text:
\section{Introduction}

Analyses of the large-scale structure of the Universe have made huge progress during the past two decades. An unprecedented wealth of data was provided by redshift surveys of larger and larger volumes out to $z\sim 1$ -- as, e.g., 2dFGRS \citep{Colless03}, SDSS/BOSS \citep{Alam17} WiggleZ \citep{Blake11b} and VIPERS \citep{Guzzo14} -- with a further order-of-magnitude increase expected soon from new projects like DESI \citep{DESI} and Euclid \citep{Laureijs11}. In this scenario, the availability of large numbers of simulated mock redshift surveys or weak-lensing maps has become a crucial requirement, as to assess both precision and accuracy of the recovered cosmological parameters. This is producing a huge effort in the community, with the goal of improving fast simulation schemes as an alternative to full $N$-body. This is motivated not only to save computational costs, but also because an improvement of these techniques could help to shed light on the statistical properties of the Cold Dark Matter (CDM) density field.

One widespread technique for the production of such catalogues is based on the observation that the evolution of the density field can be approximated with an exponential growth starting from Gaussian initial conditions, thus resulting in a lognormal distribution \citep{Coles91}. Lognormal fields are very convenient as they can be simulated having an arbitrary correlation function that can be always written in terms of the correlation of an underlying Gaussian field, and a PDF resembling that of the density field (e.g. \cite{Agrawal17,Favole20}). Nonetheless, there is growing evidence that the PDF of the density field does not exactly follow a lognormal distribution, and could have a significant effect on the estimate of cosmological parameters \citep{Uhlemann19}. With the exception of the lognormal case, the simulation of arbitrary random fields has made little progress in cosmology. The main purpose of this paper is to introduce an improvement over the limitation of the lognormal approximation, with a scheme that allows for the simulation of a random field with an arbitrary PDF and arbitrary two-point correlation structure. Similarly to the lognormal case, where a Gaussian field is mapped onto a lognormal one, the new algorithm optimizes for the Gaussian power spectrum to be used to generate many realizations of a field that can be mapped onto a target field. The underlying idea is based on the so-called translation process theory, detailed in \cref{sec:TP}. The algorithm we implement was proposed by \cite{Shields11} in the context of one-dimensional time-series simulations, and it is dubbed Iterative Translation Approximation Method (ITAM). We extend and adapt this scheme to the case of interest of 3D simulation grids. It can be regarded as a generalization of the lognormal case to distributions where no analytical formulae are available. Unlike previous schemes implemented in cosmology, the algorithm is not based on the realizations of the field itself on the grid, but it works at the level of the correlation function, thus saving computational cost and being fairly independent of the resolution, which can be chosen appropriately. Our results are reported in \cref{sec:Results}, where we also compare the covariance matrix obtained from these improved realizations against those of the lognormal model and from $N$-body simulations. The method preserves the overall structure of the covariance matrix, as the lognormal field does, with a marginal improvement over it. We also find an analytical justification of why this structure is preserved, by arguing that it is set by the an exponential growth of structures.

In the last part of this work, \cref{sec:lagscheme}, as a further test of the algorithm we apply it to another random field, the Lagrangian displacement-divergence \citep{Bouchet1995}. This is the fundamental field for simulation schemes based on Lagrangian techniques \citep{Monaco16}, which are often used for the fast generation of mock catalogues \citep{Blot19}. Indeed they are theoretically sound, since they are based on approximate solutions for the CDM evolution, but they have some limitations in reproducing the property of the density field, such as the the amount of correlation at small scales \citep{Chuang15}, the skewness of the PDF of the matter distribution \citep{Neyrinck15}, and the cross correlation with the actual dark matter evolution that is far from ideal \citep{Munari17}. We use ITAM to simulate a displacement-divergence field with a given power spectrum and PDF, accurately reproducing the $N$-body results. We assess how much information the PDF and the power spectrum convey and compare it with other analytical techniques used in the literature. We find that the displacement field generated by ITAM performs similarly to other standard techniques, with a minor improvement manifested in more-collapsed filaments, and walls. These results suggest that more accurate Lagrangian schemes are accessible if one resorts to non-local transformations of the initial density field or higher order polyspectra.

\section{Translation Process Theory}\label{sec:TP}
\subsection{Basics}
It is now well-known that the initial conditions of the Universe closely resemble a Gaussian field. This distribution, under the influence of gravity, gives rise to complex features that are not completely characterized by the two-point correlation function alone \citep{Bernardeau01}. The evolved matter density field is nonlinear, referring both to the fact that it cannot be computed from linear perturbation theory, and that it cannot be obtained as the linear superposition of Gaussian processes. An important insight in the statistics of matter clustering was provided by the observation that the distribution of galaxies follows approximately a lognormal distribution. A lognormal field can be extrapolated as a solution from the continuity equation of the CDM evolution under the Zel'dovich approximation (ZA) \citep{Coles91}. Several works showed that applying a logarithmic transform to the nonlinear density field renders it more Gaussian at the level of the one-point distribution function \citep{Colombi94} and it can improve the constraints on cosmological parameters \citep{Neyrinck09,Repp17}. A similar effect is obtained by Gaussianizing the field, namely by rank ordering the nonlinear density at the voxel level, and then mapping it onto a Gaussian PDF \citep{Weinberg92,Neyrinck11}, without making the assumption of lognormality for the density field.

Based on these evidences, both of these approaches have been extensively used in the literature in the inverse sense: an initial Gaussian field is transformed into a lognormal field \citep{Agrawal17} or to arbitrary distributions \citep{Shirasaki17}. Fast simulations generated with such a heuristc approach have limitations. In the former case they have the exact correlation function but incorrect PDF, and in the latter they have the exact PDF but incorrect correlation function. To be more formal, let us consider a Gaussian stationary stochastic process $\delta_g(\pmb{x},\tau)$. An example of such a process is the matter density field, where stationarity is guaranteed by translational invariance. $\delta_g(\pmb{x},\tau)$ can be mapped onto the target non-Gaussian variable $\delta_{ng}(\pmb{x},\tau)$ through a monotonic transformation
\begin{equation}
	\delta_{ng}(\pmb{x},\tau) = g[\delta_g(\pmb{x},\tau)].
\end{equation}

This kind of transformation is \textit{local}, as it is a one-point mapping from a distribution to another that depends only on the local value of the field, and does not explicitly depend on its coordinates. The transformation of a Gaussian variable by means of a local nonlinear transformation has been dubbed elsewhere as a translation process \citep{Grigoriu84}, and it has the property that the new correlation structure is also translation-invariant and thus it is still a stationary stochastic process.
The most precise one-point mapping that performs the transformation from $\delta_g$ to $\delta_{ng}$ is obtained by matching their cumulative distribution functions (CDFs) $\mathcal{F}_g$ and $\mathcal{F}_{ng}$, namely:
\begin{equation}\label{eq:g}
\delta_{ng} = g[\delta_g] \equiv \mathcal{F}^{-1}_{ng} \left[ \mathcal{F}_g [\delta_g] \right],
\end{equation}
where $\mathcal{F}^{-1}$ denotes the inverse CDF. It is standard practice to use this relation to transform between two distributions. It simply consists of matching the rank-ordering of $\delta_{ng}$ to the one of $\delta_g$, and it can be generalized to arbitrary distributions (see e.g. \cite{Leclercq13}). However, there is no control over the resulting correlation structure of the nonlinear field, as we are going to address now. We start by computing the expected first two moments resulting from a monotonic transformation of this kind \citep{Grigoriu95}:
\begin{equation}
	\mu = \int_{-\infty}^{+\infty} g(\delta_g) \phi(\delta_g) d\delta_g,
\end{equation}
\begin{equation}
	\sigma^2 = \int_{-\infty}^{+\infty} (g(\delta_g)-\mu)^2 \phi(\delta_g) d\delta_g,
\end{equation}
and the correlation function
\begin{equation}\label{eq:Grigoriu}
	\begin{split}
		\xi(r) &\equiv \langle \delta_{ng}(\pmb{x})\delta_{ng}(\pmb{x}+\pmb{r}) \rangle =\\
		&\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (g(\delta_g)-\mu) (g(\delta'_g)-\mu) \phi(\delta_g,\delta'_g,\rho(r)) d\delta_g d\delta'_g,
	\end{split}
\end{equation}
where $\delta_{g}=\delta_{g}(\pmb{x})$, $\delta'_{g}=\delta'_{g}(\pmb{x}+\pmb{r})$ and we made use of the univariate and bivariate normal distributions
\begin{equation}
	\phi(x)= (2 \pi)^{-1/2} \exp{(-x^2/2)},
\end{equation}
\begin{equation}
	\phi(x,y,\rho(r))= \frac{1}{\sqrt{2 \pi(1-\rho^2(r))}} \exp{ \left(\frac{-(x^2+y^2-2 \rho^2(r) x y)}{2 \pi(1-\rho^2(r)) } \right)}.
\end{equation}

\Cref{eq:Grigoriu} explicitly connects the correlation function $\rho(r)=\xi_g(r)/\xi_g(0)$ of the pre-translation Gaussian field $\delta_g$ with the new correlation function $\xi_{ng}(r)$ of $\delta_{ng}$. This equation can always be used in a forward sense, i.e. for a given monotonic transform $g$ and a Gaussian variable $\delta_g$ we can compute the new correlation function. In practice, we would like to solve the \textit{inverse problem}: we want to find the linear correlation $\xi_g$ which allows for the generation of the pre-translation field $\delta_g$ that, under transform \cref{eq:g}, is mapped onto a field $\delta_{ng}$ with prescribed correlation $\xi_{ng}$. While the new field $\delta_{ng}$ has by construction the exact PDF, there is no reason why the resulting correlation matches the desired target. This is possible only if we can solve \cref{eq:Grigoriu} for the correlation structure $\xi_g$ of the pre-translation field in terms of $\xi_{ng}$. To clarify, one explicit example where this is instead possible is the case of the lognormal density field. The monotonic transformation in this case is written explicitly as
\begin{equation}\label{eq:ln}
	\delta_{L}(\pmb{x}) + 1 = e^{\delta_{g}(\pmb{x})-\sigma^2/2},
\end{equation}
where $\sigma$ is the variance of $\delta_{g}$, which guarantees that the mean of $\delta_{L}$ is zero. With this transformation, \cref{eq:Grigoriu} can be solved exactly, yielding
\begin{equation}\label{eq:ln2}
	\xi^{L}_{ng}(r) = e^{\xi_g(r)}-1.
\end{equation}

The advantage of a lognormal field is clear: from the previous equation we can always solve for $\xi_{g}$ for any given $\xi^{L}_{ng}$ (if $1+\xi^{L}_{ng}>0$).

