking the land-speed record for a steam-powered car with an average speed of 139.8mph in the vehicle InspirationGETTY IMAGES
Charles Burnett III did not really care what vehicle he drove as long as it was either fast or powerful. At Newtown Park, his country pile near Lymington, Hampshire, classic roadsters shared parking spaces with Chieftain tanks and other military vehicles, which Burnett would drive along the country lanes close to his estate — much to the surprise of passers-by.
Scion of the Garfield Weston family, one of the wealthiest in Canada, Burnett also kept a collection of historical aircraft, including a Harvard military trainer, which could take off from a Second World War airstrip that he had restored in the grounds of his home.
In this multimillionaire's playground Burnett would bring his "toys" out to play during lavish parties. He once staged re-enactments of…


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Q: vs-code terminal vs linux terminal shows different node --version I upgraded node with "nvm" in kali-linux system:
 nvm install node

This installs latest node version. nvm ls displays this
default -> v14.15.1
       v15.14.0
       v16.2.0
       v16.3.0
         system
      v16.3.0
node -> stable (-> v16.3.0) (default)
stable -> 16.3 (-> v16.3.0) (default)
iojs -> N/A (default)
unstable -> N/A (default)

I set the default
  nvm alias default v16.3.0

I close all terminals and this sets up v16.3.0 in terminal. However when I check on vs-code node --version I m still gettting "v14.15.1". I also run nvm alias default v16.3.0 on vs-code, restart the vs-code but it still shows "v14.15.1"

redpajama_set_name: RedPajamaStackExchange


As a youth-run organization, ACCESS encourages schools and youth organizations to get involved in raising awareness and raising funds for education projects in the developing world.
While offering Educational Presentations and Educator Resources to elementary and secondary schools, ACCESS also works to raise funds and increase involvement in the post-secondary level. Events from two Ontario university campuses are highlighted below, recognizing their efforts and encouraging future involvement from schools.
In the fall semester of 08-09, Wilfred Laurier University in Waterloo, Ontario hosted two events in support of ACCESS.
Drawing much excitement from students was the Throw Whip Cream at Your Profs event. On another day, there was a Starbucks Coffee Sale on campus, with all proceeds benefiting ACCESS. The funds raised have been donated directly towards ACCESS projects in the developing world.
These fundraisers were organized by XLerate Business Club, a student-driven organization focused on developing key skills that are critical for success both in university and the business world. It is Laurier's largest student organization, providing resources such as seminars and leadership opportunities.
The Residence Council at 89 Chestnut Residence, which houses University of Toronto and OCAD students among others, fundraised for ACCESS in the in the 08-09 academic year.
The funds were raised through a Floor Competition, with residents on 24 floors competing by donating coins into jars assigned to each floor. As per the competition's rules, paper bills deducted points when placed in another floor's jar - as an incentive to donate more. Details were posted online here.
The Residence Council also dedicated an additional fundraising event to ACCESS. For Valentine's Day, Candy Grams were sold and delivered to students within the residence in February 2009.
Looking for ways to fundraise for ACCESS this year?


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Name and shame solution
PUBLISHED: 12:38 12 October 2007 | UPDATED: 23:06 28 May 2010
A SURPRISE call to the newsroom from Councillor Martin Curtis to confirm that he is the elected representative temporarily deprived of his driving licence through the totting up procedure. Quick resolution, there, I felt of Brakespeare s dilemma of last w
A SURPRISE call to the newsroom from Councillor Martin Curtis to confirm that he is the elected representative temporarily deprived of his driving licence through the totting up procedure.
Quick resolution, there, I felt of Brakespeare's dilemma of last week about naming names.
Martin says the six-month ban, now happily ended, was not something he had sought to hide and wondered why Brakespeare's interest had been aroused.
Of course he knew the answer before posing the question. The information had come from 'friends' curious to step up interest in the Tory campaign to find a replacement for Malcolm Moss.
"I did wrong," says Cllr Curtis. "I really can't say anything else. It just astonishes me you didn't know about it."
His worst 'sin', he reckoned, was to be caught on the A1 at 1am doing 81 mph in a 70 mph limit.
"I'm thinking of adding to my blog- something like 10 things I have learned from over the past two years," he added cheerily.
And just to refresh my memory of his newly-discovered prowess for running, I should add he did this year's Great North Run in 1 hr, 57min, a full 20 minutes faster that his performance a year ago.
And at the end of the month he's off to Dublin to compete in the marathon there.


