text:
\section{Introduction}
\label{sec-introduction}
A growing body of work has demonstrated that the properties of communities of interacting species can be studied using tools of statistical mechanics, with the role of the thermodynamic limit being played by the large number of species. Such high-diversity communities, with tens to thousands of species, are ubiquitous and can be found anywhere from microbes in the gut to plants in a rain forest \cite{faust_microbial_2012}. Most of the works have focused on the properties of fixed-points of the dynamics, and much less is known about the dynamics themselves, in particular when they never reach a fixed-point. \\
Dynamical mean-field theory (DMFT) is a useful theoretical framework which has been often used in the past to study complex stochastic dynamics of interacting degrees of freedom (spins, agents, neurons, ...) \cite{sompoZip_dmft, cugliandolo1993analytical, galla_replicator, opperPRL,sompoPRL}. In this work we develop DMFT for models
of ecosystems formed by a large number of interacting species \cite{biroli_marginal, buninPRE, galla_rLV}.
In the limit of large ecosystems, interactions between different species are commonly modeled by taking random interaction strengths \cite{may_theoretical_2007,barbier_assembly}. The resulting model consists in generalized Lotka-Volterra equations with random couplings. This leads to interesting problems of statistical physics, similar to ones encountered in the theory of disordered systems. Yet, there are a number of crucial differences; in particular an ecosystem is driven by non-conservative forces, hence its dynamics cannot be mapped in general to the one of a physical system in thermal equilibrium. This leads to complex dynamical regimes which have been discussed in other fields before, mainly in neural networks \cite{sompoPRL} and game theory \cite{galla_replicator, opperPRL} (see also \cite{berthier2000two}).\\
Solving numerically the equations corresponding to DMFT represents a major difficulty due to the retarded friction kernel
and the non-linearity of the equations. In the past this obstacle
has been solved---actually circumvented---only for simplified (spherical or truncated) spin-glass models
for which DMFT equations greatly simplify and reduce to closed integro-differential equations on correlation and response functions of local degrees of freedom \cite{kim2001dynamics}.
To the best of our knowledge, a procedure to numerically integrate DMFT is still missing (with the exception of
\cite{eissfeller_numDMFT} that was restricted to the case of stationary dynamics and Ising spins), especially
one able to analyze the complex dynamics relevant for ecosystems.
In this work, following ideas developed for DMFT of strongly correlated quantum systems \cite{georges_dmft}, we develop a generic numerical scheme to solve DMFT. Our method lays foundations for the study  of high-diversity ecological dynamics, but
also provides general tools that can be applied to problems beyond ecology, for example in
the fields mentioned above \footnote{Our code is available in a public gitHub repository \cite{felixroy_implements_2019}.}.

We focus on the generalized Lotka-Volterra model of ecosystems.
We first present a derivation of DMFT based on the dynamical cavity method \cite{parisi_spin}, which is a more intuitive procedure compared to the usual ones based on generating functional formalism, such as Martin-Siggia-Rose-DeDominicis-Janssen \cite{galla_replicator, galla_rLV, eissfeller_numDMFT}.
We then detail our numerical approach for solving DMFT, and show concrete examples of its implementation.
This allows us to test the method, and illustrate its ability to describe and characterize complex dynamics involving chaos and aging.
We finally conclude by discussing further directions and possible future applications.

\section{The random Lotka-Volterra model}

In this section, we introduce the random Lotka-Volterra (rLV) model \cite{buninPRE}, which describes the dynamics of interacting species. We present its phase portrait in the limit of a large number of species.

\subsection{Definition and notations}
The ecosystem consists of $S$ species. Each species $i$ is characterized by its population $N_i(t)$ which is
a positive continuous variable at all times $t$. In the absence of interactions each species may grow until saturation (e.g., due to limitations on resources).
The impact of other species is modeled through bilinear interactions. A small immigration rate $\lambda_i$ is added so that new individuals arrive to the ecosystem from the outside. The dynamical equations read:

\begin{equation*}
\forall i=1..S, \quad \frac{d N_i}{dt}=\frac{r_i}{K_i} N_i (K_i - N_i) - N_i \sum_{j \neq i} \alpha_{ij} N_j + \lambda_i
\end{equation*}

The different parameters are the intrinsic growth rates $r_i$ of the species, their single-species population size (carrying capacities) $K_i$ in the environment and the interaction matrix $\alpha$. Within our convention, a positive coefficient $\alpha_{ij}$ indicates that the presence of species $j$ is deleterious to the species $i$, due to predation or competition over resources.

For clarity of presentation, in this article we mainly discuss the case where all $r_i$ and $K_i$ are set to unity, and the immigration rate $\lambda$ is uniform. The analytical and numerical tools described can be used more generally. Immigration will act as a regularization of the problem. All results will be derived with infinitesimal but finite immigration rate $\lambda>0$. We will separately discuss in the last section the case without immigration. The elements of the interaction matrix $\alpha_{ij}$ are i.i.d. Gaussian random variables with moments:
$$\overline{\alpha_{ij}}=\mu/S, \qquad \overline{(\alpha_{ij}-\overline{\alpha_{ij}})^2}=\sigma^2/S, \qquad \overline{(\alpha_{ij}-\overline{\alpha_{ij}})(\alpha_{ji}-\overline{\alpha_{ji}})}=\gamma\sigma^2/S $$

The scaling with $S$ ensures a proper large $S$ limit. In this limit, the model becomes characterized by three parameters only: the average strength of interaction $\mu$, the variety of interactions $\sigma$, and their symmetry $\gamma$. More specifically, $\gamma$ ranges from -1 (fully antisymmetric case, where all interactions are of predation-prey type) to 1 (fully symmetric case, where an energy can be defined).

For a real ecosystem with given size $S$, the parameters $\mu$, $\sigma$ and $\gamma$ can be statistically computed from the interaction matrix $\alpha$. Our result then stands for this ecosystem with the relevant parameters value. From numerical simulations, we find that ecosystems with $S>200$
are well described by results obtained in the "thermodynamic" limit $S\rightarrow \infty$.

\subsection{Phase diagram}

In the large-$S$ limit, three different dynamical phases are found \cite{buninPRE}; see \fref{fig:phaseP}.

\begin{figure}
\centering
\begin{minipage}{.7\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig1a_phaseP_gam0.pdf}
\end{minipage}%
\begin{minipage}{.3\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig1b_phaseP_varyGam.pdf}
\end{minipage}
\caption{ Phase diagram, taken from \cite{buninPRE}. Left: At $\gamma=0$, the transition from Unique Equilibrium to Multiple Attractors is independent from $\mu$ and lies on the line $\sigma=\sqrt{2}$. Right: For generic $\gamma$ the transition lies on the line $\sigma=\frac{\sqrt{2}}{1+\gamma}$. Increasing the symmetry $\gamma$ shifts the transitions towards lower variance and stronger interactions. This result shows how predation-prey relations may stabilize an ecosystem.}
\label{fig:phaseP}
\end{figure}

\begin{itemize}

\item {\it Phase I: Unique Equilibrium}.
In this regime, corresponding to small $\sigma$,
the ecosystem displays only one stable equilibrium. Whatever the initial conditions, each species asymptotically ends up with a given number of individuals which is always the same (it can be zero as some species go extinct). This equilibrium state is stable to local and global perturbations. On \fref{fig:1eq_traj} we display the dynamics of an ecosystem in this phase: each line represents the time evolution of the population of one species.

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig2a_1eq.pdf}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig2b_1eq_unicity.pdf}
\end{minipage}
\caption{Time evolution of 100 species in the Unique Equilibrium phase:
$(\mu,\sigma,\gamma, \lambda|S)=(4,1,0,10^{-10}|100)$.
Left:
After a transient time, each species reaches a final population value which is stable.
Right:
Dynamical evolution of three species (e.g., `sheep', `rabbits' and `foxes') out of all the species. We show different trajectories obtained starting from different initial conditions. They always converge to the same equilibrium value (black dot) independent of the initial conditions, demonstrating the stability and uniqueness of this equilibrium.}
\label{fig:1eq_traj}
\end{figure}

\item {\it Phase II: Multiple Attractors}. When the variability in the interactions $\sigma$ is increased, the single stable fixed point loses its stability,
and the system is left with a huge number of (possibly unstable) equilibria. This phase exhibits a complex dynamics with chaos (or aging dynamics for $\gamma=1$). An example of such dynamics can be seen on \fref{fig:chaosUG_traj}.

\item {\it Phase III: Unbounded  Growth}. When the average interaction is negative enough ($\mu < -1$), the interactions are cooperative enough to have a beneficial effect on any given species that overrides the single-species saturation.
If we fix a higher $\mu$ and increase the standard deviation $\sigma$, at some point a small community of species will have cooperative interactions stronger than their own saturation and this subgroup of species will thus grow without bound, even though all the other species will die out.
This explains the existence of phase III also for $\mu>-1$ for a large-enough  $\sigma$. An example of such dynamics is displayed in \fref{fig:chaosUG_traj}. It should be noted that the divergence occurs as a finite time explosion of the ecosystem.
The unbounded growth is a pathology of the model that could be cured by a saturation stronger than quadratic.

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig3a_chaos.pdf}
  \label{fig:test1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig3b_UG.pdf}
  \label{fig:test2}
\end{minipage}
\caption{Left: Time evolution of 100 species in the Multiple Attractors phase: $(\mu,\sigma,\gamma,\lambda | S) =(4,2,0,10^{-10}|100)$. The trajectories do not display any simple behaviour; at some points, the system seems to relax to a fixed point before realizing that it has some unstable directions, and the dynamics starts again. Right: Time evolution of 100 species in the Unbounded Growth phase: $(\mu,\sigma,\gamma,\lambda | S) =(4,4,0,10^{-10}|100)$. A large proportion of species present a divergence of their population, while the other ones die out.}
\label{fig:chaosUG_traj}
\end{figure}

\end{itemize}
The borders between phases can be computed analytically: I/II and I/III are exact, but II/III is only approximate \cite{buninPRE}. They are shown on \fref{fig:phaseP}.