More recently, \cite{Bel16} (their appendix C) solved the inverse problem by Hermite-expanding the integral \cref{eq:Grigoriu} to linear order in $\xi_g$. This gives a linear relation $\xi_{ng} \propto \xi_{g}$, from which one can always find $\xi_{g}$. This approach is also the basis for the algorithm developed in \cite{Baratta20} for generating mock galaxy catalogues. While it is an excellent approximation to describe the non-Gaussian correlation smoothed at quasi nonlinear scales, it is not optimal if one is interested in the nonlinear regime (\cref{fig:correlations}). Deviations from the linear relation become apparent when considering a density field smoothed at scales smaller than $8$ {$h^{-1}$Mpc}, when $\xi_{ng}(0)<1$. These deviations might be relevant at smoothing scales of the order of $5$ {$h^{-1}$Mpc}.

Apart from these exceptions, one might be interested in the case where the translation transform is not known analytically, like in the case of the density field PDF. Even assuming that the one-point mapping is known and that \cref{eq:Grigoriu} can be solved analytically, it might still not possible to solve explicitly for the Gaussian correlation in a non-perturbative form. Therefore, the inverse problem can be seen as an optimization problem for the correlation structure $\xi_g$ of the pre-translation field. In the following we present an algorithm that can always perform this optimization to solve the inverse problem in a very general way.

\subsection{Iterative Translation Approximation Method}
In order to solve the inverse problem for the general case when no analytical solution is available, several numerical procedures were proposed in the past. The very first one was suggested by \cite{Yamazaki88}, which we comment briefly upon, since it is the only one that received some attention in cosmology. In particular the Yamazaki-Shinozuka (YS) algorithm has been discussed thoroughly in \cite{Vio01,Brown13}. The YS scheme is an iterative procedure that works directly on sample realizations of fields. Given a nonlinear sample realization of a field, one can measure the corresponding PDF, and consequently the translation transform $g$ of \cref{eq:g}. One then needs to initialize an arbitrary random Gaussian field, on which the transformation $g$ is applied. The translated field by construction has the prescribed one-point distribution, but it generally has a different power spectrum from the desired one, because of the arbitrariness of the pre-translation power spectrum.

In order to match a target power spectrum, one needs to optimize the pre-translation field so that it gets mapped more precisely onto the target nonlinear field. Optimizing the Gaussian field is a cumbersome task, and YS attempted to solve it by iteratively updating the amplitude of the modes of the pre-translation field by a factor proportional to the amplitude of the modes of the target field. The same phases of the zero-iteration Gaussian modes are used, and the newly generated Gaussian field gets translated and compared to the target field. This procedure is repeated until the amplitude of the modes of the translated field are close to the amplitude in the target field. Unfortunately, the YS optimization has an important limitation: the updating scheme based on the actual modes of the target realization introduces non-Gaussian correlations on the amplitudes of the pre-translation Gaussian modes. This causes the pre-translation field to be non-Gaussian \citep{Deodatis01}. Furthermore, working at the level of sample realizations of the fields makes the result realization-dependent. Many different refinements to ameliorate this problem have been explored, and they are reviewed and discussed in \cite{Bocchini08}. For the sake of this work, we instead focus on a recent scheme that elegantly addresses both problems.

This scheme was proposed in \cite{Shields11}, where the discussion is developed for time series simulations. We extend and adapt the suggested scheme to three-dimensional
grid simulations. The goal is the same as outlined in the YS scheme: we choose a target PDF $\mathcal{P}(\delta_{ng})$ and we initialize an arbitrary Gaussian power spectrum $P^{(i)}_g(k)$ for the zero iteration $(i=0)$. This spectrum can be converted into a correlation function by means of an Hankel transform:
\begin{equation}\label{eq:WK}
	\xi^{(i)}_g(r) = \int_0^{\infty} \frac{dk}{2 \pi^2} P^{(i)}_g(k) k^2 \frac{\sin{(kr)}}{kr}.
\end{equation}

This correlation function is normalized ($\rho_g=\xi_g(r)/\xi_g(0)$), and inserted into \cref{eq:Grigoriu} to compute the new correlation function $\xi_{ng}$. To perform the integration, we need the monotonic mapping \cref{eq:g}. This mapping is automatically set once $\mathcal{P}(\delta_{ng})$ has been specified, and in practice one does not need directly the PDF, but only the corresponding CDF. After computing $\xi_{ng}$, one applies the inverse transform:
\begin{equation}\label{eq:inverseWK}
	P^{(i)}_{ng}(k) = 4 \pi \int_0^{\infty} d r \xi^{(i)}_{ng}(r) r^2 \frac{\sin{(kr)}}{kr}.
\end{equation}

This power spectrum can be directly compared with the target power spectrum $P(k)$. If the power fails to converge, we update the Gaussian power spectrum to be used in the next iteration as
\begin{equation}\label{eq:update}
	P^{(i+1)}_g(k) = \left(\frac{P(k)}{P^{(i)}_{ng}(k)}\right)^{\beta} P^{(i)}_g(k).
\end{equation}

The scheme is summarized in \cref{fig:FlowChart}. The convergence criterion can be based on the relative difference between the target power spectrum $P(k)$ and the ITAM output $ P_{ng}(k)$ over all the modes of the box
\begin{equation}\label{eq:stop}
	\epsilon^{(i)} = \sqrt{ \frac{\sum_j \left(P^{(i)}_{ng}(k_j) - P(k_j) \right)^2 }{\sum_{j} P^2(k_{j}) } } < \alpha,
\end{equation}
where $\alpha$ and $\beta$ are arbitrary constants. This stopping criterion is chosen to be a global one, because there is a limit on the convergence that can be achieved, which is not known a priori. Namely for a given $\mathcal{P}(\delta_{ng})$ and $P(k)$, $\epsilon$ cannot be made arbitrarily small. This happens because the PDF and the correlation function may not be compatible. There are two kinds of incompatibilities \citep{Vio01}: one arises when the target correlation has values that lie outside the range that can be reached by the correlation function corresponding to the chosen translation transform $g$. The other incompatibility arises when the target correlation is not positive definite, which is a necessary requirement for a well defined correlation function. The updating scheme \cref{eq:update} is such that a strictly positive target power spectrum must correspond to a strictly positive pre-translation spectrum. Clearly this holds true also when a Gaussian smoothing is adopted, so that the result of ITAM is always a well defined correlation funtion. Even in the case when the first kind of incompatibility is present, ITAM still provides the best one could do if the goal is to make a fast simulation with a translation process (see \cite{Shields11} for some examples).

The speed and accuracy of the convergence depend on $\beta$, which reflects the change in the updated Gaussian power. Too-high values for $\beta$ could result in an overshoot with respect to the target. Likewise, values too small could render the updating ineffective. We find that is better to choose values such as $\beta \sim 0.2 - 1.5$ and see how small $\epsilon$ can be made, and finally select the one which delivers the smallest value. This allows to select $\alpha$ so that ITAM stops when $\epsilon<\alpha$. Alternatively, ITAM can stop when the relative change $\Delta \epsilon/\epsilon$ between consecutive iterations becomes small. This is independent of how small $\epsilon$ is, and reflects more closely the fact that the algorithm converged as well as it could to the target. This fine-tuning procedure needs to be performed only once.

Ultimately, the output of ITAM is the power spectrum $P_g(k)$ that allows realizations of pre-translation Gaussian fields to be transformed into the non-Gaussian ones by means of the translation transform \cref{eq:g}. ITAM can be applied to any stationary random field of interest in a three dimensional box; we provide the code publicly\footnote{\url{https://github.com/tos-1/ITAM}}.

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/FlowChart}
	\caption{Diagrammatic representation of ITAM algorithm.\label{fig:FlowChart}}
\end{figure}

\section{Application to the Matter Density field}\label{sec:Results}

\subsection{Method}
As a demonstration, we now investigate the use of ITAM in the cosmological case of the matter density field. The target power spectrum can be generated using one of the many useful codes available in the literature. We choose to generate the target nonlinear power spectrum with \textsc{HALOFIT} \citep{Takahashi12}, implemented in \textsc{CLASS} \citep{Blas11}.

As a next step, we need a template PDF for our target field. The task of providing a comprehensive analytical approximation to the PDF of the density field is a difficult one. \cite{Klypin17} shows how the PDF depends on the simulation box size, resolution, redshift, smoothing scale and cosmological parameters. In \cite{Uhlemann16} an analytical derivation of the PDF for the quasi nonlinear scales is presented, and in \cite{Uhlemann19} its complementarity with the power spectrum on cosmological parameter constraints is shown. Both \cite{Uhlemann16} and \cite{Shin17_PDF} argue that the PDF deviates significantly from the lognormal distribution, and it is better approximated by a skewed lognormal distribution. This result is confirmed by \cite{Repp18}, who show that a Generalized Extreme Value distribution is a better fit to the matter field PDF when transitioning to nonlinear smoothing scales. Despite these efforts, there is still no general scheme that provides a fitting function, implemented in a publicly available software and with arbitrary resolution, so we prefer to use the exact PDF measured from simulations. While this could bias the results towards a specific realization rather than a template average, it is still good enough as a proof of concept.

We take two simulations as references. The first is the Millenium run (ML) \citep{Millenium}, a simulation of size $500$ {$h^{-1}$Mpc}, with $2160^3$ particles, and with a WMAP cosmology $h=0.73$, $\Omega_b = 0.045$, $\Omega_m = 0.205$, $\Omega_{\Lambda}=0.75$, $n=1$ and $\sigma_{8}=0.9$. We use a $256^3$ nearest-grid-point density grid, provided by the ML database. We also consider an $N$-body simulation with a smaller box size (hence SB), of size $256^3${$h^{-1}$Mpc}\ and $256^3$ particles. The simulation was run with Gadget \citep{Gadget}, starting at an initial redshift of $z=50$, and with initial conditions set by second-order Lagrangian perturbation theory (2LPT). The fiducial cosmology assumed is a vanilla $\Lambda$CDM cosmological model, with $h=0.7$, $\Omega_b = 0.046$, $\Omega_{\rm CDM}=0.25$, and $\sigma_{8}=0.8$. The amplitude of the modes of the initial conditions is set to be precisely the ensemble-mean power spectrum, following the suggestion of \cite{Angulo16}, to have a better convergence on the ergodic properties. While fixing the variance does not improve the convergence on the ensemble-mean PDF, it was shown not to bias it for smoothing scales as small as $0.5$ {$h^{-1}$Mpc}\ \citep{Klypin20}. For both simulations, we sample the matter field by the grid resolution, which is $R_s = 1.95 \ h^{-1} Mpc$, $R_s = 1.0 \ h^{-1} Mpc$ for the ML and SB simulations, respectively. The smoothing is performed with a Gaussian kernel
\begin{equation}\label{eq:smoothing1}
	\delta(k,\sigma_s) = \delta(k) e^{\frac{-k^2 R_s^2}{2}},
\end{equation}
and consequently the target power spectrum has to be smoothed as well
\begin{equation}\label{eq:smoothing2}
	P(k,\sigma_s) = P(k) e^{-k^2 R_s^2}.
\end{equation}

For the algorithm to converge accurately, it is important to test beforehand that the variance of the PDFs is consistent with that computed from their target power spectra, as this could result in an offset of the normalization of the power spectrum. The variance of the PDFs can be directly computed as the cell density variance of \cref{eq:smoothing1} in real space, while the variance of the correlation function is simply \cref{eq:WK} computed at $r=0$, with the addition of the smoothing Gaussian kernel in the integration. In case there is an offset, the correlation function should be rescaled to make them consistent.