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2014 Texas Natality Narrative
2014 Natality Narrative
In 2003, HHS Secretary Tommy Thompson approved the revision to the U.S. Standard Certificate of Live Birth and encouraged all states to adopt it. The process involved in this revision, as well as details of what was revised, can be found at http://www.cdc.gov/nchs/nvss/vital_certificate_revisions.htm.
Texas adopted the new U.S. Standard Certificate of Live Birth in 2005. This revision includes changes to items such as onset of prenatal care, maternal smoking history, race/ethnicity, etc. For details regarding race/ethnicity computation for birth data, see Table 44.
Births and Birth Rates
There were 399,482 live births to Texas residents in 2014, an increase of 3.1 percent (12,372 more births) from 2013, and an increase of 4.5 percent (18,041 more births) from 2004.
(Figure 4 data)
From 2004 through 2014, the proportion of births to white* mothers increased from 39.7 percent of all births in 2004 to 41.1 percent in 2014. At the same time, the proportion of births to Hispanic mothers decreased from 49.2 percent of births in 2004 to 47.4 percent in 2014. The number of Hispanic births has exceeded the number of white* births since 1998. The proportion of births to black mothers increased from 11.0 percent in 2004 to 11.5 percent in 2014.
* Includes women of other and unknown race/ethnicity.
The 2013 crude birth rate of 14.6 births per 1,000 Texas residents is the lowest on record. The crude birth rate has dropped from 17.0 in 2004 to 14.8 in 2014. The rate for whites* dropped slightly from 12.6 in 2004 to 12.5 in 2014. The crude birth rate for blacks declined from 16.4 in 2004 to 14.8 in 2014. The Hispanic rate also declined from 23.9 to 17.7.
(Figure A data)
The general fertility rate for Texas, which is the number of live births to Texas residents per 1,000 women ages 15 through 44, was 70.5 in 2014. In 2004, it was 76.7 (Figure B). Like crude birth rates, race/ethnicity-specific general fertility rates have been highest among Hispanics and lowest among blacks (Figure B). In 2004, the general fertility rates of whites*, black, and Hispanics were 61.2, 68.0, and 100.1, respectively. In 2014, fertility rates for white* females increased to 65.8 while the rates for Black and Hispanic females decreased to 64.9 and 77.0, respectively.
(Figure B data)
Mother's Age and Father's Age
Around half (53.2 percent) of Texas resident live births in 2014 were to mothers 20 to 29 years of age, and around one-quarter (24.4 percent) were to mothers 30 to 34 years old (Table A). Mothers aged 10-17 accounted for 2.8 percent of the births in 2014, compared to 3.2 percent in 2013. The percentage of mothers aged 18 and 19 also decreased from 6.7 percent in 2013 to 6.1 percent in 2014. The percentage of births to mothers 30-34 increased slightly from 23.4 percent in 2013 to 24.4 percent in 2014 and births to mothers aged 35-39 likewise increased from 10.7 percent in 2013 to 11.0 percent in 2014.
Table A.1. Percent of Live Births** by Mother's Age Groups and Race/Ethnicity
Texas Residents, 2014
White*
4.8 9.6 12.0 8.8
12.8 9.7 9.8 11.0
** Denominator for percent excludes missing data.
Note: due to rounding, percents may not sum to 100%
Table A.2. Percent of Live Births for Selected Mother's Age Groups by Race/Ethnicity
Mothers' ages ranged from 11 to 60 years; the mean mother's age was 27.6 years. Among birth certificates that included information on the father's age (86.3 percent of all birth certificates), fathers ranged in age from 13 to 79. The mean father's age was 30.4 years.
Age-specific birth rates, calculated as the number of live births per 1,000 women in the specified age group, were highest among women aged 25-29, followed by women aged 20-24 for all races combined (Table B). Hispanic women aged 20-24, followed by Hispanic women aged 25-29 and black women aged 20-24 had the highest age-specific birth rates for individual race/ethnicities. White women were the only race/ethnicity whose highest age-specific birth rate occurred in the 25-29 age group. Age-specific birth rates for Hispanics were higher than rates for whites or blacks at every age group except 10-14 years and 30-34 years. Whites in the youngest three age groups had the lowest fertility; blacks had the lowest fertility in the remaining age groups.
Table B. Age-Specific Birth Rates1 by Race/Ethnicity
(Live Births Per 1,000 Women in the Race/Ethnicity and Age Group)
79.7 111.6 125.9 105
120.1 112.2 126.8 112
112.5 79.8 92.9 100.3
All Ages^
^ The general fertility rate. Numerator includes mothers of unknown ages; denominator is women 15-44 years of age.
Note: Rates were calculated using population data from the Texas State Data Center. Births of unknown race/ethnicity, and of racial/ethnic groups other than white, black or Hispanic were included with white for the purposes of calculating birth and fertility rates.