The symmetric case $\gamma=1$ is special in the sense that the Multiple Attractor phase is not a chaotic one, but rather a spin glass one \cite{biroli_marginal}: the dynamics gets slower and slower as the system approaches marginally stable states.  In this case, a kind of physical energy can be defined and serves as a Lyapunov function. The dynamics corresponds to a gradient descent in a rough energy landscape.

\section{The Dynamical Mean Field Theory}

In this section, we derive the Dynamical Mean Field Theory (DMFT) using the dynamical cavity method \cite{parisi_spin}. For simplicity, we first present the derivation in the simplest case of random Lotka-Volterra model, then we extend the result to more general models, and finally we explain a numerical method to solve the DMFT equation. We also checked the relevance of the description by comparing the DMFT results with direct simulations, increasing the size $S$ of the ecosystem.

\subsection{Derivation via the cavity method}
For simplicity, DMFT is first derived with the simplest random Lotka-Volterra model, presented in equation \eref{eq:rLV}. Our approach holds in more general cases, we will present its generalization in \sref{sec:DMFTGen}. We start from the Lotka-Volterra equations:

\begin{equation}
\forall i =1..S, \quad \quad \dot{N_i}= N_i (1 - N_i - \sum_{j \neq i} \alpha_{ij} N_j + h_i(t))
\label{eq:rLV}
\end{equation}
where we have added an external field $h_i$ that will be necessary to define the response of the system to a perturbation. The initial conditions are sampled from a product measure: $P\{N_i(t=0)\}=\prod_{i=1}^SP(N_i(t=0))$. For instance, we generally use a uniform distribution in $[0,1]$ for simulation purposes.

The main steps of the derivation are the following:
\begin{enumerate}
\item For given parameters $\mu$, $\sigma$, $\gamma$ and system size $S$, consider a system whose interactions and initial conditions are drawn for the $N_{i=1..S}$ species;
\item Following the dynamics according to equations \eref{eq:rLV} defines the trajectories $N_{i=1..S}(t)$;
\item We add a new species $N_0$, and therefore draw its initial condition $N_0(0)$ and the interactions $\alpha_{i0}$ and $\alpha_{0i}$ for $i=1..S$;
\item If $S$ is large enough, the impact of this new species on the previous trajectories is a small perturbation and therefore we only consider linear response for the trajectories $\tilde{N}_{i=1..S}(t)$ in the presence of species `0':
$$\tilde{N}_i(t)=N_i(t) - \sum_{j =1..S} \int_0^t \left. \frac{\delta N_i(t)}{\delta h_j(s)}\right|_{h=0} \alpha_{j0} N_0(s) ds$$
The partial derivative are to be understood in a functional sense. We introduce the notation $\chi_{ij}(t,s)=\left.\frac{\delta N_i(t)}{\delta h_j(s)}\right|_{h=0}$
\item We plug these new trajectories in the equation for $N_0$:
$$\dot{N_0}=N_0 (1 - N_0 - \sum_{i \neq 0} \alpha_{0i} \tilde{N}_i + h_0(t))$$
We introduce the matrix $a_{ij}$: $\alpha_{ij}=\mu/S+\sigma a_{ij}$, so that $a_{ij}$ is a Gaussian with zero mean and $1/S$ variance, verifying in addition $\overline{a_{ij}a_{ji}}=\gamma/S$. All the sums $\sum_i$ stand for $\sum_{i=1..S}$, so the interaction term reads:
\begin{eqnarray}
\fl \sum_j \alpha_{0j} \tilde{N}_j=\frac{\mu}{S} \sum_i {N}_i(t)  -\frac{\mu}{S} \sum_{ij} \int_0^t \chi_{ij}(t,s) \left(\frac{\mu}{S} +\sigma a_{j0}\right)  N_0(s) ds \nonumber \\ + \sigma \sum_i a_{0i} {N}_i(t) - \sigma \sum_{ij} a_{0i} \int_0^t \chi_{ij}(t,s) \left( \frac{\mu}{S}+\sigma a_{j0} \right) N_0(s) ds
\label{eq:detailEq}
\end{eqnarray}

\item We take the large S limit and analyze the statistical properties of all terms. The main idea is that by construction $N_{i=1..S}$ are independent from $\alpha_{i0}$ and $\alpha_{0i}$, therefore one can use central-limit-like arguments. Henceforth, the notation $\langle .\rangle$ refers to the average over the couplings $a_{ij}$ and initial conditions $N_i(0)$. We will detail the procedure for the response function term as an example. We start from $\sum_{ij} a_{0i} \chi_{ij}(t,s) a_{j0}$. We consider that the different $\chi_{ij}(t,s)$ are random functions that will depend on the initial conditions $N_{i>0}(0)$ and the interaction matrix $a_{ij>0}$, but are otherwise independent from $a_{j0}$ and $a_{0i}$.
We first treat the diagonal part. According to the central limit theorem and up to second order contribution, the term $\sum_{i} a_{0i} \chi_{ii} a_{i0}$ will converge towards its average:
$$S \langle  \chi_{ii} a_{i0} a_{0i}\rangle = S \langle  \chi_{ii}\rangle\langle  a_{i0} a_{0i}\rangle=\gamma\langle  \chi_{ii}\rangle$$
We now focus on the non-diagonal part. Its average is zero because $\langle a_{0i} a_{j0}\rangle_{i \neq j}=0$. To determine the scaling of its fluctuations we evaluate the variance of its single components obtaining $\langle \chi_{ij}^2\rangle_{i \neq j} \langle a_{j0}^2 a_{0i}^2\rangle_{i \neq j}$. It can be shown by perturbation theory in the strength of interactions that $\chi_{ij}$ is of order $S^{-1/2}$ for $i\neq j$ \cite{parisi_spin} (see \ref{si:scaling}). Regrouping the scalings, we obtain that $\sum_{i\neq j} a_{0i} \chi_{ij} a_{j0}$ behaves as:
\begin{eqnarray*}
\fl \ S(S-1)\langle \chi_{ij}\rangle_{i \neq j}\langle a_{j0} a_{0i}\rangle_{i \neq j} + \sqrt{S(S-1)}\sqrt{\langle \chi_{ij}^2\rangle_{i \neq j}}\sqrt{\langle a_{j0}^2 a_{0i}^2\rangle_{i \neq j}}Z \\
\sim 0 + S\frac{1}{\sqrt{S}}\frac{1}{S}Z
\end{eqnarray*}
where $Z$ is a centered standard Gaussian. This shows that the non-diagonal term induces corrections of order $S^{-1/2}$ and can therefore be neglected in the large-S limit. After careful evaluation of all terms in equation \eref{eq:detailEq} according to the same procedure, we get:

\begin{equation*}
\fl \dot{N_0}=N_0 \{ 1 - N_0 - \mu \langle  N_i(t)\rangle  - \sigma \eta(t) \\ + \gamma \sigma^2 \int_0^t \langle  \chi_{ii}(t,s)\rangle N_0(s) ds + h_0(t) \}
\end{equation*}

where $\eta(t)$ is a Gaussian noise with zero mean and covariance $\langle \eta(t)\eta(s)\rangle_\eta = \langle  N_i(t) N_i(s)\rangle$.

\item Since nothing differentiates $N_0$ from any other species, we obtain the self-consistent equation that leads to dynamical mean field theory:

\begin{equation}
\dot{N}=N \{ 1 - N - \mu m(t)  - \sigma \eta(t) + \gamma \sigma^2 \int_0^t \chi(t,s) N(s) ds + h(t) \}
\end{equation}

where $\eta$ is a Gaussian noise with zero mean and correlator $C(t,s)$, and $m(t)$, $C(t,s)$ and $\chi(t,s)$ are given functions. They are self-consistently determined with the relations:

\begin{equation}
\left\{
\eqalign{
	m(t)&=\mathbb{E}[N(t)] \cr
	C(t,s)&=\mathbb{E}[N(t)N(s)] \cr
	\chi(t,s)&=\mathbb{E}[\left.\frac{\delta N(t)}{\delta h(s)}\right|_{h=0}]
}
\right.
\label{eq:closure}
\end{equation}
In these definitions, the averages $\mathbb{E}[.]$ are now taken with respects to the noise trajectories $\eta$ and the initial condition $N(0)$. Therefore, the equation is self-consistent in the three following functions: the \textbf{average population} $m(t)$, the \textbf{correlator} of the noise $C(t,s)$ and the \textbf{averaged response function} $\chi(t,s)$.

\end{enumerate}

To sum up, we started from an $S$-body deterministic system of differential equations, and ended up with a one-body stochastic self-consistent differential equation \footnote{The derivation is similar to the one of the Langevin equation from Newtonian dynamics \cite{zwanzig2001nonequilibrium}, with the extra-ingredient that the bath corresponds to the rest of the system whose behavior can
be self-consistently obtained from the one of $N_0$.}.
It has been mathematically proven \cite{benarous_convInLaw} for spin glasses that when $S\to \infty$, there is a convergence in law between the statistics  of the two descriptions.  We expect that this holds true for our class of models as well, due to the similarity of both the equations and the method.

An important additional remark is that the DMFT is valid as long as we consider times that do not diverge with system size $S$. Otherwise, one cannot neglect terms vanishing with $S$ as we did.