The power spectrum must be smoothed, to limit the variance of the field; without smoothing, \cref{eq:WK} would diverge when evaluated at $r=0$. Alternatively, the smoothing can be avoided by considering the PDF of the field sampled at the grid resolution. In this case, there is an implicit filtering to be considered, i.e. a sharp cut-off of the target power spectrum at the Nyquist mode of the box. In the following section we will show how ITAM performs in both cases.
\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/CJ_rel_diff_spectra}
	\caption{ Results of ITAM for the mapping \cref{eq:CJ}. The top panel shows the power spectrum achieved by ITAM, while in the bottom its relative difference with respect to the target is shown (blue line). The pre-translation Gaussian spectrum that results from the optimization process is also shown, and its relative difference with respect to the Gaussian one predicted by \cref{eq:ln2} is plotted in the bottom (yellow line). The smoothing scale considered is 1 {$h^{-1}$Mpc}. \label{fig:CJ_rel_diff_spectra} }
\end{figure}

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/log_rel_diff_spectra}
	\caption{ Results of ITAM for the mapping \cref{eq:CJ_offset}. The top panel shows the nonlinear power spectrum that can be achieved with ITAM (see legend). In the bottom it is shown the relative difference with respect to the target (dashed blue line). The \textit{expected itam} curve shows the theoretical prediction for the nonlinear spectrum computed from \cref{eq:ln4}, with its relative difference with respect to \textit{itam} shown in the bottom panel (solid red line). The pre-translation spectrum is offset with respect to the translated one (\cref{eq:ln4}). The smoothing scale considered is 1 {$h^{-1}$Mpc}. \label{fig:log_rel_diff_spectra}}
\end{figure}

\subsection{Results}
Before applying ITAM to the case of the exact PDF measured from $N$-body simulations, we apply it by assuming a lognormal PDF for the target field, for which we can derive the exact solution of the inverse problem. To run ITAM we first initialize a pre-translation Gaussian power spectrum. The choice is not relevant as long as it is a sensible one -- e.g. one could choose the very same target spectrum or the linear density power spectrum for the initialization. Following the scheme outlined in \cref{fig:FlowChart}, we transform this power spectrum in real space to compute \cref{eq:Grigoriu}, for which we need to specify the translation transform \cref{eq:g}, namely the combination of the CDF of a zero-mean Gaussian variable and the inverse CDF for the associated lognormal
\begin{equation}\label{eq:CJ}
	\delta^L_{ng} = e^{-\sigma^2/2} \mathcal{F}_L^{-1}\left[\mathcal{F}_g[\delta_g]\right]-1.
\end{equation}

The prefactor is necessary to adjust the inverse CDF of the lognormal to map onto a zero mean field. This mapping effectively corresponds to \cref{eq:ln}, as we show in \cref{fig:CJ_rel_diff_spectra}.
As a separate test, we implement a transform where we ignore the prefactor $e^{-\sigma^2/2}$, so that the mapped field is a lognormal field with nonzero mean. We can compute the expected mean $\mu$ that must be subtracted in \cref{eq:Grigoriu} by considering that in this case the lognormal transform is
\begin{equation}
	\delta^L_{ng} +1 = e^{\delta_g},
\end{equation}
whose expected mean is
\begin{equation}
	\mu = e^{\sigma^2/2} - 1,
\end{equation}
so that the corresponding mapping to a zero mean lognormal field is
\begin{equation}\label{eq:ln3}
	\delta^L_{ng} = e^{\delta_g} - e^{\sigma^2/2}.
\end{equation}

In analogy to \cref{eq:CJ}, the translation transform we implement in this case is
\begin{equation}\label{eq:CJ_offset}
	\delta^L_{ng} +1 = \mathcal{F}_L^{-1}\left[\mathcal{F}_g[\delta_g]\right]-e^{\sigma^2/2}+1,
\end{equation}
which is still set by matching the CDF of the Gaussian to the CDF of the lognormal, knowing that the expected mean is $\mu=e^{\sigma^2/2}-1$. While this lognormal field does not correspond to a density field, having $\delta^{L}_{ng}<-1$, it is still useful to consider, as we can use \cref{eq:ln3} to compute the expected correlation function to make a comparison with the numerical results of ITAM. The expected correlation function is
\begin{equation}\label{eq:ln4}
	\xi^L_{ng}(r) = e^{\sigma^2} (e^{\xi_{g}(r)} -1)
\end{equation}
(see \cite{Xavier16} for a derivation of correlations for general lognormal fields). In this case we cannot solve explicitly for the pre-translation Gaussian correlation, as also the prefactor depends on the Gaussian correlation via $\sigma^2=\xi_g(0)$. However, we can still check whether the optimized pre-translation correlation function satisfies \cref{eq:ln4}; this is shown in \cref{fig:log_rel_diff_spectra}. It is worth noticing that in both cases examined above, $\sigma$ varies during the optimization process as well, as it is the standard deviation of the pre-translation field.\\
\indent We proceed to examine the results of ITAM when the input PDF is measured from the SB and ML simulations. We already know that the nonlinear density field has zero mean, so that in order to fully specify the translation transform \cref{eq:g}, it is enough to use the Gaussian CDF together with a lookup table of the inverse CDF of the smoothed target field. ITAM can be run after both the initialization and target power spectra have been chosen. The result of the optimization can be seen in real space in \cref{fig:correlations}, which shows that it is equivalent to solve $\xi_{ng}$ from \cref{eq:Grigoriu} as a function of $\xi_g$. An example of the PDF of a density field generated with ITAM is shown in \cref{fig:PDFs}, compared with the approximation of a lognormal realization.\\
\indent
In \cref{fig:Gauss_rel_diff_spectra} we compare the output power spectrum that can be obtained by ITAM relative to the target. We can see that for both PDFs measured from our two simulations, SB and ML, percent accuracy is reached up to half of the Nyquist mode of the box, with a more precise result for the SB simulation. This is due to the fact that the convergence to the target spectrum is affected by the smoothing scale. The steep cutoff introduced by the smoothing can be seen in the drop of the residuals plotted at the bottom of \cref{fig:Gauss_rel_diff_spectra}. In the case the PDF is measured from a simulation, the choice of the smoothing scale cannot be arbitrary, but depends on the mass assignment scheme used to interpolate the density on the grid, and on the sampling rate of the simulation. We find that smoothing at the scale of the sampling rate of the simulation guarantees the best convergence for the optimization.
In the case we use the PDF of the field sampled at the grid resolution, the overall convergence becomes less accurate, but the target power spectrum is recovered with increased precision up to the Nyquist mode, as shown in \cref{fig:shapcoff}. It seems that there is an interplay between large scale and small-scale power: the larger the small scale power, the more difficult the global convergence to the target, perhaps because a PDF with a larger tail is more difficult numerically. For the sake of precision cosmology, we think the Gaussian smoothing is preferable: one can always employ the PDF from a high-resolution simulation to compensate for the loss of signal at smaller scales, and eventually downsample the grid to reach a better accuracy up to the Nyquist mode for the fast realizations.\\
\indent To prove that the Gaussian filtering does indeed work for smaller smoothing scales than those considered until now, we run ITAM in the case of the lognormal transform \cref{eq:CJ} examined before. In this case we are not limited by the accuracy of the sampled PDF, as it is analytical, and we can adopt arbitrary smoothing scales, shown in \cref{fig:rel_resolution}. This confirms that ITAM can in principle be pushed to smaller scales than the ones we examined with finer grids and higher-resolution PDFs. ITAM is independent of the resolution, which means that its output power spectrum can be used to make realizations at the desired sampling rate, so that is always possible to adjust the grid in order to recover a realization accurate up to the Nyquist mode.\\
\indent From \cref{fig:Gauss_rel_diff_spectra} we notice that the pre-translation Gaussian spectrum output from ITAM has an offset with respect to the target power spectrum, even on large scales. This offset can also be seen in \cref{fig:correlations} in the Gaussian correlation, and it is due to the fact that we did not add a scaling factor, which is present in the case of \cref{eq:CJ}. In fact, for the actual PDF of the density field we ignore what is the correct rescaling, but one can see that it is related to the skewness of the distribution. While in the lognormal case the skewness is set by specifying the variance of the underlying Gaussian field, distributions more skewed than the lognormal require additional parameters to characterize them \citep{Shin17_PDF,Repp17}, and they break this one to one correspondence. From \cref{fig:Gauss_rel_diff_spectra} one can see that a more skewed distribution results in an increased offset, namely a scaling factor that has to be smaller than $e^{-\sigma^2/2}$ of \cref{eq:CJ}. Ultimately, this rescaling is only convenient for visualization purposes, but it is not relevant for the aim of generating nonlinear fields, as ITAM can correctly take it into account in the optimization process. We caution that these pre-translation fields should not be taken to be physically meaningful for cosmological purposes.

\begin{figure}
	\includegraphics[width=0.43\textwidth]{images/correlations}
	\caption{Results of the optimization process of ITAM in real space. The target power spectrum and the N-body PDF from the SB simulation were both smoothed at 1 {$h^{-1}$Mpc}. The result for the lognormal PDF was also obtained by running ITAM with the transform \cref{eq:CJ}. At this smoothing scale, we can see that a linear relation $\xi_{ng}(r) \propto \xi_{g}(r)$ holds for $\xi_g \lesssim 1$, which corresponds to $r\gtrsim6$ {$h^{-1}$Mpc}. \label{fig:correlations}}
\end{figure}

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/PDFs}
	\caption{PDF of an ITAM density field realization compared to a lognormal density field realization, and to the expected target PDF (dashed line). The smoothing scale for both realizations is 1 {$h^{-1}$Mpc}.\label{fig:PDFs}}
\end{figure}