Overall, 58.1 percent of mothers reported being married. However, there were large differences in marriage rates across age groups and race/ethnicities (Table C). In general, white mothers were most likely to be married and black mothers least likely to be married. Very few mothers aged 14 and younger were married. The likelihood of being married generally increased with the mother's age for all race/ethnicities, although marriage rates for most mothers dropped slightly at age 40 and up.
Table C. Percent Married** by Mother's age and Race/Ethnicity
18.2 3.5 13.7 13.4
^ Numerator and denominator include mothers of unknown ages.
Beginning with 2001 data, Vital Statistics annual reports do not include tables or data based on the Kessner Index. (These data may still be obtained by request.) Please see the sources at the end of this chapter for more information regarding strengths and weaknesses of various prenatal care utilization indexes 1, 2 . The Kessner index data in Table 13 has been replaced with data on the mother's marital and educational status.
This annual report still includes data on the month in which prenatal care began because it is one of the indicators suggested by the CDC's Healthy People 2020 initiative 3. However, due to implementation of a new birth certificate for 2005 data, onset of prenatal care within the first trimester is not directly comparable to years prior to 20054.
The percentage of mothers beginning prenatal care in the first trimester decreased from 62.5 percent in 2013 to 61.6 percent in 2014 (Table D). Increasing from 4.9 percent in 2013 to 5.2 percent in 2014, many mothers did not receive prenatal care in 2013. White* women were more likely than black or Hispanic women to have early onset of prenatal care.
Table D. Percent of Live Births**
by Trimester Prenatal Care Began,
5.5 10.0 7.8 7.1
No Care
Note: due to rounding, percents may not sum 100%.
Place of Delivery and Birth Attendant
Nearly all Texans who gave birth in 2014 did so in a hospital (98.5 percent). A small number occurred in licensed birthing centers (1.1 percent) or at home (0.6 percent).
Physicians delivered 95 percent of infants born to Texas residents. Certified nurse-midwives attended 3.1 percent of all births and lay midwives attended 0.8 percent. The remainder, about 0.8 percent, were delivered by other types of attendants, such as EMS workers or relatives.
Low Birth Weight and Very Low Birth Weight
There were 32,661 low birth weight (<2,500 grams) infants born to Texas residents in 2014, which is 8.2 percent of live births (Table E).
The risk of giving birth to a low birth weight infant remains much higher for black mothers (13.4 percent) than for Hispanic mothers (7.5 percent) or white mothers (7.5 percent). The incidence of low birth weight infants is higher among the youngest and oldest mothers. In 2014, 10.6 percent of births to mothers ages 14 and younger and 12.7 percent of births to mothers ages 40 and older were low birth weight.
A total of 5,718 births (1.4 percent of live births) were very low birth weight (<1,500 grams). This rate is the same as the rate in 2004 (1.4 percent). As with low birth weight, the risk of having a very low birth weight infant was highest for black mothers (2.9) and mothers aged 40 and up (2.7). The percentage of births to women 40 and over is up from 2004, when 2.3 percent of the women in that age group gave birth to very low birth weight infants.
Table E.1. Low Birth Weight and Very Low Birth Weight by Mother's Race/Ethnicity
Race/
VLBW
Percent of
LBW
1,981 1.2 12,244 7.5
1,351 2.9 6,161 13.4
Table E.2. Low Birth Weight and Very Low Birth Weight by Mother's Age,
534 1.5 3,179 9.1
1,282 1.3 7,852 7.9
272 2.7 1,283 12.7
3 10.3 4 13.8
Other Birth Characteristics
In 2014, male infants accounted for 51.2 percent of all births (204,503) and female infants accounted for 48.8 percent (194,979). The majority of all births were either first children (37.8 percent) or second children (30.8 percent). Third children accounted for 18.2 percent of all births, and fourth children 8.0 percent.
There were 386,398 singleton births, accounting for 96.7 percent of all births. Twin births accounted for 3.1 percent of all births. All other multiple births occurred in 0.1 percent of all births.
Kogan MD, Martin JA, Alexander GR, Kotelchuck M, Ventura SJ, Frigoletto FD. The changing pattern of prenatal care utilization in the United States, 1981-1995, using different prenatal care indices. JAMA , 279:1623-1628.
Alexander GR, Kotelchuck M. Quantifying the adequacy of prenatal care: a comparison of indices. Public Health Rep 1996 Sep-Oct;111(5):408-18.
http://www.healthypeople.gov/2020/topicsobjectives2020/objectiveslist.aspx?topicId=26#.
Technical Appendix from Vital Statistics of the United States, 2004. Natality in the Documentation of the Detail Natality Public Use File for 2004
Links to external sites are intended to be informational and do not have the endorsement of the Texas Department of State Health Services. Also, these sites may not be accessible to persons with disabilities.
2014 Annual Report Table of Contents