\subsection{DMFT equation for a general class of models}
\label{sec:DMFTGen}

The derivation above can be performed almost identically in more general cases. The only additional subtlety is that we use the fact that the correlation $\langle N_i(t)N_j(t) \rangle$ scales as $S^{-1/2}$ for $i\neq j$, as can be shown by perturbation theory in the strength of interactions \cite{parisi_spin} (see \ref{si:scaling}). Below, we just present the result for a general class of dynamics with a generic and species-dependent response function $R_i(N_i)$, non-linear $p$-body interactions due to $I_i(N_i)$, $J(N_j)$ and a species scaled thermal noise $f_i(N_i)\xi_i(t)$.

\begin{equation}
\fl \dot{N_i} = R_i(N_i) + I_i(N_i) \left( \sum_{1\leq j_2 < ,.. < j_p \leq S} \alpha^i_{j_2,..j_p} J(N_{j_2})..J(N_{j_p}) + h_i(t) \right) + f_i(N_i)\xi_i(t)
\label{eq:generalModel}
\end{equation}
where $\xi_i$ is a Gaussian white noise, with variance $2\omega^2$. The $i$-dependence of the functions denotes the possible presence of random parameters for each species. For instance, in the general Lotka Volterra case, $R_i(N_i)= r_i/K_i(K_i-N_i)$ where the $r_i$ and $K_i$ respectively correspond to species-dependent growth rates and carrying capacities, that we will treat as random variables sampled from given distributions. The coupling tensor satisfies $ \alpha^i_{j_1,..j_p}=0$ if there exists $k$ such that $i=j_k$, so as not to interfere with the self-interaction $R_i(N_i)$. Otherwise, its cumulants are taken as:

$$\overline{\alpha^i_{j_1,..j_p}}=\mu\frac{p!}{2S^{p-1}} \qquad \overline{(\alpha^i_{j_1,..j_p})^2}_{con}= \sigma^2 \frac{p!}{2S^{p-1}}  \qquad \overline{\alpha^i_{j_1,..j_p} \alpha^{j_k}_{j_1..i..j_{k-1}j_{k+1}..j_p}}_{con}=\gamma \sigma^2 \frac{p!}{2S^{p-1}}$$
where the notation $\overline{X}_{con}$ denotes the connected average of $X$, {\it i.e.} when subtracting their average to the elements. Because of the constraint $1\leq j_1 < ,.. < j_p \leq S$, when considering the cross correlation, there is only one place for the upper index $i$ to go down.

Within this set-up, the DMFT equation for a given species reads:
\begin{eqnarray}
\fl \dot{N}=R_i(N) +I_i(N)\left( \mu m + \sigma \eta + \gamma \sigma^2 \frac{p(p-1)}{2} \int_0^t \chi(t,s) C(t,s)^{p-2} J(N(s)) ds + h \right) \nonumber \\
 + f_i(N)\xi
\label{eq:DMFT_all}
\end{eqnarray}
where $\eta$ is a Gaussian noise with zero mean and covariance $\frac{p}{2} C(t,s)^{p-1}$, and $\xi$ is a Gaussian white noise, with variance $2\omega^2$. Using subscripts for the different times, we obtain the self-consistent closure:

\begin{equation}
\left\{
\eqalign{
	m(t)&=\mathbb{E}[J(N_t)]^{p-1}\\
	C(t,s)&=\mathbb{E}[J(N_t)J(N_s)]\\
	\chi(t,s)&=\mathbb{E}[J'(N_t)\left.\frac{\delta N_t}{\delta h_s}\right|_{h=0}]
}
\right.
\end{equation}
where the average $\mathbb{E}[.]$ is now taken with respects to the initial condition distribution, the distribution of species-dependent parameters in the functions $R_i$, $I_i$ and $f_i$, the noise trajectory $\eta$ and the thermal noise $\xi$.

It should be stated that the DMFT we derived with the dynamical cavity technique can also be obtained using generating functional technique of Martin-Siggia-Rose-DeDominicis-Janssen \cite{galla_replicator, galla_rLV, eissfeller_numDMFT}.

\subsection{Solving numerically the DMFT equation}
\label{numSolve}

It is difficult to solve numerically a self-consistent equation where the self-consistency applies to functions. We implemented a strategy which works as pictured in \fref{fig:numScheme}. In this section, we write down in details the methodology of the algorithm. The different steps of the program are the following:

\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.5]{fig4_numericalScheme.pdf}
\caption{Sketch of the numerical scheme for solving the DMFT equation.}
\label{fig:numScheme}
\end{figure}

\begin{enumerate}
\item We start from initial guesses for the correlator matrix $C(t,s)$, the average vector $m(t)$, and the response matrix $\chi(t,s)$. For instance we tried a random average vector for $m$,  diagonal or random positive symmetric matrices for $C$, and lower triangular random matrices for $\chi$ (since $\chi$ is a causal function);
\item Using the correlator, we can sample a Gaussian path as a simple multivariate Gaussian random variable with covariance matrix $p/2C(t,s)^{p-1}$. We draw many ($\#_{traj}$) such Gaussian paths.
\item For each path, we use our guesses $m(t)$ and $\chi(t,s)$ to numerically integrate the DMFT equation where the initial condition is sampled according to the wanted distribution. We used the uniform measure on $[0,1]$ for example. For each Gaussian path, we get a different population trajectory.
\item From these trajectories, we compute the updated values of the average population vector, the correlator matrix and the response matrix (see below), using the self-consistent closure:

\begin{equation*}
\left\{
\eqalign{
	m^{new}(t)&=\mathbb{E}_{paths}\left[ J(N_t) \right] ^{p-1}\\
	C^{new}(t,t')&=\mathbb{E}_{paths} \left[ J(N_t)J(N_t') \right]\\
	\chi^{new}(t,t')&=\mathbb{E}_{paths} \left[ J(N_t)\int ds \; C^{-1}(t',s)\; \eta(s)\right] \mathbf{or} \; \mathbb{E}_{paths} \left[ \chi_i(t,t')\right]
}
\right.
\end{equation*}

\item We update softly the set of functions: $X^{updated}=(1-a)X+aX^{new}$ with $X$ being respectively $m$, $C$ and $\chi$. The soft reinjection is necessary for the algorithm to converge, and not jump erratically from functions to functions. A reinjection parameter $a=0.3$ seems to be a good choice.

\item We start a new iteration of the loop, with the updated set of functions.

\end{enumerate}

The convergence of the algorithm is of exponential form in the number of iterations, and is independent of the initial set of functions. On \fref{fig:convergenceObs}, we show an example of such a convergence.

\begin{figure}[hbtp]
\centering
\includegraphics[width=.6\linewidth]{fig5_iterativeObservables.pdf}
\caption{We show the convergence of different observables as a function of the number of iterations: the final mean abundance $m(t_f)$, the final same-time correlation $C(t_f,t_f)$ and the integrated response kernel $\chi_{int}=\int_0^{t_f} ds \; \chi(t_f,s)$. For comparison, dotted black lines represent the analytic stationary cavity solutions. The relative errors to the stationary cavity solutions are below $2\%$. As the DMFT observables are computed from a finite number of trajectories $\#_{traj}$, there is always some residual fluctuations. The solver was run with rLV DMFT with parameters $(\mu,\sigma,\gamma,\lambda)=(10,1/2,-1,10^{-4})$ in the Unique Equilibrium phase. The parameters of the program are: reinjection rate $a=0.3$, final time $t_f=40$, discrete time steps $\tau=0.1$, final number of trajectories to average upon $\#_{traj}=10^5$.}
\label{fig:convergenceObs}
\end{figure}

Now, let us explain point (iv) in more detail. Obtaining the average population vector, and the correlator matrix from the trajectories is a trivial procedure: one just needs to average. Evaluating the response function $\chi$ is instead more tricky. We studied two different complementary, or alternative, procedures. The first one consists in using Novikov's theorem \cite{novikov} (or Stein's lemma) in the statistical field formulation in order to obtain:
\begin{equation}
\label{eq:Girsanov}
\chi(t,s)=\sigma^{-1}\mathbb{E}[J(N_t)\int dx \; C^{-1}(s,x) \; \eta(x)]
\end{equation}
In this formulation, $C^{-1}$ denotes the matrix inverse of $C$. The detailed derivation is presented in \ref{si:girsanov}. This expression is easy to implement, however it is sometimes too greedy for numerics. For instance, in our problem with multiplicative noise, the number of DMFT trajectories to average upon in order to obtain a satisfactory estimate for the response function is too high. We thus derive another relation, by directly applying $\frac{\delta}{\delta h(t')}$ to \eref{eq:DMFT_all}. In this way, for each trajectory $i$, we can compute the response function $\chi_i(t,t')$ \textit{via} temporal integration, and eventually average over trajectories to obtain $\chi(t,t')$. This procedure is less greedy in terms of needed trajectories, however it is of higher numerical complexity in the number of time steps. The details are given in \ref{si:tempIntegResponse}. In \ref{si:responseSchemes}, we sum up and compare the adequacy of the two methods.

The algorithm we presented here can still be improved in several ways. More specifically, when the response function is needed (when $\gamma \neq 0$), the above algorithm is quite expensive numerically. The complexity of the algorithm might be reduced by proceeding in time slices. Indeed, instead of trying to make the observables converge for the whole time interval $[t_0,t_f]$, it should be numerically more efficient to make them converge on $[t_0,t_1]$, then on $[t_1,t_2]$ using the already converged result of $[t_0,t_1]$, and so on.

The details of the numerical implementation are in \ref{si:numStrategy}, and a public gitHub repository with the corresponding Python programs is available \cite{felixroy_implements_2019}.

\subsection{Numerical check of the results}

We checked that the numerical solution of DMFT is consistent with the one from direct simulations. More specifically, we sample $\#_{instances}=200$ interaction matrices and initial conditions for an ecosystem of size $S$, run the deterministic dynamics, and aggregate the observables by averaging over the $S$ species and $\#_{instances}$ realizations. This is what we call direct simulations. In the Unique Equilibrium phase, the agreement is excellent. In \fref{fig:compDirectSim}, we show the comparison between DMFT and direct numerical simulations in the Multiple Attractors phase. As $S$ increases the direct simulations observables converge at all times to the one from DMFT. It is surprising however that the direct simulations are so different from DMFT for $S=200$. We reckon it is related to the fact that at finite $S$, when sampling the interaction matrix with parameters in the Multiple Attractor phase, there is a non-zero probability to get an interaction matrix that describes an Unbounded Growth ecosystem. This problem makes it difficult to have clean data using direct simulations, and underlines the relevance of DMFT analysis.