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/Gauss_rel_diff_spectra}
	\caption{The top panel shows the pre-translation spectra and the nonlinear spectra output from ITAM, for both the simulations. The bottom panel shows the relative difference with respect to the target nonlinear spectra. The smoothing scale introduces a cut-off that reduces the convergence on smaller scales. Also, the increased smoothing scale adopted in the ML simulation (2 {$h^{-1}$Mpc}) with respect to the SB simulation (1 {$h^{-1}$Mpc}) manifests itself as a smaller offset between the pre-translation spectrum and the target spectrum, thus showing that it is due to the skewness of the distribution. \label{fig:Gauss_rel_diff_spectra}}
\end{figure}

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/residuals_sharp_coff.png}
	\caption{Residuals of the ITAM power spectrum with respect to the target, in the case of the PDF from the SB simulation. The field has not been filtered, while a sharp cut-off for the target power spectrum in correspondence with the Nyquist mode of the box is adopted. While in the unsmoothed case ITAM is able to recover the signal better up to the Nyquist mode, the global convergence gets worse with respect to the Gaussian smoothing case. \label{fig:shapcoff}}
\end{figure}

\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/rel_resolution}
	\caption{Tests of ITAM in the case of a lognormal target PDF for a simulation of box size 1000 {$h^{-1}$Mpc}, by varying the smoothing scale $R_s$. \label{fig:rel_resolution}}

\end{figure}

\subsection{Covariance Matrix from ITAM}
Having optimized for the linear power spectrum, we can generate many samples of Gaussian fields to transform into the target density field. The skewness and the kurtosis of the PDF are related to integrals of the bispectrum and the trispectrum, so we expect the ITAM scheme to provide a small improvement over the standard lognormal case for the corresponding higher-order polyspectra. We test this hypothesis at the level of the covariance matrix of the power spectrum, defined as
\begin{equation}
	C_{ij} = \frac{1}{N-1} \sum^{N}_{n=1} \left( P(k_i) - \overline{P}(k_i) \right) \left( P(k_j) - \overline{P}(k_j) \right),
\end{equation}
where $N$ is the number of realizations, and $\overline{P}(k_i)$ is their average power spectrum, estimated for each simulation as
\begin{equation}
	\overline{P}(k) = \frac{1}{N_k} \sum_{|\pmb{k}| \in k} |\delta(\pmb{k})|^2,
\end{equation}
where $N_{k}$ is the number of independent modes in the sum. For the PDF, we use the ML simulation, as it has a bigger volume. To validate our covariance matrix, we choose as a theoretical model the phenomenological description found by \cite{Neyrinck11_cov}. There, the matter power-spectrum covariance from $N$-body simulations was found to be accurately approximated by
\begin{equation}\label{eq:covariance}
	C_{ij} = \delta_{ij} 2 \frac{P^2(k_i)}{N_{k_i}} + \alpha P(k_i) P(k_j),
\end{equation}
hence we refer to this model as $\alpha$ \textit{model}. This model was later revisited by \cite{Mohammed14,Carron15}. In \cite{Neyrinck11_cov} the $\alpha$ parameter was fitted against the covariance of a suite of $N$-body from the Coyote Universe simulations \citep{Lawrence10}, but its value could also be estimated directly from the cell density variance of each realization as $\alpha={\rm Var}(\sigma^2_{\rm cell})/\langle\sigma^2_{\rm cell}\rangle^2$, with $\sigma^2_{\rm cell}$ being the cell density variance of the field, and $\langle\sigma^2_{\rm cell}\rangle$ the corresponding ensemble average over the realizations. In order to make a comparison we generate $5000$ realizations, both using ITAM and lognormal approximation, and we compute their power spectra, their covariance and the corresponding $\alpha$ parameters.

\begin{figure*}
	\includegraphics[width=0.95\textwidth]{images/off_diagonal}\caption{ Cut through the correlation matrix for the three approximations examined; the fixed mode is indicated at the top of each panel. \label{fig:off_diagonal}}
\end{figure*}

We compare the structure of the covariance matrices by analyzing both the off-diagonal (\cref{fig:off_diagonal}) and diagonal elements (\cref{fig:diagonal}). We can see that either cases retain a similar shape to the $\alpha$ model, with a minor improvement of the ITAM case over the lognormal one. In \cref{fig:Tij}, both the lognormal and ITAM follow rather well the $N$-body fit provided by the $\alpha$ model, shown in terms of the non-Gaussian contribution to the covariance. This is confirmed by the $\alpha$ values that we measured for either approximation schemes in two ways, both by employing $\alpha={\rm Var}(\sigma^2_{\rm cell})/\langle\sigma^2_{\rm cell}\rangle^2$ at the level of sample realizations, and by fitting the functional form \cref{eq:covariance} to the covariance matrices. In the case of the lognormal, we find $0.0009$ for the former and $0.0014$ for the latter method, while in the case of the ITAM scheme we have $0.0013$ and $0.0014$ respectively, to be compared against the reference value $0.0011$ found by fitting \cref{eq:covariance} to the covariance of the Coyote Universe simulations \citep{Lawrence10} used by \cite{Neyrinck11_cov}.
\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/diagonal}\caption{
		Diagonal elements of the covariance matrix for the three approximations examined in the text. \label{fig:diagonal}}
\end{figure}

\subsection{Relation to multiplicative bias model}\label{subsec:bias}
\cite{Neyrinck11_cov,Carron15} show that the approximation to the covariance provided by \cref{eq:covariance} is exact for a $b \delta$ field, where $\delta$ is a Gaussian field with power spectrum $P(k)$ and $b$ is a multiplicative `bias' field, i.e.\ a locally constant factor multiplying the density field, which varies from patch to patch, such that $\langle b^2 \rangle=1$ and $\langle b^4 \rangle-\langle b^2 \rangle^2= \alpha$.

The fact that translation processes (ITAM, and the lognormal models) approximate the covariance in \cref{eq:covariance} suggests that the bias fluctuations can be ascribed to the monotonic mapping of the initial density fluctuations. The non-Gaussian components of ITAM and lognormal covariances retain a very similar structure to the $\alpha$ model approximation, as shown in \cref{fig:Tij}. Moreover, the covariance structure of an $N$-body density field can be largely removed by Gaussianizing or taking the log-density of the field (e.g. fig. 2 of \cite{Neyrinck11_cov}). In the case of the lognormal field, the exact computation can be performed, and we can attempt to see whether the form \cref{eq:covariance} is an approximation for its covariance.

We already considered the relation between lognormal and Gaussian fields (\cref{eq:ln}) in terms of their correlation functions (\cref{eq:ln2}). It is possible to generalize the relation between the correlation functions of the two fields to arbitrary orders \citep{Coles91}:
\begin{equation}\label{eq:highcorr}
\begin{split}
	\langle (\delta^L_{ng}(\pmb{x}_1) + 1)&...(\delta^L_{ng}(\pmb{x}_n) + 1) \rangle =\\
	&\exp{\left( \sum_{i<j}^n \xi_g(|\pmb{x}_i-\pmb{x}_j|) \right)} = \prod_{i<j}^n \left[1+\xi(|\pmb{x}_i-\pmb{x}_j|) \right],
\end{split}
\end{equation}
where $\xi$ is now used to refer to the two-point correlation function of the lognormal field. We also know that the three-point correlation of the lognormal field can be written as
\begin{equation}\label{eq:3point}
\begin{split}
	\langle (\delta^L_{ng}(\pmb{x}) + 1)&(\delta^L_{ng}(\pmb{x}+\pmb{r}_1) + 1)(\delta^L_{ng}(\pmb{x}+\pmb{r}_2) + 1)) \rangle=\\
	&\Gamma(\pmb{r}_1,\pmb{r}_2) + \xi(r_1)+ \xi(r_2)+ \xi(r_{12})+1,
\end{split}
\end{equation}
where $\Gamma$ is the connected three-point correlation function. By combining \cref{eq:highcorr} and \cref{eq:3point}, we can solve for the connected component
\begin{equation}\label{eq:connected3point}
\begin{split}
	\Gamma(\pmb{r}_1,\pmb{r}_2)&= \xi(r_1)\xi(r_2)+\xi(r_1)\xi(r_{12})+\xi(r_2)\xi(r_{12})\\&+\xi(r_{1})\xi(r_{2})\xi(r_{12}).
\end{split}
\end{equation}

In the same way we compute the fourth order correlation, which depends not only on the two-point correlation function, but also on the three-point connected component
\begin{equation}\label{eq:4point}
\begin{split}
\langle(\delta^L_{ng}(\pmb{x}) &+ 1)(\delta^L_{ng}(\pmb{x}+\pmb{r}_1) + 1)(\delta^L_{ng}(\pmb{x}+\pmb{r}_2) + 1))(\delta^L_{ng}(\pmb{x}+\pmb{r}_3) + 1)) \rangle\\&=
	1 + \xi(r_{1}) + \xi(r_{21}) + \xi(r_{2}) + \xi(r_{31}) + \xi(r_{32}) + \xi(r_{3})\\ &+ \xi(r_1)\xi(r_{32})+\xi(r_{21})\xi(r_{3})+\xi(r_2)\xi(r_{31})\\ &+ \Gamma(\pmb{r}_1,\pmb{r}_2)+\Gamma(\pmb{r}_1,\pmb{r}_3)+\Gamma(\pmb{r}_2,\pmb{r}_3)+\Gamma(\pmb{r}_{21},\pmb{r}_{31})\\ &+\Delta(\pmb{r}_1,\pmb{r}_2,\pmb{r}_3).
\end{split}
\end{equation}

Solving for the connected part of the four-point correlation function by retaining only the lowest order terms $O(\xi^3)$, we obtain
\begin{equation}
\begin{split}
\Delta(\pmb{r}_1,\pmb{r}_2,\pmb{r}_3) &= \xi(r_1)\xi(r_2)\xi(r_3) + \xi(r_1)\xi(r_2)\xi(r_{31})\\ &+ \xi(r_1)\xi(r_2)\xi(r_{32}) + 13 \textnormal{ perm.} + O(\xi^4).
\end{split}
\end{equation}

The same equation is obtained in \cite{Joachimi11} in the context of the two-dimensional convergence field (eq. B11 in appendix B2). The Fourier transform of the four-point connected component $\Delta$ corresponds to the trispectrum $T(\pmb{k}_1,\pmb{k}_2,\pmb{k}_3)$. The trispectrum contribution to the covariance is obtained by considering its parallelogram configurations, yielding
\begin{figure*}
	\includegraphics[width=0.95\textwidth]{images/Tij}\caption{ Non-Gaussian part of the covariance matrix, $C^{NG}_{ij} = C_{ij}-2\delta_{ij}P^2(k_i)/N_{k_i}$. The black lines denote the isocontours. \label{fig:Tij}}
\end{figure*}
\begin{equation}
\begin{split}
T(k_{1},k_{2}) &= P(k_1)P(k_2) \left[\phantom{\frac{|}{|}} 2 P(k_{1}) + 2 P(k_{2}) \right. \\&\left. + 2P(k_1+k_2)+2P(|k_1-k_2|) \right.\\ & \left. +\frac{P(k_1)}{P(k_2)}P(k_1+k_2) + \frac{P(k_1)}{P(k_2)}P(|k_1-k_2|) \right.\\ & \left. + \frac{P(k_2)}{P(k_1)}P(k_1+k_2) +\frac{P(k_2)}{P(k_1)}P(|k_1-k_2|) \right] + O(P^4).
\end{split}
\end{equation}

We obtained that the trispectrum contribution to the covariance of a lognormal field scales as $T(k_1,k_2) \propto P(k_1)P(k_2)f(k_1,k_2)$. If we speculate that $f(k_1,k_2)$ is approximately constant, this matches the non-Gaussian form of the $\alpha$-model covariance. While this far from a proof, it suggests that the ansatz of \cref{eq:covariance} may be justified in view of the exponential growth of structure. In fact, a more accurate fit to the covariance matrix measured from $N$-body simulations still can be written as $C(k_1,k_2) = P(k_1)P(k_2)g(k_1,k_2)$, namely a non trivial modulation of equation \cref{eq:covariance} \citep{Klypin18}.