Shopping for a motor coach charter company to transport your group can be a confusing and uncertain experience for consumers. Prices and services vary from company to company, and healthy competition exists for your business.
Selecting a company on the basis of price alone could lead to disappointment down the road. There are some unscrupulous operators who exploit a competitive advantage gained through non-compliance with the laws that regulate the motor coach industry. Canada's motor coach industry is proud of its exemplary safety record and continuously strives to provide the highest level of safety, service, and value to consumers. Transport Canada has praised the bus industry for its remarkable safety record as compared to other modes of travel.
While safety and compliance pays, it also "costs". Responsible, legal, and reputable coach companies make considerable investments to recruit and train suitable drivers, monitor driver condition and performance, inspect and maintain their vehicles to prescribed standards, obtain required insurance coverage, comply with hours of service limits, etc. Unfortunately there are a few companies that don't. These companies can operate, albeit for a limited time, by cutting corners and shaving operating expenses in areas of safety, maintenance, insurance, and evading other government regulations.
What is your company's policy regarding the number of hours a driver can work? Can you affirm that our driver will be in full compliance with the Canadian (and U.S.) rules throughout the course of our trip?
Does your company have a carrier identification number issued by the province/territory?
Could you fax to me a recent copy of your carrier performance abstract?
A Carrier Abstract is a document issued by the province/territory available to the general public summarizing a company's safety performance for a two-year period.
Does your company have an Operating Licence issued by the province/territory that authorizes the service being offered? If yes, what is the licence number?
What is the name of your insurance company? What is your policy number?
Provincial/territorial laws require a public vehicle operator to carry $8 million in public liability insurance for buses.
The US DOT number can be verified through the US DOT website at www.safersys.org/ In addition, insurance required by US DOT for US travel (US $5 million) can be verified on this web site. The US DOT number must be displayed on the side of the coach.
Are your drivers enrolled in a drug and alcohol-testing program as required by US DOT? Can you provide proof?
Be sure to check CVOR numbers on the Ministry of Transportation (MTO) website.
Be wary of companies that will not provide references attesting to their safety performance and service quality.
Ask questions to satisfy yourself that the company conducts regular driver training and evaluation.
Determine the age of the coach if that's important to your group. Motor coaches are solid and well-made vehicles that have a much longer service life than cars, so older equipment that has been well cared for can indeed provide safe, clean, and reliable transportation. Be sure to ask about amenities that are important to your group, such as video and sound systems, lavatory, reclining seats, wheelchair lifts, luggage storage, air conditioning, etc.
Even the most modern, well-maintained coaches can experience an unforeseen mechanical failure. Reputable and experienced companies have 24-hour back-up measures and contingency plans in place to minimize disruption and delays en-route.
Legal, responsible, and safe companies have nothing to hide and will proudly show you their maintenance and operations facility. If you're unsure whether the company's services will meet your needs, ask to drop by to see before you buy.


Home » Where do I find the surface altitude of an L1B2 SOM grid location?
Where do I find the surface altitude of an L1B2 SOM grid location?
The value is stored in the MISR Ancillary Geographic Product (AGP), of which there are 233 unique files, one for each of the World Reference System-2 (WRS-2) paths. Like the L1B2 products, these are divided into blocks, and the fields within the "Standard" grid of the AGP are provided at 1.1-km sampling. The average elevation over a 1.1-km grid sample are provided in the "AveSceneElev" field. AGP files can be obtained from the MISR Order and Customization Tool and through the ASDC Data Pool.