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig6a_m_directSim_and_dmft.pdf}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig6b_q_directSim_and_dmft.pdf}
\end{minipage}
\caption{ Comparison of the observables $m(t)$ (Left) and $C(t,t)$ (Right) between direct simulations varying the ecosystem size $S$, and DMFT predictions in dotted red line. It shows the convergence in law towards DMFT as $S$ increases. The parameters of the simulations are $(\mu,\sigma,\gamma,\lambda) =(4,2,0,10^{-4})$, in the Multiple Attractors phase.}
\label{fig:compDirectSim}
\end{figure}

\section{Application to the random Lotka-Volterra model}

From the general case of equation \eref{eq:DMFT_all}, we recover the random Lotka-Volterra DMFT by taking:
\begin{equation*}
\left\{
\eqalign{R_i(x)&=x(1-x)+\lambda\\
		I_i(x)&=-x\\
		J(x)&=x\\
		f_i(x)&=0
}
\right.
\end{equation*}

From this we obtain:
\begin{equation}
\dot{N}=N \{ 1 - N - \mu m(t)  - \sigma \eta(t) + \gamma \sigma^2 \int_0^t \chi(t,s) N(s) ds + h(t) \} + \lambda
\label{eq:DMFTrLV}
\end{equation}
The self-consistent closure is that of Eq. \eref{eq:closure}. In this section, we show how to get back the stationary results \cite{buninPRE} from DMFT, and we study the stability of such stationary solution. All the following results are valid for $\lambda>0 $ where the limit $\lambda \to 0$ is subsequently taken.

\subsection{How to recover the stationary results}
If the ecosystem parameters belong to the Unique Equilibrium phase, each species will eventually reach a final population value and stops changing. We describe this final state using DMFT. The one-species stochastic process becomes time-independent, so the derivative is zero, the average $m(t)$ converges to a number $m(\infty)$, the population $N(t)$ and the Gaussian noise $\eta(t)$ converge to random variables $N(\infty)$ and $\eta(\infty)$. As the process is stationary, we treat the memory kernel as time-translational invariant $\chi(t,s)=\chi(t-s)$ and therefore:
$$\int_0^t \chi(t,s) N(s) ds=\int_0^t \chi(u) N(t-u) du \to_{t\to \infty} \int_0^\infty \chi(u) du N(\infty)$$
Introducing the integrated memory kernel $\chi_{int}=\int_0^\infty du \; \chi(u)$, the DMFT equation \eref{eq:DMFTrLV} finally converges to:
\begin{equation}
0=N_\infty \{ 1 - N_\infty - \mu m_\infty  - \sigma \eta_\infty + \gamma \sigma^2 \chi_{int} N_\infty + h \}
\label{eq:DMFT1eq}
\end{equation}
\begin{equation}
\label{eq:closEqRS}
\left\{
\eqalign{
		m_\infty&=\mathbb{E}[N_\infty]\\
		\chi_{int}&=\mathbb{E}[\frac{\delta N_\infty}{\delta h}]\\
		\mathbb{E}[\eta_\infty^2]&=\mathbb{E}[N_\infty^2]
}
\right.
\end{equation}

From equation \eref{eq:DMFT1eq}, $N_\infty$ can either be 0 or $(1-\gamma \sigma^2 \chi_{int})^{-1}(1 - \mu m_\infty  - \sigma \eta_\infty )$. By a simple linear stability analysis performed on the real system of $S$ species (see \ref{siSec:stability_dead}), it can be shown that the 0 solution is linearly unstable when the other solution is positive. Therefore, we obtain:

\begin{equation}
N_\infty = \max \left( 0,  \frac{1 - \mu m_\infty  - \sigma \eta_\infty }{1-\gamma \sigma^2 \chi_{int}} \right)
\label{eq:rsGuy}
\end{equation}
so the random variable $N_\infty$ follows a Gaussian distribution, truncated for negative abundances. We write the closed system of equations in \ref{siSec:1eq}, and show that we end up with the same system as the one from \cite{buninPRE}. From it, all observables can be computed numerically as a function of the parameters $(\mu,\sigma,\gamma)$: the fraction of species coexisting at the fixed-point, the mean abundance $N$ of species that survive and the mean response function. Some further analytical results can be derived as well, such as identifying in parameter space the boundary between the Unique Equilibrium phase and the Unbounded Growth phase. However, this analysis is only exact  when we are in the Unique Equilibrium phase. It becomes approximate in the Multiple Attractors phase.

\subsection{Dynamical stability and the transition line to Multiple Attractors}

We now describe the loss of stability of the Unique Equilibrium solution when increasing the variability $\sigma$ of interactions: a dynamical phase transition takes place. The setup follows the one of \cite{opperPRL}: starting in the Unique Equilibrium phase, we let the system reach an equilibrium point, then add some small field $h(t)$ which we will take as a Gaussian white noise with covariance $\overline{h(t)h(s)}=\sigma_h^2\delta(t-s)$, and see how the system responds in perturbation theory. In order to do so, we consider the DMFT equation \eref{eq:DMFTrLV}, linearize it around a stationary solution, and perform a Fourier analysis \cite{opperPRL}. The detailed calculations are presented in \ref{si:stability_1eq}. Introducing $\tilde{X}$ the Fourier transform of $X$, we obtain the small frequency expansions for both the connected correlator $C_c(t,t')=\mathbb{E}[\eta(t)\eta(t')] - \mathbb{E}[\eta_\infty^2]$ and the response function:

\begin{eqnarray}
\label{eq:smallOmExpC}
&\tilde{C_c}(\omega) = \frac{\sigma^2_h}{ \frac{(1-\gamma\sigma^2 \chi_{int})^2}{\phi} -\sigma^2 +|\omega| \frac{|\chi_{int}|}{\phi^2} \pi p_+(0)}\\
&\tilde{\chi}(\omega) = - \frac{\phi}{1-\gamma \sigma^2 \chi_{int}} +i|\omega| \log|\omega| \frac{p_+(0)}{(1-\gamma \sigma^2 \chi_{int})^2 - \gamma \sigma^2 \phi}
\label{eq:smallOmExpK}
\end{eqnarray}
where $\phi$, $\chi_{int}$ and $p_+(0)$ are properties of the Unique Equilibrium we started from. They correspond respectively to the fraction of surviving species, the integrated response to perturbations, and the value in $0^+$ of the surviving species' distribution. They can be computed using the stationary cavity equations in \ref{siSec:1eq}.

Different things should be noted about these expansions. First, it can be checked that we obtain the same zeroth order condition for the response function as in the stationary cavity study: $\tilde{\chi}(\omega=0)=\chi_{int}$. Secondly, the correlator initially behaves as $(a+|\omega|)^{-1}$ which corresponds to a temporal decay as $1/t^2$. But a change of behaviour is displayed when zeroth order term $a$ goes to zero: we observe a $1/f(\sim1/\omega)$ correlation spectrum, which is an indicator of the chaotic transition \cite{opperPRL}. Indeed, with this criterion we find the same transition in parameter space as the one from random matrix theory (the line $\sigma_c=\frac{\sqrt{2}}{1+\gamma}$ in the phase portrait in \fref{fig:phaseP}). Surprisingly, the response function instead does not exhibit a transition at $\sigma_c$, except for $\gamma=1$ where the fluctuation-dissipation theorem establishes a direct link between the correlation function and the response function. More complex response functions might be needed to locate the transition in the general case.

\section{Numerical solution for the random Lotka-Volterra model}

In this section, we present some numerical results for the random Lotka-Volterra DMFT, and show the consistency of both analytics and numerics. The aim is to illustrate the quality of the DMFT results, and present a first description of the dynamical phases (a more complete one will be presented elsewhere).

\subsection{Result in the Unique Equilibrium phase}

We focus on the correlator $C(t,t')=\mathbb{E}[N(t)N(t')]$. In the Unique Equilibrium phase, it reaches a plateau as each trajectory converges to a random constant. Moreover, the value of the plateau coincides with the stationary cavity observable $q$. This is indeed the case, as pictured on \fref{fig:numPlateau}. The convergence to the stationary solution is a good check of the validity of our numerical strategy. It is shown more precisely on \fref{fig:convergenceObs}.

\begin{figure}[hbtp]
\centering
\includegraphics[scale=0.7]{fig7_correlator_1eq.pdf}
\caption{Numerical correlator of plateau type, for rLV DMFT with parameters $(\mu,\sigma,\gamma,\lambda)=(4,1,0,10^{-4})$ below the onset of chaos. The parameters of the program are the same as in \fref{fig:convergenceObs}.}
\label{fig:numPlateau}
\end{figure}

\subsection{Result in the Multiple Attractors phase}

In the Multiple Attractor phase we expect a different behaviour. The system does reach a time-translational invariant (TTI) chaotic state. This means that the one-time observables (the mean population $m(t)$, the proportion of alive species $\phi(t)$, or the equal-time correlation $C_{\sigma}(t,t)$) converge to a constant, and the two-time observables become functions of the time difference: $C_\sigma (t,t')=C_\sigma (t-t')$. If we focus on large times, we expect a relaxing behaviour for the correlator, as the trajectory decorrelates from itself when it explores the phase space along the chaotic attractor. We observe this phenomenon in the numerical solutions. Moreover, the TTI state depends on how deep in the Multiple Attractors phase the system is. On \fref{fig:chaos_TTI}, we show the dependence on $\sigma$ of the TTI correlation $C_\sigma (t-t')$, rescaled as follows. These functions $C_\sigma (\tau)$ starts at a TTI value for the equal-time correlation $C_\sigma (0)$, then as the trajectories decorrelate from themselves $C_\sigma(\tau)$ relaxes towards a TTI final value $C_\sigma(\infty)$ over a timescale $a_\sigma$. We therefore plot $\frac{C_\sigma(t-t')-C_\sigma(\infty)}{C_\sigma(0)-C_\sigma(\infty)}$. We also denote $Q_\sigma=C_\sigma(0)-C_\sigma(\infty)$ the amplitude of the decorrelation. It is representative of the chaos strength, and this is an order parameter for the chaotic transition. On \fref{fig:chaos_orderParam} we show the dependence of both the chaos strength $Q_\sigma$ and the time scale $a_\sigma$ as a function of the chaotic depth $\sigma-\sigma_c$. As expected, the chaos strength $Q_\sigma$ increases and the chaos time scale $a_\sigma$ decreases with the chaotic depth. Our results show that chaos emerges through a {\it second-order out of equilibrium dynamical phase transition}. A first attempt to obtain critical exponents is shown in \fref{fig:chaos_orderParam}. A more thorough study will be presented elsewhere.