\section{Application to the Displacement Field}\label{sec:lagscheme}
In this section we apply the ITAM simulation scheme to another random field, the divergence of the Lagrangian displacement field $\psi=\nabla \cdot \pmb{\Psi}$. This application not only serves as a proof of concept of the algorithm, but it is worth considering also in view of the relevance of $\psi$ to perform fast simulations of the matter density field \citep{Munari17}. Indeed, most of these schemes are based on approximations of $\psi$ to describe the evolution of dark matter particles in simulations. Thanks to ITAM, we can accurately reproduce $\psi$ measured from a simulation by retaining only its PDF and correlation structure, thus allowing us to elucidate the information content they bear compared with other semi-analytical techniques available in the literature. We first briefly summarize some of these analytical approximations for $\psi$.

\subsection{2LPT}
The evolution of preassureless particles evolving in an expanding Universe is given by the solution of \citep{Bernardeau01}
\begin{equation}\label{eq:fundamental}
	\frac{d^2}{d \tau^2} \pmb{x} + \mathcal{H} \frac{d}{d \tau} \pmb{x} = -\nabla \Phi(\pmb{x}),
\end{equation}
where we used $d \tau = dt / a$, $\mathcal{H}= H/a$. $H$ and $a$ represent the Hubble rate and the scale factor respectively, while $\Phi$ is the peculiar gravitational potential. In the Lagrangian picture of structure formation one solves for the displacement field $\pmb{\Psi}$, namely the mapping between Lagrangian coordinates $\pmb{q}$ onto Eulerian coordinates $\pmb{x}$:
\begin{equation}\label{eq:master}
	\pmb{x(q)} = \pmb{q} + \pmb{\Psi}(\pmb{q}).
\end{equation}

In Lagrangian perturbation theory one seeks a perturbative solution for \cref{eq:fundamental} of the form
\begin{equation}
	\pmb{\Psi}= \pmb{\Psi}^{(1)}+\pmb{\Psi}^{(2)}+ ...
\end{equation}

In practice we do not solve for the displacement directly, but we solve for its divergence $\psi^{(1)} = \nabla \cdot \pmb{\Psi}^{(1)}(\pmb{q},\tau)$ \citep{Buchert92, Bouchet1995}. Specifically, the famous ZA is the first-order solution
\begin{equation}\label{eq:zeld}
	\psi^{(1)} = -\delta^{(1)}(\pmb{q},\tau) = -\delta^{(1)}(\pmb{q}) D_1(\tau),
\end{equation}
where in the last equality we exploited the fact that the growth function $D_1$ at linear order is separable from the spatial part, and is given by the standard equation
\begin{equation}
D_1'' +\mathcal{H} D_1'= \frac{3}{2}  \mathcal{H}^2 \Omega_m(\tau) D_1(\tau).
\end{equation}

The ZA is often used to generate initial conditions of $N$-body simulations. Indeed, by having an initial Gaussian power spectrum we can generate a primordial density field $\delta^{(1)}(\pmb{q})$, which can be evolved to the actual displacement by integrating \cref{eq:zeld}. This is possible only under the assumption that the field is \textit{irrotational}, namely there exists a scalar potential $\phi^{(1)}(\pmb{q})$ such that
\begin{equation}
\pmb{\Psi}^{(1)} = -\nabla \phi^{(1)}.
\end{equation}

The assumption of irrotationality has been studied thoroughly by \cite{Chan2014}, and has been confirmed to be accurate up to $k \sim 1 \ h/Mpc$, even at $z=0$. To increase the precision of the initial conditions, the perturbative solution is found at higher orders, for example at second order:
\begin{equation}\label{eq:2lpt}
	\psi^{(2)}(\pmb{q},\tau) = \frac{D_2(\tau)}{2 D^2_1(\tau)} \sum_{i \neq j} \left( \phi^{(1)}_{i,i} \phi^{(1)}_{j,j} - \phi^{(1)}_{i,j} \phi^{(1)}_{j,i}\right),
\end{equation}
where $D_2$ is the second order growth factor. Fitting functions for the growth factors are respectively \citep{Carrol1992}
\begin{equation}
	D_{1}(a) \simeq \frac{5}{2} \frac{a \Omega_m(a)}{\Omega_m(a)^{4/7}-\Omega_{\Lambda}(a)+(1+\Omega_m(a)/2)(1+\Omega_{\Lambda}(a)/70)},
\end{equation}
and \citep{Bouchet1995}
\begin{equation}
	D_2 \simeq -\frac{3}{7} D^2_1.
\end{equation}

Also for the second order we can assume irrotationality, which allows us to define the second order displacement potential
\begin{equation}
    \pmb{\Psi}^{(2)} \equiv \nabla \phi^{(2)}.
\end{equation}

To summarize, the displacement in 2LPT is:
\begin{equation}
	\pmb{x} = \pmb{q} - \nabla \phi^{(1)}(\pmb{q},\tau) + \nabla \phi^{(2)}(\pmb{q},\tau).
\end{equation}

2LPT is ubiquitous in implementations of fast simulation schemes, being the basis for constrained realizations of density field \citep{Kitaura13,Jasche13,Leclercq15}, for fast mock catalogue realizations \citep{Monaco13,Stein18}, or hybrid approaches \citep{Tassev13}.

\subsection{{\scshape muscle}}
Despite its success in the perturbative regime, 2LPT becomes inaccurate when density perturbations grow large, at high resolution and low redshift \citep{Matsubara08a}.
In this non-perturbative regime, 2LPT produces extreme particle overcrossings at high density, and also overestimates densities at low densities, even producing overdense clumps at what should be extreme underdensities  \citep{Sahni1996, Neyrinck13}.

An improvement over the ZA and 2LPT approaches was proposed by \cite{Neyrinck13}. He adapted the Eulerian spherical approximation formula introduced by \cite{Bernardeau1994b}, to a form previously found by \citep{Mohayaee1996}, for the evolution of isolated spherical perturbations to describe the Lagrangian evolution of particles:
\begin{equation}\label{eq:sc}
    \psi_{sc}(\tau) =  3 \left[ \left(1-\frac{\delta^{(1)}(\tau)}{\gamma} \right)^{\gamma/3} -1 \right].
\end{equation}

The parameter $\gamma$ here is related to the spherical-collapse density; following e.g.\ \citet{Neyrinck13}, we adopt a value $\gamma=3/2$. This formula belongs to the broader family of local Lagrangian mappings \citep{Protogeros97}, as it links the value of the density at position $\pmb{q}$ and time $\tau$ with the new value at the same position in Lagrangian coordinates. The spherical collapse formula improves the cross correlation with $N$-body simulation at the level of smaller scales, but it fails to reproduce the large scale displacements of 2LPT.   This impediment was overcome by \cite{Kitaura13_ALPT}, with Augmented Lagrangian Perturbation Theory (ALPT). Here, an interpolating smoothing kernel is a applied to the divergence displacement in order to combine the spherical collapse at small scales with 2LPT at large scales. This method requires a fitting parameter for this separation scale. Providing an alternative, \cite{Neyrinck15} proposed a non-perturbative, parameter-free technique that has a similar performance to ALPT. This approach is called MUltiscale Spherical ColLapse Evolution ({\scshape muscle}), where the condition for collapse is checked in the initial density field on increasingly larger smoothing scales. If at some smoothing radius $R_c$ it occurs $\delta(\pmb{q},R_c)>\gamma$, $\psi(\pmb{q})=-3$ is set at that voxel, indicating a halo particle as it is observed in $N$-body simulations \citep{Neyrinck13}. The multi-scale approach of {\scshape muscle}\ accounts for the void-in-cloud problem, which leads to retrieving the large scale displacement, and can be summarized as follows
\begin{equation}\label{eq:muscle}
\psi_{msc}(\pmb{x})=
\begin{cases}
3\left[ \left(1-\frac{\delta_l}{\gamma} \right)^{\gamma/3} -1. \right] \ \ \ \ \delta_l(R)<\gamma, \ \forall R, \\
\\
-3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \delta_l(R_{c}) > \gamma.
\end{cases}
\end{equation}

We have updated the previous code for these schemes, putting it in a new github repository\footnote{\url{https://github.com/tos-1/MUSCLE}}.
\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/psi_PDFs}\caption{PDFs of $\psi$ for various schemes. The peak at $\psi \sim -3$ is associated with collapsed halo particles. In {\scshape muscle}, halo particles give a Dirac-delta function at $\psi = -3$ (see \cref{eq:muscle}). \label{fig:psi_PDFs}}
\end{figure}
\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/psi_transfer}\caption{ Transfer function of $\psi$, defined as the ratio of the power spectrum of $\psi$ generated from the fast simulation scheme over the one measured from the $N$-body realization. \label{fig:psi_transfer}}
\end{figure}
\begin{figure}
	\includegraphics[width=0.45\textwidth]{images/psi_crossntransfer}\caption{ The top panel shows the cross correlation between the full-gravity matter field and those resulting from the approximations. The bottom panel shows the transfer function of the matter power spectra. \label{fig:psi_crossntransfer}}
\end{figure}
\begin{figure}
		\includegraphics[width=0.45\textwidth]{images/voboz_psi_density_PDFs.png}\caption{ PDF of the matter fields generated by the divergence displacement fields listed in the legend. These densities have been measured with the Voronoi tessellation code in \textsc{voboz/zobov}. \citep{Neyrinck05}\label{fig:voboz_psi_densities_PDFs}}
\end{figure}

\subsection{Translation Process for Lagrangian Simulations}
Ultimately, all previous techniques map the initial density field onto the displacement divergence field without explicitly depending on the coordinates. This kind of mapping is a local Lagrangian mapping \citep{Protogeros97}, and it falls in the same category of the mappings considered by translation process theory, being a local nonlinear transformation. More precisely, the ZA and spherical collapse mappings (\cref{eq:zeld,eq:sc}) are exactly monotonic, while 2LPT and {\scshape muscle}\ (\cref{eq:2lpt,eq:muscle}) use non-local information. In light of translation process theory, we could attempt to optimize for the pre-translation Gaussian field that is most adapted to be transformed onto $\psi$, and we can check whether this yields an improvement in encoding the evolution of the matter density field with respect to these techniques.   In order to generate an ITAM realization of $\psi$, we measure the target power spectrum $\langle \psi(\pmb{k}) \psi^{*}(\pmb{k}') \rangle \equiv (2 \pi)^3 P_{\psi}(k) \delta(\pmb{k}+\pmb{k}')$, and $\psi$'s PDF, from the SB simulation. In using both from a single realization, the results may depend on peculiarities in that realization. The limitation is due to the lack of an available software to produce accurate fitting functions for the power spectrum and PDF of $\psi$. This introduced dependence is not a problem, as long as one has the aim of isolating the contribution from the PDF and the correlation of $\psi$ to the specific realization examined, to assess how much information they convey with respect to the exact result. To emulate the final $\psi$ field, we set the phases of the pre-translation Gaussian field to the phases of the initial density field of the realization.
We do not smooth the measured $\psi$ field, because both the power spectrum and the PDF are measured at the grid level, and one can check that the $\psi$ field variance is already consistent with its power spectrum variance integrated up to the Nyquist frequency.

In \cref{fig:psi_PDFs}, we can see that the PDF of the $\psi$ realization from ITAM matches the one of the $N$-body simulation (much better than {\scshape muscle}, and particularly 2LPT), as it should by the virtue of the translation transform \cref{eq:g}. Likewise, the Lagrangian-space $\psi$ power spectrum of ITAM matches that of the $N$-body simulation better than the other prescriptions, as shown in \cref{fig:psi_transfer}. The two standard simulation schemes have a mismatch, with 2LPT diverging the most at small scales, and {\scshape muscle}\ departing the most on large scales. This confirms the detailed analysis of the power spectra of Lagrangian schemes conducted by \cite{Chan2014}, where it is also found that these techniques come short of matching the power spectrum measured in $N$-body simulations. Another benchmark of the goodness of a fast simulation scheme is the similarity between the resulting matter density and the $N$-body matter density. This is generally quantified by means of the cross correlation spectrum, that for two fields A and B is
\begin{equation}\label{eq:cross}
	X(k) = \frac{ \langle \delta_{A}(k)\delta^*_{B}(k) \rangle }{P_A{(k)} P_B{(k)}}.
\end{equation}

The cross correlations are plotted in the top panel of \cref{fig:psi_crossntransfer}. As we can see, the {\scshape muscle}\ implementation still performs better than ITAM, which in turn is better than 2LPT, at least on smaller scales. Examining this further, 2LPT is almost a local approach, in the sense that it considers the information of each voxel and its nearest neighbours (\cref{eq:2lpt}). ITAM by construction is non-local, as it involves the two-point correlation function. {\scshape muscle}\ is also non-local, but the non-locality is coming from a smoothing procedure on various scales centered around each voxel (\cref{eq:muscle}). Considering that the one-point mapping of ITAM is very well approximated by the spherical collapse formula (e.g. fig. 5 of \cite{Neyrinck15}), it seems that the multi-scale information is more relevant than the non-local information of the two-point correlation function. In fact from \cref{fig:psi_densities} we can see that halo particles are more accurately tagged by {\scshape muscle}, rather than ITAM. However, when one examines the particle positions plotted in \cref{fig:lagslice}, it is apparent that ITAM is collapsing more filaments and walls when compared to {\scshape muscle}.
This improvement appears also in the bottom panel of \cref{fig:psi_crossntransfer}. The matter power spectrum of density field generated through ITAM on $\psi$ is closer to the $N$-body case. In \cref{fig:voboz_psi_densities_PDFs}, ITAM also manages to capture the highest densities of any approximation, quantifying the high filament compactness that ITAM manages in \cref{fig:lagslice}. Overall, it seems that the improvement, despite not being substantial, could pave the way for a new and more accurate fast simulation scheme, as e.g. a combination of {\scshape muscle}\ and ITAM. We leave this investigation for a future study.
\begin{figure*}
	\includegraphics[width=0.85\textwidth]{images/psi_densities}

	\caption{A 1-{$h^{-1}$Mpc} -thick Lagrangian slice showing the $\psi$ field in various schemes. {\scshape muscle}\ performs better than ITAM in locating the patches of haloes in Lagrangian space. \label{fig:psi_densities}}
\end{figure*}

\begin{figure*}
	\includegraphics[width=0.7\textwidth]{images/lagslice}
	\caption{A 1-{$h^{-1}$Mpc} -thick Lagrangian slice showing the Eulerian position of particles for the simulation schemes considered in the text. ITAM performs better than {\scshape muscle}\ in collapsing filaments and walls. \label{fig:lagslice}}
\end{figure*}

\section{Summary \& Conclusions}\label{sec:Conclusions}
In this work we have adapted the ITAM algorithm, first suggested by \cite{Shields11} for one-dimensional time simulations, to the three-dimensional case of spatial fields (\cref{fig:FlowChart}). ITAM optimizes for an effective Gaussian power spectrum to generate many realizations of a field that can be mapped onto the target field under a simple monotonic transform. The target field produced in this way has prescribed PDF and power spectrum, which are the only required inputs of the code. Using as a reference the PDF and the power spectrum of an $N$-body simulation, we are able to generate many realizations of a nonlinear field that match the density field at least at the level of the one-point and two-point correlation functions, thus improving over the usual lognormal assumption. We compared the covariance of the ITAM field against the lognormal field, showing that the accuracy of ITAM is comparable to the lognormal case, or even closer to the expected covariance (\cref{fig:diagonal,fig:off_diagonal}). For the first time, we also motivate analytically a common ansatz used for a phenomenological description of the covariance matrix, showing that its form is a result of the Eulerian growth of structure such as an exponential growth in the case of a lognormal field (\cref{subsec:bias}). In fact, this functional form is found directly from examining the trispectrum of the lognormal field.    As a last example, we apply ITAM to the case of the bi-modal PDF of the Lagrangian displacement field (\cref{fig:psi_PDFs}). As expected, we find that the displacement field generated by ITAM is more faithful to the exact one measured from the $N$-body simulation than available semi-analytical schemes (\cref{fig:psi_transfer}), as well as being effective in generating a more accurate density field (\cref{fig:voboz_psi_densities_PDFs,fig:lagslice}). This result suggests that further improvements for Lagrangian schemes are not expected to come from one-point mappings of the initial density field, but are to be found in multi-scale or non-local approaches.

\section*{Acknowledgements}
FT acknowledges financial support by ASI Grant No. 2016-24-H.0 and thanks the "Dipartimento di Fisica Aldo Pontremoli" of the University of Milano, and the Department of Theoretical Physics at the University of the Basque Country in Bilbao for hospitality during the development of this work.
LG, BRG and FT acknowledge financial support by grant MIUR PRIN 2015 "Cosmology and Fundamental Physics: illuminating the Dark
Universe with Euclid". MCN is grateful for funding from Basque Government grant IT956-16. FT is grateful to Michael D. Shields for clarifications on ITAM and acknowledges useful discussions with Raul Angulo, Cora Uhlemann, Enzo Branchini and Carmelita Carbone.

\section*{Data availability}
The codes for ITAM and {\scshape muscle}\ simulation schemes are available at \url{https://github.com/tos-1/ITAM} and \url{https://github.com/tos-1/MUSCLE} respectively.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


Your dentist says it is time to get your tooth removed and you are scared, well do not be.
As a dentist, our main aim is to preserve your natural teeth and keep them healthy, sparkling and strong for as long as possible. However, in some situations, it may be in your best interest to have your tooth extracted.
There are many reasons why a dentist may suggest that you require an extraction.
You may have a tooth that has been brutally damaged by trauma or decay; or your child has a stubborn baby tooth simply won't fall out of its own accord, even though it's past time for it to go; or perhaps an impacted wisdom tooth that may cause trouble for you later on; you may have a nasty infection or require teeth to be removed to allow for braces treatment.
Rest assured that we make sure our patients are completely numb before proceeding with this and any other treatment. During an extraction it is normal for patients to feel pressure, unfortunately there is nothing that can be done about this, but there should be no pain. If at any time during a procedure you are not feeling comfortable or feel any pain, simply raise your hand and we will stop work immediately and do everything possible to help you.
Though surgery may sound frightening, it is very common and often less painful than the discomfort associated with damaged or impacted wisdom teeth or a toothache.
The first step in any extraction is an X-ray to assess the position of the impacted wisdom or decayed tooth and the condition of the surrounding bone. This will allow any possible complications to be known beforehand. Then your dentist will take a detailed medical to ensure that you are fit enough to undergo the surgery.
Thereafter your mouth will be numbed with local anaesthesia to avoid discomfort during the procedure. On some occasions, if you are particularly nervous, an oral sedative may be given before the procedure to help you relax and manage the procedure more calmly.
The dentist will test to ensure the anaesthetic is fully working and then proceed to gently remove the tooth. Afterwards a piece of sterile gauze will be placed in your mouth to help control the bleeding and your dentist will explain to you the aftercare instructions You will also be given spare gauze to take home and all the instructions will be written down to help you remember.
Dentists are finding more and more patients are having problems with their wisdom teeth. This is due to the fact that our jaws are getting smaller as we eat more refined and processed food, it is a simple case of evolution. They are called wisdom teeth as they usually don't start coming through until we are aged between 16 to 25, when we are theoretically older and wiser.
Often there is not enough room for the wisdom tooth to come through, so they will try to erupt at all sorts of odd angles. They can put pressure on the tooth next to them, poke into your cheek or sometimes remain hiding in the bone. Many times, patients will start to experience problems when food gets trapped next to them, so it is important to make sure you keep your wisdom tooth and the gum surrounding them nice and clean. More often than not wisdom teeth that cause problems will need to be extracted.
To determine the exact position of your wisdom tooth, and how it is sitting in relation to the surrounding nerves and adjacent teeth, your dentist will require you to have a full mouth x-ray, called an OPG (Orthopantomogram).
This will help them see any potential issues that may arise during the extraction. The wisdom tooth extraction can either be the easiest or most difficult tooth for a dentist to extract. For the most part a general dentist can safely remove your wisdom tooth, but the more difficult extractions will need to be carried out by a specialist.
So, if you are having wisdom tooth problems or think you need to have a tooth removed, contact Passion Family Dental North Lakes at Unit 4/ 6 Endeavour Boulevard, North Lakes, QLD 4509 or call (07) 3465 1199 to learn more or schedule a visit today.