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig8a_correlatorTTI_sigma.pdf}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig8b_qTTI_sigma.pdf}
\end{minipage}
\caption{Left: Time evolution of the TTI correlation $C_\sigma(t-t')$ varying $\sigma$. Those are DMFT numerical results with parameters $(\mu, \gamma, \lambda)=(10,0,10^{-4})$. We checked that the system indeed reaches TTI, $t'=200$ is enough here. More precisely, we show the rescaled TTI correlator $\frac{C_\sigma(t-t')-C_\sigma(\infty)}{C_\sigma(0)-C_\sigma(\infty)}$, in order to see the dependence of the chaotic time scale $a_\sigma$ with $\sigma$. This time scale decreases with $\sigma$. In order to have a quantitative approximation for $a_\sigma$, we use a Lorentzian fit; an example of such is the dotted black curve. Right: $\sigma$ dependence of the TTI equal-time correlator $C_\sigma(0)$. The red line indicates the chaotic transition. In orange dots, we show for comparison the analytical static cavity results. In the Unique Equilibrium phase, the DMFT and static cavity results coincide. In the Multiple Attractors phase, they diverge from each other, but the static cavity remains a good approximation for a relevant chaos depth. Note that $C_{\sigma}(0)>0$ for all $\sigma$.}
\label{fig:chaos_TTI}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig9a_QTTI_sigma.pdf}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=\linewidth]{fig9b_chaosTimescaleTTI_sigma.pdf}
\end{minipage}
\caption{Left: Chaos strength $Q_\sigma=C_\sigma(0)-C_\sigma(\infty)$ as a function of $\sigma$. It is zero in the Unique Equilibrium phase, and non-zero in the Multiple Attractors phase. The red line corresponds to the chaotic transition $\sigma=\sigma_c$. The inset is a log-log plot $Q(\sigma - \sigma_c)$. The behaviour seems to agree with a critical exponent around 2.4: $Q \sim (\sigma - \sigma_c)^{2.4}$. Right: Chaos time scale $a_\sigma$ as a function of $\sigma$. It is non-zero in the Multiple Attractors phase, and should diverge as we approach the chaotic transition. The red line corresponds to the chaotic transition $\sigma=\sigma_c$. These values are only approximate, based on basic lorentzian fit from \fref{fig:chaos_TTI}. They do not allow us to extract a critical exponent.}
\label{fig:chaos_orderParam}
\end{figure}

We recall that we have considered small but finite immigration. Dynamics without immigration is different, as we discuss below.

\subsection{Aging dynamics without immigration}
We now consider the effect of the absence of immigration on the chaotic dynamics. The main issue is that
chaos induces fluctuations that can drive species to extinction in absence of immigration and, hence, potentially kill chaos itself. The sustainability of chaotic dynamics without immigration is therefore far from being granted, actually a very different dynamical behavior can be present when $\lambda=0$.
Here we show that this is indeed the case for $\gamma=0$. In \fref{fig:aging}
we compare the correlation functions, normalized by its equal time value, obtained by DMFT for $\gamma=0$ with and without immigration. In the former case (left panel), it is clear that a stationary chaotic state establishes as $C(t,t')$ becomes a function of $(t-t')$ at large times. On the contrary, without immigration (right panel), $C(t,t')$ shows the aging behavior characteristic of glassy system: the correlation function is not a function of $t-t'$ and displays a relaxation that is slower the older is the system. This is a nice illustration of how our numerical implementation of DMFT allows to unveil the existence of different and complex dynamical behaviors. \\
A detailed understanding of the aging chaotic behavior shown in \fref{fig:aging}, its dependence on the degree of asymmetry $\gamma$, and a thorough analysis of how and when chaos fades away is left for a future work.

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
\includegraphics[width=\linewidth]{fig10a_noAging.pdf}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
     \includegraphics[width=\linewidth]{fig10b_aging.pdf}
\end{minipage}
\caption{Aging phenomenon without immigration in DMFT. In the chaotic phase, we show the decay of the rescaled correlation $C(t,t')/C(t',t')$ as a function of $(t-t')$, varying $t'$. The parameters are $(\mu,\sigma,\gamma)=(10,3,0)$ Left: With immigration $\lambda=10^{-4}$, the system reaches a TTI state, there is no dependence on the age of the system $t'$. Right: Without immigration $\lambda=0$, the relaxation of the correlation does depend on the age of the system $t'$; the older the system, the longer it takes to relax. The parameters of the program are the same as in \fref{fig:convergenceObs}.}
\label{fig:aging}
\end{figure}

\section{Conclusion}
In summary, we have presented a general derivation of DMFT for models of ecosystems based on the dynamical cavity method.
We have implemented and tested our numerical method for generalized Lotka-Volterra models of ecosystems
and showed that it can capture complex dynamics such as chaos and aging.
Future works will be devoted to a thorough analysis of these complex dynamical regimes, and also
to improvements of our algorithm along the ways discussed in this paper.  \\
The main contribution of our work is the development of a numerical method
to solve DMFT that can be used for many different systems characterized by stochastic dynamics
and by a large number of degrees of freedom. One important potential application is to the dynamics
of interacting particle glassy systems in the limit of infinite dimensions for which mean-field dynamical
equations were derived recently \cite{kurchan2016,infDim_agoritsas}.
\ack

This work was partially funded by Capital Fund Management - Fondation pour la Recherche, and Simons Foundation collaboration
Cracking the Glass Problem (No. 454935 to G. Biroli). G. Bunin acknowledges support by the Israel Science Foundation (ISF) Grant no. 773/18.
This research was supported in part by the National Science Foundation under Grant No. PHY-1748958.

We thank A. Altieri, J.-P. Bouchaud, D.S. Fisher, T. Galla, A. Manacorda, V. Ros and F. Zamponi for useful interactions on this topic.

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
Photo Credit: The Woodlands EDP
The Woodlands Area Economic Development Partnership Honors Elected Officials
By Ashley Byers | March 14, 2022
The Woodlands Area Economic Development Partnership (EDP) recently held its 24th annual Elected Officials Reception at The Westin at The Woodlands on Thursday, March 10, 2022. For over two decades, the EDP has hosted a reception honoring elected officials at the local, county, state, and federal levels. Over 150 partners and guests attended to show their appreciation to those who serve our community and region in the various roles.
In addition to honoring the elected officials in attendance, a special presentation was made to honor Congressman Kevin Brady, who will retire from Congress at the end of his 13th term later this year. Mr. Brady has served in the U.S. House since 1996 and has been a centrical part in supporting the EDP, which was founded in 1997.
Photo by The Woodlands EDP; Congressman Kevin Brady
"It is our pleasure and honor to rightly thank our elected officials for their tireless dedication to enhancing our community, supporting our local businesses, as well as encouraging new businesses to locate to our area. Over the many years, our members and staff look forward to hosting this high-profile event" said Gil Staley, CEO.
A special thank you to the Platinum Level Sponsors who make hosting this reception a success: Caldwell Companies, Entergy Texas, The Howard Hughes Corporation®, Petroleum Wholesale, The Woodlands Township, Truist Bank, Trustmark Bank and Waterway Wealth Management.
Partners in attendance were A Strong CPA, Amegy Bank, Avion Wealth, Caldwell Companies, Calvetti Ferguson, Consolidated Communications, Corporate Incentives, Cypressbrook Company, Ellard Insurance & Risk Management, Entergy Texas, First Financial Bank, Ikard & Company, LLC, Linscomb & Williams, Lone Star College System, Memorial Hermann The Woodlands Medical Center, PNC Bank, Sam Houston State University, Soules Insurance Agency, St. Luke's Health, Stibbs & Co. Attorneys, P.C., Tachus Fiber Internet, Teal Construction, The Howard Hughes Corporation®, The Strong Firm P.C., The University of Texas MD Anderson Cancer Center, The Woodlands Area Chamber of Commerce, The Woodlands Christian Academy, The Woodlands Township, Tough Law Firm, PLLC, Truist Bank, Trustmark Bank, Waterway Wealth Management, Westwood Magnolia Parkway Improvement District, Woodforest Development, Inc., Woodforest National Bank and Wright Business Technologies.
Photo by The Woodlands EDP; Elected Officials with EDP CEO, Gil Staley, L to R Front Row: Jay Guerrero, Steve Toth, Brandon Creighton, Betty Russo, Melanie Bush, Ryan Gable, Kevin Brady, Chris Grice, Paul Bond, Dawn Candy, Frances Planchard, Skeeter Hubert. L to R Back Row: Jason Nelson, Dr. Ann Snyder, Theresa Wagaman, Dr. Shelley Sekula-Gibbs, Paul Carroll, EDP Chairman of the Board, Avion Wealth, Sharlene Valdez, Mary Ann Turner, Ray Sanders, Amy Tucker, and Lisa Michalk.
The Woodlands Area Economic Development Partnership (EDP) is a non-profit organization dedicated to promoting the economic development of The Woodlands Area through the support of existing businesses and the recruitment of new businesses that can bring job opportunities and capital investment to the area.
The EDP is a public/private partnership with 96 funding partners active in the organization. For further information on the EDP, log on to www.edpartnership.net or call (281) 363-8130.
Source: The Woodlands EDP
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Set up accounts receivable for each invoice.
Establish a history record for each order.
Make all appropriate accounting entries.


text:
De zevenster (Trientalis europaea) is een plant uit de sleutelbloemfamilie (Primulaceae). Het is een plant die groeit in bossen op vochtige, zure en humeuze grond.

Het is een tenger plantje, dat in Noord-Azië, Noord-Europa en Noordwest-Amerika voorkomt. De plant wordt 5–20 cm hoog en heeft kruipende wortelstokken. Aan de rechtopstaande stengel zitten een paar smalle blaadjes met aan de top een bladrozet van langwerpige, zwak getande bladeren.

De zevenster bloeit in mei tot juli met witte (soms roze) langgesteelde bloemen met zeven (soms zes of acht) kelkbladen, kroonbladen en meeldraden, vandaar de naam zevenster.

Plantengemeenschap
De zevenster is een kensoort voor het beuken-eikenbos (Fago-Quercetum).

Externe link

Sleutelbloemfamilie
zevenster
zevenster

redpajama_set_name: RedPajamaWikipedia


text:
Extinction Rebellion protesters stop traffic in Nottingham city centre during march
A samba band called The Band of Outlaws played during the peaceful procession that stopped buses and cars
Ben ReidDigital News Correspondent
Climate activists Extinction Rebellion brought traffic to a standstill in the streets of Nottingham city centre as they marched to deliver letters to MPs.
The protesters disrupted city centre traffic between noon and 1pm on Thursday (June 25) in order to make the deliveries, accompanied by a loud ensemble of drums while carrying flags and banners.
The letters demand local MPs speak out about "the government's failings to reach its own climate targets."
The route began at Wellington Circus by Nottingham Playhouse at the offices of Lilian Greenwood, MP for Nottingham South and continued onto Maid Marian Way and along Upper Parliament Street ending at the office of Tom Randall, MP for Gedling on King Edward Street.
A samba band called The Band of Outlaws played during the peaceful procession as marchers kept to social distancing along the Nottingham roads.
Pictures and video taken by Nottinghamshire Live shows buses and cars being stopped on the usually busy streets as the activists made their message clear in the baking 30 degree heat - "Inaction = death".
Climate activists Extinction Rebellion in Nottingham city centre on June 25 (Image: Nottingham Post/Ben Reid)
Olympian canoeist, Etienne Stott of The Meadows took part in the walk and told Nottinghamshire Live: "The climate and ecological emergency dwarfs the Covid crisis in terms of potential for misery and suffering of the most vulnerable people around the world."
He added: "Today is a stark warning. The committee on climate change which reports on the government's progress towards climate targets has told the government that they are failing and leaving us exposed to the risks of a change of climate.
"It's really important we call out our MPs and get them to hold the government to account.
"The Covid crisis is really difficult and is causing huge issues for people and it's really hard to see it on the news but the climate and ecological emergency dwarfs it in terms of potential for misery and suffering of the most vulnerable people around the world.
"We've really got to get this message through. Time is running out.
"This is an issue that any sane, sensible or responsible person would see as a great risk to the future and the future they care about.
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"It's not about left-wing or right-wing, it's about all of us being in this really difficult situation."
The protest was part of UK wide action today in which MPs will receive letters delivered by Extinction Rebellion.
Prior to the march, the group said: "With banners, drum beat and brass, the socially distanced march is happening because the government's own Committee on Climate Change (CCC) reports today to say that the government's actions fall massively short of meeting its own legally-binding target to achieve net-zero greenhouse gas emissions by 2050, a target most scientists consider to be woefully inadequate."


text:
Persons using assistive technology might not be able to fully access information in this file. For assistance, please send e-mail to: mmwrq@cdc.gov. Type 508 Accommodation and the title of the report in the subject line of e-mail.
Mobility Limitation Among Persons Aged >40 Years With and Without Diagnosed Diabetes and Lower Extremity Disease --- United States, 1999--2002
Diabetes increases the risk for mobility limitation, especially among older persons (1,2). Lower extremity disease (LED), which includes peripheral arterial disease (PAD) and peripheral neuropathy (PN), also increases the risk for mobility limitation (3,4). To assess the prevalence of mobility limitation among persons with diagnosed diabetes, persons with LED, and persons with both or neither condition, CDC analyzed data from the National Health and Nutrition Examination Survey (NHANES) 1999--2002 for adults aged >40 years. This report summarizes the preliminary findings, which indicated that the national prevalence of mobility limitation is higher among persons with either diagnosed diabetes or LED than those without the conditions, and that adults with both conditions have a higher prevalence of mobility limitation than those with either condition alone. Monitoring the prevalence of diabetes, LED, and associated risk factors and identifying effective LED prevention strategies will help reduce the burden of mobility limitation in the United States.
NHANES is an ongoing, cross-sectional survey of representative samples of the U.S. civilian noninstitutionalized population. The survey uses a complex multistage probability design. Data are collected through in-person interviews and medical examinations. During 1999--2002, NHANES participants were asked, "Other than during pregnancy, have you ever been told by a doctor or health professional that you have diabetes or sugar diabetes?" Participants who answered "yes" to this question were classified as having diagnosed diabetes; if the answer was "no" or "borderline," participants were classified as not having diagnosed diabetes. LED was defined as the presence of either PAD (ankle/brachial blood pressure ratio <0.9), PN (one or more insensate areas, on the basis of monofilament testing of foot sensation), self-report of foot/leg ulcers, or technician-observed toe or foot lesion or amputation. Further details of these measures have been described previously (5). Mobility limitation was determined on the basis of participants' response to the questions, "How much difficulty do you have walking for a quarter of a mile; walking up 10 steps without resting; and walking from one room to another on the same level?" Participants who responded "some difficulty," "much difficulty," or "unable to do" to one or more of the three questions were classified as having mobility limitation; participants who responded "no difficulty" to all three questions were classified as having no mobility limitation. Prevalence of mobility limitation was calculated for adults with and without diagnosed diabetes and LED by age and sex for participants aged >40 years who had complete data in the interview and examination variables of interest (n = 4,689); of the 6,059 persons aged >40 years who received the health examination, 1,370 (23%) were excluded from the analysis because of missing data. All reported percentages and 95% confidence intervals (CIs) were estimated using examination weights and taking into account the complex sampling design. Data were age-adjusted to the 2000 U.S. standard population using the age groups 40--59 years, 60--74 years, and >75 years. Logistic regression analysis was used to assess the association of diabetes status and LED status with mobility limitation, including whether an interaction existed between diabetes status and LED status in their associations with mobility limitation, after adjusting for demographic characteristics.
The age-adjusted prevalence of mobility limitation among adults with diagnosed diabetes was greater than for those without diagnosed diabetes overall (27% and 16%, respectively) and in each age and sex group. The age-adjusted prevalence of mobility limitation among those with LED was also greater than for those without LED overall (26% and 15%, respectively) and in each age and sex group (Table).
Overall, adults with diagnosed diabetes but without LED had a similar prevalence of mobility limitation as adults with LED but without diagnosed diabetes (23% and 25%, respectively). Those with diagnosed diabetes and LED had a prevalence of mobility limitation greater than those with either condition alone and almost three times greater than those with neither condition (39% and 14%, respectively).
In a logistic regression model that included both diagnosed diabetes and LED, after data were adjusted for age, sex, and race/ethnicity, the odds of mobility limitation were greater for adults with diagnosed diabetes (odds ratio [OR] = 2.0; CI = 1.4--3.0) than adults without diagnosed diabetes; the odds of mobility limitation were also increased among adults with LED (OR = 2.3; CI = 1.7--2.9) compared with adults without LED. No statistically significant interaction between diabetes status and LED status existed (i.e., the relative odds of mobility limitation for persons with diagnosed diabetes or LED were additive overall and not modified by the presence of both conditions).
Among persons with mobility limitation, the most frequently reported mobility limitations were related to walking a quarter mile and walking up 10 steps without resting (Figure). Among those who had both diagnosed diabetes and LED, 33% reported difficulty walking a quarter of a mile and difficulty walking up 10 steps; 6% reported having difficulty walking from one room to another on the same level, which is the most severe form of mobility limitation analyzed.
Reported by: MS Eberhardt, PhD, S Saydah, PhD, R Paulose-Ram, PhD, National Center for Health Statistics; M Tao, PhD, EIS Officer, CDC.
Editorial Note:
Diabetes has long been identified as one of the major factors associated with mobility limitation (6,7). The findings in this report suggest a statistically significantly higher prevalence of mobility limitation among adults aged >40 years who had diagnosed diabetes compared with those without diagnosed diabetes. The cross-sectional design of NHANES does not permit an investigation into the causal pathway for mobility limitation; however, research has indicated that PN and PAD, as well as other diabetic complications (e.g., vision loss) or comorbidities (e.g., obesity, cardiovascular disease, or arthritis), are predictors of mobility limitation among persons with type 2 diabetes (2). Studies have also demonstrated that LED has an independent effect on mobility among older persons (3,4). Consistent with previous findings, this report indicates substantially higher percentages of mobility limitation among adults aged >40 years with LED but without diabetes, compared with those with neither condition.
In this sample of the noninstitutionalized U.S. population, only a small percentage (6%) of those with diagnosed diabetes and LED reported difficulty moving from one room to another on the same level, which can impair a person's ability to perform activities of daily living. A larger percentage reported impaired ability to walk a quarter mile or climb 10 steps without resting. Such physical limitations can affect a person's ability to live independently or participate in community life and might decrease well being.
The findings in this report are subject to at least three limitations. First, NHANES does not include institutionalized persons, such as those in long-term--care facilities, a population less healthy and more likely to have functional limitations. Second, the sample size was not sufficiently large to analyze additional factors related to mobility limitations (e.g., comorbidities) through bivariate analysis. Finally, among the 6,059 persons aged >40 years who received the health examination, 1,370 (23%) were excluded from the analysis because of missing data; because persons with missing data were older and more likely to have diagnosed diabetes, the prevalence of mobility limitation is probably underestimated.
As the U.S. population ages and the prevalence of diabetes increases, LED and its health consequences, including chronic ulcers in feet or legs, amputations, and mobility limitations, will become increasing public health concerns. Proper foot care is one example of preventive care that might help reduce the prevalence of LED and mobility limitations. CDC's national diabetes surveillance data for 2003 indicated that only 67% of persons with diabetes reported receiving an annual foot examination, even though 88% reported having an annual doctor visit (8). CDC collaborates with state health departments and communities to prevent and manage LED to minimize its impact on mobility. The National Diabetes Education Program has developed materials related to LED and foot care for persons with diabetes and their health-care providers (available at http://www.ndep.nih.gov/campaigns/feet/feet_overview.htm).
Volpato S, Ferrucci L, Blaum C, et al. Progression of lower-extremity disability in older women with diabetes: the Women's Health and Aging Study. Diabetes Care 2003;26:70--5.
Bruce DG, Davis WA, Davis TM. Longitudinal predictors of reduced mobility and physical disability in patients with type 2 diabetes: the Fremantle Diabetes Study. Diabetes Care 2005;28:2441--7.
McDermott MM, Liu K, Greenland P, et al. Functional decline in peripheral arterial disease: associations with the ankle brachial index and leg symptoms. JAMA 2004;292:453--61.
Resnick HE, Vinik AI, Schwartz AV, et al. Independent effects of peripheral nerve dysfunction on lower-extremity physical function in old age: the Women's Health and Aging Study. Diabetes Care 2000;23:1642--7.
Gregg EW, Sorlie P, Paulose-Ram R, et al. Prevalence of lower extremity disease in the U.S. adult population >40 years of age with and without diabetes. 1999--2000 National Health and Nutritional Examination Survey. Diabetes Care 2004;27:1591--7.
Gregg EW, Beckles GLA, Williamson DF, et al. Diabetes and physical disability among older US adults. Diabetes Care 2000;23:1272--7.
Guccione AA, Felson DT, Anderson JJ, et al. The effects of specific medical conditions on the functional limitations of elders in the Framingham study. Am J Public Health 1994;84:351--8.
CDC. Age-adjusted rates of annual dilated eye exam, daily self-monitoring of blood glucose, foot exam in the last year, and doctor visit for diabetes in the last year per 100 adults with diabetes, United States, 1994--2003. Atlanta, GA: US Department of Health and Human Services, CDC; 2005. Available at http://www.cdc.gov/diabetes/statistics/preventive/fX.htm.
Use of trade names and commercial sources is for identification only and does not imply endorsement by the U.S. Department of Health and Human Services.
References to non-CDC sites on the Internet are provided as a service to MMWR readers and do not constitute or imply endorsement of these organizations or their programs by CDC or the U.S. Department of Health and Human Services. CDC is not responsible for the content of pages found at these sites. URL addresses listed in MMWR were current as of the date of publication.
Disclaimer All MMWR HTML versions of articles are electronic conversions from ASCII text into HTML. This conversion may have resulted in character translation or format errors in the HTML version. Users should not rely on this HTML document, but are referred to the electronic PDF version and/or the original MMWR paper copy for the official text, figures, and tables. An original paper copy of this issue can be obtained from the Superintendent of Documents, U.S. Government Printing Office (GPO), Washington, DC 20402-9371; telephone: (202) 512-1800. Contact GPO for current prices.
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Date last reviewed: 11/22/2005
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Everybody's Talkin' Nilsson
How The Beatles' George Harrison coined the word "grotty" and deconstructed influencer culture in 1964
George Harrison in A Hard Day's Night. Picture: United Artists
George Harrison's movie-stealing scene in A Hard Day's Night famously gave the world the word "grotty" but it also did so much more.
A Hard Day's Night was a cinema landmark, and not just as the first Beatles movie.
Rock and pop artists had been in films before, but while most were quick and cheap cash-ins, Richard Lester's musical comedy was a critical as well as a commercial smash.
Charting 36 fictional hours in the life of the Fab Four just as Beatlemania was exploding, it showed off the lads' abilities as talented actors as well as musicians.
QUIZ: How well do you know the Beatles' song lyrics?
John, Paul, George and Ringo display a dry wit and tremendous ability to play themselves on screen, which isn't nearly as easy as it sounds.
As well as some frankly incredible musical numbers, from the opening 'A Hard Days Night' chase to the closing 'She Loves You' concert, the narrative is stitched together by an incredibly sharp script from Alun Owen, putting together endlessly quoteable bits of dialogue and laugh-out-loud vignettes.
And maybe the very best moment of the film finds The Beatles in a TV studio when George Harrison wanders off into the office of adman Simon Marshall, played by an uncredited Kenneth Haigh (who had previously played Jimmy Porter in Look Back in Anger on the stage).
Read more: Paul McCartney's infamous angry 'Long and Winding Road' letter after tinkering is a must-read
George is asked by Simon to give his opinion on some new shirts. "Well, not your REAL opinion, naturally. It'll be written out for you, and you'll learn it," Simon says.
"Now, you'll like these. You'll really 'dig' them. They're 'fab', and all the other pimply hyperboles."
So in 1964, George is basically being asked to be an influencer. All that's missing is the #ad hashtag. But George doesn't seem too keen to play ball.
George Harrison eyes up the "grotty" shirts. Picture: United Artists
Read more: The inside story of why The Beatles really broke up
"I wouldn't be seen dead in them," he murmurs. "They're dead grotty."
And while we all use the word grotty today, the 1964 novelisation of A Hard Day's Night was the first time the word was ever used in print.
Years later in Anthology, George remembered: "Alun Owen made that up; I didn't. People have used that word for years now. It was a new expression: grotty – grotesque."
George Harrison steals the show in A Hard Day's Night. Picture: United Artists
Gold's Hall of Fame: George Harrison
Adman Simon was naturally not too impressed.
"Make a note of that word and give it to Susan. It's rather touching, really. Here's this kid, giving me his utterly valueless opinion, when I know for a fact that within a month, he'll be suffering from a violent inferiority complex and loss of status because he isn't wearing one of these nasty things!" he explodes.
"Of course they're grotty, you wretched nit! That's why they were designed! But that's what you'll want."
Simon Marshall is shocked by George's honesty. Picture: United Artists
Read more: 'Eleanor Rigby' by The Beatles: The making of the dark pop classic
And when George says he doesn't care about being replaced, Simon predicts the 1980s with the killer line: "And that pose is out too, Sonny Jim. The new thing is to care passionately and be right-wing."
And things only go downhill from then, when Simon dangles the opportunity of meeting Susan Canby, his brand's professional trendsetter and "resident teenager". "You'll have to love her, she's your symbol."
George instead dismisses her as "that posh bird who gets everything wrong" and "a drag – a well-known drag".
So, effectively coining the word "grotty" AND dismantling influencer culture in less than three minutes, all the way back in 1964. Not bad for The Quiet Beatle.
More from The Beatles
See more More from The Beatles
The Beatles final rooftop performance is coming to cinemas as a one-hour special
Does the policeman who shut down the last ever Beatles gig have any regrets?
'Rocket Man' by Elton John: The making of the stellar space ballad