text:
Students create full-size pirate captain's cabin TV set
Rachel Smith January 6, 2021
Production design students at The Northern School of Art have been exploring the golden age of piracy to create a full-size TV studio set of a pirate ship captain's cabin.
The atmospheric interior of the 17th Spanish galleon has been painstakingly crafted by a team of second year students as part of their Research, Design and Construction module for the School's BA (Hons) degree in Production Design for Stage and Screen at the university-level campus in Hartlepool.
With a brief to build a set that reflected the richness and detail of the interior of the cabin of a captain who has spent many years sailing the seas, plundering a wide range of treasures, the students filled the set with an exotic mix of delights.
The students created the stunning set, from the initial concepts through to research into the period to create historically accurate final designs and the build itself, in just fourteen weeks, finishing just before they left for the Christmas break.
It has been completed to industry-standard suitable for 4k filming and could be used in a real TV studio.
It has also been created sustainably, using materials recycled from previous sets in accordance with the film and TV industry's albert sustainability guidelines.
Working against the backdrop of the pandemic and the second lockdown provided an extra challenge for the students who adapted to work in socially-distanced ways.
Student Holly Speakman, from Hartlepool, said: "It's been an incredible opportunity to work on a full-scale set and as a group. The experience will really help when we get into the industry."
Kerry O'Connor, from Thornaby, pictured below second on right, agreed saying she had chosen the course because she "liked to create big-scale things" and Sophie Skillcorn, from Normanby, pictured below far right, said it had been a "great opportunity to also get an extra level of skills."
Francesca Pajovic, from Nottingham, added that "we had to problem-solve and find solutions as we went along which replicates how we will have to work in the industry."
Shania Hurd, from Birmingham, pictured below far on left left, found that "doing a big build has helped me discover where my skills lie," noting that the build also required them to even "make plastic look like wood!"
The breadth of skills used was also what Holly Russell, from Leeds, pictured below second from left, enjoyed saying: "I liked that we were able to explore every aspect, from the start to the end."
The School's Production Design Lecturer, Norman Austick, was impressed with what the students have achieved: "They've worked brilliantly together and didn't have a cross word in fourteen weeks. Learning how to communicate and respect each other whilst working as a team will serve them well after they leave and move into work where they will have to work with a wide range of people.
"It's been challenging working in groups in the pandemic but we have had strict protocols in place to ensure safe working and we have the benefit of not having huge student numbers so it was achievable."
The Northern School of Art's BA (Hons) degree in Production Design for Stage and Screen gives students the skills required of professionals working in the film, TV and theatre industries. Undergraduates work over three years to develop niche skills such as: prop making, sculpting, concept art and character development designed to give them a competitive edge when applying for roles after graduation.
For further information about the course visit https://northernart.ac.uk/study-with-us/hartlepool-campus/degree-fda-courses/ba-hons-production-design-stage-screen-2/
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redpajama_set_name: RedPajamaCommonCrawl


Founded in 2006, the Alliance for a Green Revolution in Africa (AGRA), is an African-led African-based organization that seeks to catalyze Agriculture Transformation in Africa. AGRA is focused on putting smallholder farmers at the center of the continent's growing economy by transforming agriculture from a solitary struggle to survive into farming as a business that thrives. As the sector that employs the majority of Africa's people, nearly all of them small-scale farmers, AGRA recognizes that developing smallholder agriculture into a productive, efficient, and sustainable system is essential to ensuring food security, lifting millions out of poverty, and driving equitable growth across the continent.
AGRA is rolling out a new 5-year strategy to catalyze and sustain an Inclusive Agricultural Transformation in Africa to increase incomes and improve food security for 30 million farmers. Under this strategy, AGRA will deliver through an approach that simultaneously catalyzes change at farmer level, strengthens input and output market systems and puts government at the center to enable and champion private-sector-led agricultural growth at national level.
To support this transformation, AGRA is seeking to recruit an experienced and exceptional individual to the role of Development Cooperation Officer – West Africa.
The role will be based in Accra, Ghana with regular engagement and support to AGRA West African countries, including particularly Ghana, Burkina Faso, Mali and Nigeria. The role will also provide support to other AGRA focus regions and countries from time to time as requested for top priorities.
• Reporting to the Head – Development Cooperation, the Development Cooperation Officer- West Africa will contribute to the development and implementation of AGRA's overall organizational resource mobilization and development cooperation strategy.
• S/he will ensure proper relationship building while garnering a strong support base for AGRA across a wide range of development partners, including bilaterals, multilaterals and private foundations.
• Working closely with the Development Cooperation Unit and Program teams, s/he will support the identification of funding opportunities, coordinates development and presentation of proposals to potential funders and supports the team to ensure compliance to AGRA and donor requirements for accountability, retention and upgrading of strategic partnerships after funding agreements are secured.
• S/he also works closely with the Development Cooperation Unit and Program teams to understand the development partners landscape in the focus countries and identify and advance opportunities for strategic leverage, coherence, and coordination of resources from bilaterals, multilaterals and private foundations that will result in greater impact aligned with AGRA's strategic mission of inclusive agricultural transformation.
• Contribute to the development of a comprehensive short, medium and long-term resource mobilisation and development cooperation strategy for sustainable growth and flexibility.
• Identify opportunities and working with teams to develop quality fundable concepts and proposals that will expand AGRA's current support base.
• Provide technical support to the preparation of concepts and proposals for donor funding and achieve pledge conversion rate and sufficient, predictable and sustainable donor support for AGRA in West Africa priority countries – Ghana, Burkina Faso, Mali, and Nigeria.
• Secure funding from bilateral and multilateral donors, philanthropic foundations, and other sources, both to AGRA directly and to leverage funds into priority country plans more broadly.
• Lead in understanding and cultivating opportunities with a portfolio of key development partners for the region as assigned to the team, including the African Development Bank, Islamic Development Bank, and key bilateral and foundation partners for the region.
• Establish and maintain strategic relationships with a portfolio of development partners and collaborators at national government, regional level, and private sector that AGRA can work with to leverage its investments in agricultural transformation.
• Support in development and management of a donor database containing critical information e.g. donor policies, funding cycles etc. in order to strengthen AGRA's institutional capacity to effectively position itself with potential donors, retain and upgrade current partners.
• Contribute to the development of regular donor updates for the board and management.
• Support AGRA's efforts to harmonize and coordinate reporting guidelines to donors across all AGRA activities to the extent possible in order to make reporting most efficient.
• Support the development and dissemination of good practices and lessons learnt in order to contribute to AGRA's institutional strengthening efforts.
• Contribute to capacity building in resource mobilisation efforts as necessary.
• A Bachelor's degree in a relevant field, whether agriculture, finance, marketing, business administration, law, economics, international relations, communications or any other. A Master's degree in a relevant field is preferred, but not required.
• A minimum of 7 years' of high-level experience in international development, including working in resource mobilization and/or business development for UN agencies and multi-lateral agencies, large NGO's or donor-aid programs.
• Experience in donor relations, resource mobilization and sound understanding of donor financing landscape and processes with a proven track record.
• Proven ability to work with donors, manage and build strategic donor relations in West Africa is essential.
• Fluency in English and French is required both written and verbal, as a working language of AGRA and key partners in West Africa region and priority countries.
Please quote the job reference number in the subject line of the application e-mail.


text:
Sen. Collins co-sponsors bill requiring presidential campaigns to report foreign contacts
She is the first Republican senator to support the measure, but Republican Majority Leader Mitch McConnell has so far blocked the bill from coming to a vote.
By Randy BillingsPortland Press Herald
Sen. Susan Collins is co-sponsoring legislation that would require presidential campaigns to report foreign contacts to federal authorities.
Collins, a Maine Republican who serves on the Senate Intelligence Committee, said in a statement Tuesday that she's co-sponsoring the Foreign Influence Reporting in Elections, or FIRE, Act, introduced by Sen. Mark Warner, D-Virginia. She's the first Republican to back the bill, according to The Hill, a newspaper and website that covers Congress.
The FIRE Act would require presidential campaigns to report any contacts with foreign nationals trying to make campaign donations or otherwise coordinate with a campaign. Such contacts would have to be reported to federal authorities, including the Federal Bureau of Investigation.
"Russia's efforts to interfere in our elections remain relentless," Collins said in a written statement. "I'm proud to join Senator Mark Warner in co-sponsoring the bipartisan FIRE Act to require presidential candidates to immediately call the FBI if they are contacted by a foreign power attempting to target our elections."
Warner posted a statement to Twitter welcoming Collins' support.
"Today I'm pleased to add @SenatorCollins as a sponsor of the FIRE Act," Warner said. "We need to make it clear for presidential campaigns going forward: If a foreign power contacts you attempting to interfere in our elections, you call the FBI. #ProtectOurElections"
The bill would require campaigns to implement a compliance system to monitor reportable foreign contacts with campaign representatives and to train all other employees and associates about their legal obligations, Collins said in a news release. Campaigns would be required to report the contacts to both the FBI and the Federal Elections Commission and preserve all relevant records.
The bill faces an uncertain future in the Republican-controlled Senate. Majority Leader Mitch McConnell has so far blocked the legislation from coming forward.
Collins' announcement comes days after the Senate Intelligence Committee issued a report saying that Russia likely targeted election systems in all 50 states during the 2016 elections.
Special Counsel Robert Mueller's report on Russian's interference in 2016 described multiple contacts between the Trump campaign and Russian nationals, including a June 9, 2016, meeting with a Kremlin-linked attorney and other Russians offering damaging information on his opponent, Hillary Clinton.
President Trump has dismissed concerns about Russian interference in U.S. elections. And in June, Trump said in an interview with ABC that he would be willing to meet and listen to foreign agents offering damaging information on a political opponent, while suggesting he may not alert the FBI about such contacts.
"I think I'd take it," Trump said, according to the interview transcript. "If I thought there was something wrong, I'd go maybe to the FBI. If I thought there was something wrong."
His remarks were criticized by both Democrats and Republicans, including Collins, who said that such contacts should be reported to the FBI.
Sen. Angus King, a Maine independent who caucuses with the Democrats, also serves on the Senate Intelligence Committee and was an original co-sponsor of the legislation. King has strongly criticized Trump for saying he would accept help from a foreign power.
"This shouldn't need to be said, and this certainly shouldn't be a partisan issue: Accepting any sort of assistance from a foreign power in an attempt to influence an election is completely and totally antithetical to America's founding principles," King said in a statement provided by his office. "Dating back to George Washington and the Federalist Papers, the independence of American elections has been a hallmark of our electoral system, and we cannot allow any foreign nations – particularly adversaries – to put a thumb on the scale of our public debate. Frankly, this should be non-controversial, because any candidate, from any party, should know that this type of behavior is wrong. It's not complicated – just call the FBI!"
Dollars from Donors: See who is contributing to Maine's U.S. Senate candidates
U.S. presidential envoy sent to Sweden for A$AP Rocky's trial
Boston's gang database made up mostly of young black, Latino men
congress, susan collins
Mitch McConnell said Trump 'fed lies' to mob about Biden election
Central Maine Obituaries


text:
Tazehabad-e Chaf (, also Romanized as Tāzehābād-e Chāf; also known as Tāzehābād) is a village in Chaf Rural District, in the Central District of Langarud County, Gilan Province, Iran. At the 2006 census, its population was 523, in 142 families.