American native Artist Anthony Kleem creates story telling pictures in a colorful primitive style that can be easily translated with threads. Pumpkin is one of a series by Kleem, each measures 3" x 9" on 14 mesh mono deluxe needlepoint canvas. We are happy to select and add your choice of wool, cotton or silk threads to help complete. Please contact us if you would like to add fibers of your choice. Available in alternate sizes.


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Q: Bootstrap carousel slide start without "active" i like to use the Bootstrap carousel slider.
Is it possible to start the slider without the active class in the first Image? The code is set with TYPOSCRIPT and i can not set the first item to active. Is there a way with JavaScript?
Thanx for help
Christian
HTML Code

A: Yes, you can add class like this (using jQuery):
$(document).ready(function () {
  $('#myCarousel').find('.item').first().addClass('active');
});

redpajama_set_name: RedPajamaStackExchange


The Gods of copyright are satiated . With the recent judgement calling for AFP and Getty Images to disburse $1,2 million to Mr Morel for copyright infringement, they have been offered reverence. Or have they ?
See, both companies had already been found guilty of copyright infringement many years ago. It has been clearly demonstrated that AFP's procedure had been wrong and by consequence, so was Getty's. This week's judgement was all and only about the penalty to be paid, not about the wrongdoing. The fault here is clearly and solely in the hands of that one over-zealous AFP photo editor who single handedly decided to take images from Twitter and distribute via their network. He was not ordered to do so, he took the decision on his own. He then apparently proceeded in making more of mess of the situation by being extremely obtuse and irresponsible. The fact that he still has a job is a testament of the power of the employees in French society. In the US, he would have been sacked immediately and would never work in the photography world ever again. But that is beyond the point.
His mistake, clearly due to the extreme pressure the wire services have to remain competitive at all cost by supplying images as fast as possible, is a clear example on how wire services – and any breaking news media- are ill equipped to deal with social media's competition. Not only he saw Twitter as a supplier of images – instead of a publisher- but somehow, he wrongly believed that an image published on Twitter would not be seen much and thus deserved to be pushed via AFP's network. He would not have done the same if it had been published on the New York Times' website not only because of very clear copyright issue but also because of its perceived audience, that image would have little or no interest for anyone else. In the breaking news business, you want to be the first one to publish an image otherwise you look like you are playing catching up.
But why is this not a great win for copyright infringement ? Because of who is involved. AFP and Getty are both image distributors and are the first to suffer from infringements. Getty has bought into advance technology, as well as massive resources, to fight continuous assaults on photography's rights. They have set up websites, organised talks and are involved with numerous international associations to educate the market on copyright legislation.Thus, this is a slap on the hand of one , if the not the most dedicated photography copyright enforcer in the world. Furthermore, in this case, they seem to have been the victims of AFP's mismanagement more then anything else. They receive thousands of images a day and one would assume that they would trust a state-owned company they have been working with for years to respect and honor copyright legislation. Once the images made it into their pipeline, because of the speed of distribution, it is already too late to stop.At the time, much was written about Morel's move to post some of his images on Twitter. Some had notoriously said he was "stupid" and deserved what happened to him and others protected his right to publish his images wherever he wanted, whenever he wanted. Seems the latter were proven right.
Sure, a cop can break the law and should be punished if he does. Getty was found guilty and is paying a hefty price for not having stronger controls in place and trusting their partners too much. You can be sure that they have changed their procedures since. AFP, who knows?Because they are a state owned company that can never fail, they might continue business as usual.
But copyright infringement continues to be rampant. Pinterest is not worried about this judgement, nor are thousands upon thousands of websites that continuously use photographs without proper approval. Flickr, or Tumblr – you pick- are currently hosting thousands, if not millions of illegal images. right now. ( including probably, ironically, Morel's images). They don't get caught, judged or punished. In fact, they are valued at billions of dollars. Do they do anything to educate or protect copyright? Nope. Nor do they care.
So yes, AFP, and by consequence, Getty should have not done what they did. I think everyone agrees on that. Should they have been punished. Probably. Should the photo community rejoice ? No. Because neither AFP nor Getty are the bad guys. They are a very far cry from being any threat to photography copyright, especially compared to social media. Sure, this judgment will make images taken from Twitter to be republished or sold less likely but it is certainly not a win for the photography world.
Hopefully Morel and his lawyers will donate some of their earnings to the victims of the earthquake in Haiti. At least, like that, some good would have come out of this.
@melchp 'The fault here is clearly and solely in the hands of that one over-zealous AFP photo editor' So not true.
Gabe Palacio liked this on Facebook.


In Stampin' Up!'s most recent catalog, you'll find exclusive stamp sets and stunning accessories unrivaled anywhere else in the stamping world.
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text:
Nodalla (Costachillea) eatoni is een insect uit de familie van de Berothidae, die tot de orde netvleugeligen (Neuroptera) behoort.

Nodalla (Costachillea) eatoni is voor het eerst wetenschappelijk beschreven door McLachlan in 1898.

Berothidae


Santiam Christian School is a private high school located in Adair Village, OR and has 611 students in grades Pre-Kindergarten through 12th. Santiam Christian School is the 9th largest private high school in Oregon and the 839th largest nationally. It has a student teacher ratio of 14.0 to 1.
You can also use the form below to search for alumni info and Santiam Christian School students.


Marking a truly astounding moment, the Philae lander, built by the European Space Agency (ESA), soft landed on comet 67P/Churyumov–Gerasimenko today, Nov. 12, while traveling some 500 million kilometers (300 million miles) from Earth at over 41,000 mph on a mission of cutting edge science to elucidate our origins.
Confirmation of Philae's momentous soft landing at the Agilkia landing site came at about 16:03 GMT/17:03 CET/11:03 a.m. EST, during a live broadcast from ESA mission control at the European Space Operations Center (ESOC) in Darmstadt, Germany.
"Philae is in great shape," announced Rosetta Flight Director Andrea Accomazzo.
"We definitely confirm the lander is on the surface and is talking to us," he said soon after landing.
Jubilation broke out in mission control when the confirmatory signals were received back on Earth after traveling at the speed of light, about 28 minutes after the history-making landing actually took place.
"We saw the touchdown signal," said Stephan Ulamec, Philae Lander Manager at the DLR German Aerospace Center.
And as if the tension wasn't already thick enough, Philae had to land on the "head" of the bizarre, two-lobed comet while it was rotating. Comet 67P measures only 4 kilometers wide (2.5 miles).
Agilkia was selected as the primary landing site only about six weeks ago, after Rosetta and Philae arrived in early August. The landing site is only about 500 meters across.
The signal confirmation was transmitted via ESA's accompanying Rosetta orbiter to ESOC. The first data from Philae was transmitted to the lander's Science, Operations and Navigation Centre at France's CNES space agency in Toulouse.
However, the exact status of Philae was still being determined several hours later.
It's not known if the lander is securely bound to the comet's surface, since its set of two harpoons apparently did not fire as planned and as was initially thought.
Indeed it may have lifted off, turned, and touched down again.
But Philae has transmitted some data gathered by several science instruments from the comet's surface, before Rosetta went below the horizon and lost contact as planned.
"ESA is the first to have done this to land on a comet," said Jean-Jacques Dordain, ESA's Director General, during the broadcast.
Comet 67P is an extremely odd, two-lobed celestial body stemming from the origin of our Solar System.
Comets could hold clues to the origin of life on Earth since they brought significant quantities of water and organic molecules to the early Earth as it was evolving.
"We are rendezvousing with our origins," noted Dordain exuberantly.
Earlier this morning, Nov .12, Europe's Rosetta orbiter dropped off the tiny attached Philae landing craft for an uncontrolled seven-hour-long descent to the surface of comet 67P, which very surprisingly looks like a rubber ducky.
Philae was released from the Rosetta mother ship into the loneliness of space at 09:03 GMT/10:03 CET/4:03 a.m. EST.
Shortly thereafter, Rosetta's high-resolution OSIRIS science camera snapped a series of amazing and eerie snapshots of the departing Philae showing all three landing legs and the antenna successfully deployed. Rosetta's OSIRIS imaging system includes both a wide-angle camera and narrow-angle camera.
Philae also snapped a mesmerizing farewell parting shot of Rosetta with the lander's CIVA-P imaging system that shows one of Rosetta's 14-meter-long solar arrays and glare from the Sun.
"We took an image shortly after release," said Jean-Pierre Bibring, a lead lander scientist and principal investigator of the CIVA instrument at the IAS in Orsay, France, during the broadcast.
ESA designed and built the Rosetta/Philae mission along with significant participation from NASA in the form of a trio of science instruments.
Before releasing Philae, the team had to make a series of five critical go/no-go decisions over the past day and this morning to confirm that both spacecraft were ready and operational, and that Rosetta had accomplished the final thruster maneuver to achieve the desired orbit orientation for lander dropoff.
Almost at the last minute, a problem was detected with the lander. An issue was found with the small thruster on top that is designed to counteract the recoil of the lander's pair of harpoons to push the lander down onto the surface.
The team elected to proceed ahead anyway. It's not known at this time if the landing thruster fired or not.
If they had aborted today's landing attempt, the next opportunity would not have come until about two weeks later, at the earliest.
The separation took place as planned at 09:03 GMT (10:03 CET) at a distance of 22.5 kilometers from the center of the comet.
Philae is about the size of a washing machine. The 100-kg lander is equipped with 10 science instruments including cameras, spectrometers, chemical analyzers, and a drill.
About two hours after separation, Rosetta established contact with Philae to confirm the vehicle health and act as a relay to Earth for all the telemetry and science data.
During the descent, Philae's ROLIS (ROsetta Lander Imaging System) camera acquired an image of the Agilkia landing site from a distance of approximately 3 kilometers from the surface.
Today's events are the culmination of a 10-year-long trek through space of over 6.4 billion kilometers (4 billion miles) from launch on March 4, 2004, to rendezvous with the comet on August 6, 2014.
Before that came another decade to plan, design, and build Rosetta and Philae.
NASA, as a mission partner, also announced the agency's congratulations.
"How audacious! How exciting! How unbelievable! -is this achievement. We should relish this moment," said Jim Green, Director of NASA's Planetary Science Division,Washington, D.C, during the broadcast.
"We congratulate ESA on their successful landing on a comet today," said John Grunsfeld, astronaut and associate administrator for NASA's Science Mission Directorate in Washington, D.C., in a statement.
"This achievement represents a breakthrough moment in the exploration of our solar system and a milestone for international cooperation. We are proud to be a part of this historic day and look forward to receiving valuable data from the three NASA instruments on board Rosetta that will map the comet's nucleus and examine it for signs of water.
The exact status of Philae is still being determined and won't be known for sure until tomorrow, when contact is reestablished with the Rosetta orbiter.
The three-legged lander was to fire two harpoons and use ice screws to anchor itself to comet's surface. Philae was programmed to collect stereo and panoramic images.
The spacecraft can also drill 20 to 30 centimeters into the comet to sample below its surface and deliver the pristine material to an on-board chemistry lab where it will to be heated for detailed analysis of organics and other materials.
Philae is both battery and solar powered. It has enough battery power to survive 2.5 days. Thereafter, it requires solar power to continue operating.
"With Rosetta we are opening a door to the origin of planet Earth and fostering a better understanding of our future. ESA and its Rosetta mission partners have achieved something extraordinary today," noted Dordain.
Rock on, Philae! The ESA is really picking up speed with its achievements.
This is an astounding feat and a testament to human ingenuity. Mankind's reach has yet again been extended and the science gleaned will provide a deeper understanding of our cosmic back yard. Let's hope for some amazing surface photos!
And the pictures today are incredible!
From Mariner 4 on, what humanity's eyes have seen – wow!


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Гомотример — белок, состоящий из трёх идентичных полипептидных субъединиц.

Примеры
 Гемагглютинин вируса гриппа

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Canadian Chamber of Commerce Canada's Resource Champions
Resolute's Roger Barber named one of Canada's resource champions
Roger Barber, our general manager – Ontario Woodlands, was named a resource champion by the Canadian Chamber of Commerce in October 2015. Canada's Resource Champions Awards were created to recognize and highlight outstanding efforts by organizations and individuals who have helped to support the country's world-class resource industry, focusing on public engagement, Aboriginal development, and innovative processes to reduce environmental impacts.
PPI Awards
Leadership in safety management earns Resolute an award
In September 2015, RISI's Pulp and Paper International (PPI) presented Resolute with an award in the managing risk category. This prize honors true innovation in a company's approach to loss prevention, safety and risk management. Resolute's winning entry highlighted the company's safety management system and focus on safety as a core value. The PPI awards are the only global awards dedicated to recognizing and celebrating excellence in the pulp and paper sector.
Canada's Clean50
Clean50 kudos for Pascale Lagacé, Jim Balik, Mylène Labrie and Alice Minville
On September 22, 2015, Pascale Lagacé, Resolute's vice president of Environment and Climate Change, was named to Canada's prestigious Clean50 awards for 2016. Pascale and the Resolute Team, which includes Jim Balik, manager, Environment – U.S. Pulp and Paper Operations, Mylène Labrie, analyst, Environment and Climate Change, and Alice Minville, senior legal counsel – with strong support from Resolute President and CEO Richard Garneau – have helped drive a cultural shift that has been embraced by all employees throughout the company. The award recognizes leaders who have made the greatest contributions to sustainable development or "clean capitalism" in Canada. This is the second year in a row that Resolute has received this honor.
Best in Biz Awards International
On August 5, 2015, Resolute received a bronze Best in Biz Award (International) for Most Socially or Environmentally Responsible Company of the Year. The honor recognized the company's world-class safety performance, the certification of nearly all facility environmental management systems to ISO 14001, transparent sustainability reporting, innovative partnerships with Aboriginal peoples, community outreach, and reduction of environmental incidents. In December 2015, Resolute also won a same-category silver award under the Best in Biz North American program. Best in Biz Awards are the only global business awards program judged solely by members of the press and industry analysts.
APA – The Engineered Wood Association Safety and Health Awards
Wood products facilities rewarded for world-class safety standards
Our Resolute-LP Larouche and Saint-Prime (Quebec) joint-venture engineered wood facilities picked up prestigious 2014 safety and health awards from APA–The Engineered Wood Association in recognition of their excellent safety records as well as commitment to maintaining an injury-free workplace and respecting world-class safety standards. The APA Safety and Health awards are based on lowest injury rates and severity, greatest improvement in reducing injuries, innovations to reduce injuries, and best company record.


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