References

Populated places in Langarud County

redpajama_set_name: RedPajamaWikipedia


At 28 years old, I've decided to learn to play the drums. I do not come from a musical family. None of my immediate family have ever played an instrument – I only dabbled.
The decision to start playing wasn't completely out of the blue – I live with an omni-musical person. My boyfriend can play an envy-inducing amount of instruments; piano, sax, bass, guitar, he sings, he produces, he even plays the drums a bit. It was almost inevitable that his passion would rub off on me. Some of my fondest memories of us together are singing loudly on road trips, all the while drumming on any surface nearby.
So, I ask him if he knows a drummer, turns out he knows one of the best in Liverpool; Dave Kelly.
Dave Kelly; The Brian Cox of music tuition.
Dave is based out of The Baltic Social in Liverpool. For those of you not familiar with this space, it's at the heart of the independent scene in Liverpool; The Baltic Triangle. Cafes, collaborative workspaces, studios and pubs are all nestled together in a hippy trifecta. Ultimately, it's the result of a bunch of people my age posing the question; where do I want to work? and building it. Dave's studio is at the heart of it.
Dave's passion for drumming is clear. He sees the drums as more than an instrument – it's a way of life. The drums teach you good posture. To listen. Discipline. Dave is also one of the only people to have given a positive response to; "Aerosmith are one of my favourite bands." Turns out he saw them recently.
When he talks about drumming, he's got the same glittering look in his eye as TV personality and sometimes scientist Brian Cox. Everything seems possible in his company. You can start a band. You are in time. The universe is expanding. The whole experience is incredible.
At the beginning of the lesson Dave poses a completely reasonable question; why do you want to play the drums? I didn't have an eloquent answer ready. But on reflection, the drums are in good company. Keith Moon from The Who. Taylor Hawkins from the Foo Fighters. Julie Edwards from Deap Vally. Viola Smith (the first recognised professional female drummer).
When I think about it, I don't know why the idea hadn't occurred to me before. You're being noisy, without being the centre of attention. You're the heartbeat of the band. You get to wear a lot of sleeveless shirts. It's a physical and mental workout. The whole thing is a riot.
In our first lesson, Dave teaches me two rudiments; a single stroke roll and a double stroke roll. It turns out, I have some rhythm. Dave is encouraging and patient. A single hit of the drum evokes a "Brilliant!" from him. And I do feel brilliant, just for making noise on a drum. I day dream band names.
For someone as opinionated as myself, I was surprised to find I was scared to make a loud sound. Everytime I hit the drum my face would make an apologetic grimace. It was like I was apologising for hitting the drum skin. Drumming enabled me to surprise myself; in the first instance, I'm not as brave as I think I am.
Learning an instrument as an adult is completely different to my experiences as a kid. First of all, I'm paying so I'm invested both financially and emotionally, and it's an hour a week where i'm totally dedicated to learning something new. It's nothing to do with writing, like so many of my other hobbies are. It is what it is and it is what I put in. It's my new paramour.


text:
Sildefiskeri er en dansk dokumentarfilm fra 1934 instrueret af Paul Hansen.

Handling
Især optaget i Julianehåbs Distrikt.
Yderligere oplysninger findes i Carl Nørresteds registrant (p.t. på Karins kontor, 2012)

Referencer

Eksterne henvisninger

Dokumentariske optagelser
Danske film fra 1934


12 mg has a 75% VG and 25% PG ratio.
Nicotine in most e liquid is freebase nicotine, and is produced by eliminating the protons, which also separates the nicotine "salts" from the nicotine to create the freebase nicotine "regular nicotine". Once the protons and other components have been eliminated and a regular nicotine has been created, the nicotine can be used in e-liquid.
Nicotine salts on the other hand, utilizes the natural salts formed in the leaf of the tobacco plant. Nicotine salt is a more stable alternative to regular nicotine, and administers more nicotine per ml of ejuice. Nicotine salts also have a quicker absorption rate of the nicotine into their bloodstream than typical eliquids. In ejuice, the nicotine absorption rate is stable and consistent, but with nicotine salts, the nicotine absorption is substantially higher at the point of inhalation, comparable to a regular cigarette. This is ideal for Vapers who find that eliquid doesn't deliver the same satisfaction that they get from a combustible cigarette, since nicotine salts have a similar absorption rate for nicotine into the bloodstream, as seen with a cigarette. With nicotine salts the throat hit is strongly intensified, which can be great for those Vapers who are making a shift from consuming a large amount of nicotine through smoking, to vaping.
Nic salts have a much faster absorption rate into your blood stream than regular e-juice and therefore your body processes the nicotine very quickly. A high nicotine content combined with high wattage can increase the nicotine absorption rate even further which can present health concerns and nicotine overdose. You should not vape nic salts at any wattage higher than 15 watts without potentially putting yourself at risk for nicotine overdose. This type of liquid is designed for HIGH ohm and LOW wattage devices. If you are unsure of whether you can use nic salts with your device PLEASE contact us before using it.


Update, September 4: An earlier version of this article indicated that NIRSA Foundation scholarships could be used for 2018 regional events, which is incorrect. The article has been updated with accurate information.
It's that time of year again, NIRSA family! Get your travel calendars out and let's start booking trips to learn, grow, and thrive this year at our 2018-2019 NIRSA regional conferences. There are exceptional professional development opportunities available to you, as well as regional scholarship support that you may be eligible for! So please, join us in a journey of learning with NIRSA's core competencies guiding educational content and NIRSA leaders at the helm to ensure every attendee enjoys their professional development experience.
The Region I Conference will be held in Portsmouth, New Hampshire by the University of New Hampshire from October 29-31. Don't miss out on the Student Lead On, Monday, October 29 from 11:30am – 4:00pm. Other preconference activities include a UNH Campus Recreation facility tour and construction process discussion, small program workshop, island bike tour, and beach clean-up community service trip!
Come "Discover the Magic" in Birmingham, Alabama this year at the Region II Conference from November 5-7. Preconference offerings include facility tours in both the Birmingham and Tuscaloosa areas, a community service project with the Community Food Bank of Central Alabama, an outdoor adventure trip to Chandler Mountain and more! This year the Mark Fletcher Walk will be taking place along the Birmingham Civil Rights Heritage Trail. Don't miss out on the Student Lead On which will be Monday, November 5 from 8:30am – 3:00pm.
This year, the Region III Student Lead On invites you to "Be the Spark. Ignite Your Future." This student-led, student-run conference will be hosted by the University of Wisconsin – Oshkosh January 31 – February 2. In the fall of 2018, Region III will be hosting the following state workshops: The Wisconsin State Workshop will be held October 3-5 in River Falls, Wisconsin by the University of Wisconsin River Falls. The University of Michigan will be hosting the Michigan State Workshop October 24-26 in Ann Arbor. The Illinois State Workshop will be October 24-26 and is set to be hosted by Northern Illinois University in DeKalb. South Bend will be home of the Indiana State Workshop November 1-2, which will be hosted by the University of Notre Dame. Finally, the Ohio State Workshop will be held in Akron, October 31 – November 2, hosted by the University of Akron.
Don't miss out on the opportunity to "Cultivate Your Potential" at this year's Region IV Conference in College Station, Texas from October 21-24. The Region IV Conference will feature preconference offerings including a ropes course experience, StrengthsQuest session, or a guided facilities tour of Texas A&M RecSports. This year the Region IV Student Lead On will run concurrently with the Region IV Conference so that there are student-focused educational sessions that students or professionals may attend throughout the conference.
Region V will not be coordinating a regional conference this year, but we would love to see you at one of our state workshops in October! North Dakota State University will be hosting the North Dakota/Minnesota State Workshop, October 4-5. The planning committee is excited to begin "Sharing Solutions" with the NIRSA community. The Colorado/Wyoming State Workshop will be hosted by the University of Wyoming from October 10-12 in Laramie, Wyoming. Creighton University will be hosting the Nebraska State Workshop on October 15.
Region VI invites you to begin "Expanding Horizons" in Bellingham, Washington from October 8-10 for the Region VI Conference. The conference will be hosted by Western Washington University and the University of Washington. Mountain biking, facility tours, a community service project, and a hiking up to Oyster Dome are a few of the many exciting preconference activities offered this year. The Student Lead On will be Monday, October 8, from 8:30am-4:00pm and will feature student-led, student-run opportunities for networking, engagement, and development.
The Canadian Student Lead On will be held in Toronto, October 11-12 and will be hosted by Centennial College. Conference proposals will continue to be accepted until September 30. The theme of the Conference is "Inspiring Your Campus, Involving Your Community." The event committee aims to inspire leaders who will ignite change and implement integrated recreational, wellbeing, and fitness programs to build a strong and healthy community.
Make sure to put your best foot forward at your regional conference, because you may find yourself representing your region at the Annual Conference in Boston!
Again this year, a unique opportunity is being offered to the NIRSA regional conference presenters and NIRSA 2019 Annual Conference attendees. The Annual Conference Program Committee has set aside seven educational session slots during the conference to be held for the best presentation from each region. To be eligible for selection, an educational session must be personally presented at a regional conference or state workshop between September 1 and November 30.
Presentations that are innovative, creative, highly-rated, well-attended, and well-received will be considered when the NIRSA 2019 Program Committee is selecting the featured regional presentation. Additionally, the availability of presenters to attend the Annual Conference will also be taken into consideration.
For more information about local or regional professional development opportunities in your area, contact your regional representative on the NIRSA Member Network.
Caroline Ciell is currently Graduate Assistant – Facility Operations at the University of Mississippi and the NIRSA Student Leader. Corrine Pruett is currently a Graduate Assistant in Facilities at the University of North Carolina at Greensboro and the NIRSA Region II Student Leader.