text:
\section{Introduction}\label{sec:intro}
During the last century the general theory of relativity, which conventionally attributes the gravitational interaction to the curvature of spacetime, encoded in the presence of the Ricci scalar \(R\) in the action, has passed most of the tests given by solar system and astrophysical observations~\cite{Bertotti:2003rm,Fienga:2011qh,Fienga:2014bvy,Linder:2010py,Olmo:2005zr,Will:2014kxa,Will:2018bme,Will:2018mcj}. Several observations in cosmology and the dynamics of large scale structures, however, are unexplained within general relativity itself, unless it is supplemented with additional, ``dark'' components~\cite{Riess:1998cb,Perlmutter:1998np,Scolnic:2017caz,Aghanim:2019ame,Aghanim:2018eyx,Aghanim:2018oex,Akrami:2019izv,Akrami:2019bkn,Akrami:2018odb,Ahumada:2019vht,Alam:2020sor}. These open questions, as well as the quest for a quantum theory of gravity, have motivated the construction and investigation of several modifications and alternatives to general relativity~\cite{Nojiri:2006ri,Nojiri:2010wj,Nojiri:2017ncd}.

While the aforementioned modifications retain the interpretation of gravity as a manifestation of curvature, this underlying geometrical interpretation of gravity is not without alternatives. Recent developments shed light on two different interpretations of the nature of gravity~\cite{BeltranJimenez:2019tjy}. One alternative is teleparallel gravity, where gravity is a manifestation of a connection on spacetime with vanishing curvature and non-metricity, but non-vanishing torsion~\cite{Aldrovandi:2013wha}. The third possibility, next to curvature and torsion, is to attribute gravity to a flat, torsion-free connection, thus making non-metricity the only non-vanishing tensorial quantity~\cite{Nester:1998mp,Adak:2005cd,Adak:2008gd,Mol:2014ooa,Adak:2018vzk,Jimenez:2019yyx,DAmbrosio:2020nqu}. Most remarkably, the Einstein-Hilbert action of general relativity can be formulated equivalently in either of these other geometries, by introducing suitable scalars \(T\) and \(Q\) from the torsion and non-metricity, respectively, which take the role of the Ricci tensor in the action. This leads to the teleparallel equivalent of general relativity (TEGR)~\cite{Maluf:2013gaa} in the case of torsion, and the symmetric teleparallel equivalent of general relativity (STEGR)~\cite{Nester:1998mp} in the case of non-metricity. Also a more general equivalent, featuring both torsion and non-metricity, has been proposed~\cite{Jimenez:2019ghw}. In these alternative formulations, equivalence is to be understood as dynamical equivalence in the sense that the field equations of all three formulations impose the same dynamics for the metric, which is present in all three geometries.

While general relativity can equivalently be formulated in terms of each of the three quantities - curvature, torsion or non-metricity - generalizations of these three approaches towards gravity are not equivalent anymore. For example, the $F(R)$ class of gravity theories~\cite{Sotiriou:2006hs,DeFelice:2010aj} is not equivalent to their corresponding $F(T)$ gravity theories~\cite{Bengochea:2008gz,Krssak:2015oua,Cai:2015emx}, where $F$ is a free function constituting the Lagrangian, which determines a particular theory within this class. The reason for this is, that the two scalars \(R\) and \(T\) differ by a total derivative, which contributes to the gravitational field equations, unless the Lagrangian function \(F\) is linear, in which case it becomes a boundary term and the theories are dynamically equivalent. For a general function \(F\), however, this difference leads to different field equations, hence to inequivalent dynamics. The same holds true for theories which introduce scalar fields, which are non-minimally coupled to the gravitational action~\cite{Fujii:2003pa,Faraoni:2004pi,EspositoFarese:2000ij,Hohmann:2018rwf,Hohmann:2018vle,Hohmann:2018dqh,Hohmann:2018ijr}. Therefore it is worth studying such kind of modifications, which are based on torsion and non-metricity, as they are essentially different from the class of modifications based on curvature.

The symmetric teleparallel formulation of gravity theories, in which non-metricity is employed as the mediator of gravity, is the least investigated so far, and only a few of its generalizations have been discussed. One possibility is to generalize the symmetric teleparallel gravity action by replacing the non-metricity scalar with an arbitrary function \(F(Q)\), following the same line of thought as the aforementioned \(F(R)\) and \(F(T)\) theories~\cite{BeltranJimenez:2017tkd}. This class of theories has been studied in particular in the context of cosmology~\cite{Jimenez:2019ovq,Lazkoz:2019sjl,Barros:2020bgg,Mandal:2020buf,Mandal:2020lyq}. Another possibility is given by generalizing the non-metricity scalar to other terms quadratic in the non-metricity tensor, known as newer general relativity~\cite{BeltranJimenez:2017tkd,BeltranJimenez:2018vdo,Hohmann:2018wxu,DAmbrosio:2020njs}. Yet another class of theories is obtained by non-minimally coupling scalar fields to non-metricity~\cite{Jarv:2018bgs,Runkla:2018xrv}. Finally, even more general theories have been proposed, such as~\cite{Conroy:2017yln,Koivisto:2018aip,Hohmann:2018xnb,Dialektopoulos:2019mtr,Koivisto:2019jra,Harko:2018gxr,Lobo:2019xwp}.

The large number of gravity theories considered to address the open questions in cosmology requires efficient tools in order to be constrained by observations. One of the most powerful tools to obtain observational bounds on modified theories of gravity is the parametrized post-Newtonian (PPN) formalism \cite{Will:1993ns,Will:2014kxa,Will:2018bme}. It has been widely used to restrict several classes of gravity theories based on curvature~\cite{Hohmann:2013rba,Hohmann:2015kra,Hohmann:2016yfd,Hohmann:2017qje,Nordtvedt:1970uv,Olmo:2005hc,Perivolaropoulos:2009ak,Scharer:2014kya} and torsion~\cite{Emtsova:2019qsl,Flathmann:2019khc,Gladchenko:1990nw,Li:2013oef,Sadjadi:2016kwj,Ualikhanova:2019ygl}, as well as to gravity theories based on multiple metrics~\cite{Clifton:2010hz,Hohmann:2013oca,Hohmann:2017uxe}. Recently, a gauge-invariant approach to the PPN formalism has been developed, which consolidates its mathematical foundation as a tool to study gravity theories by the way how they describe the geometry of spacetime~\cite{Hohmann:2019qgo}.

The purpose of this article is to apply the PPN formalism to a class of symmetric teleparallel gravity theories, thereby opening the path of post-Newtonian calculations for theories in which gravity is modeled by non-metricity. Here we consider a class of theories whose action is determined by a free function of the five parity-even scalars which are quadratic in the non-metricity tensor~\cite{Koivisto:2018aip,Hohmann:2018xnb,Dialektopoulos:2019mtr}. This class of theories is interesting since it encompasses various other theories mentioned above, including newer general relativity and $F(Q)$ gravity~\cite{BeltranJimenez:2017tkd}.

The article is structured as follows. In Sec.~\ref{sec:dynamics} we review the class of symmetric teleparallel gravity theories analyzed in this article. We state its geometric foundations and present the field equations emerging from an action which consists of a free function of the five independent contractions of the non-metricity tensor. Then we state the main assumptions we make in order to be able to apply the PPN formalism in Sec.~\ref{sec:ppn}. In Sec.~\ref{sec:solution} we calculate the post-Newtonian approximation of the field equations and solve them order by order up to the fourth velocity order, where we will be assuming the standard PPN gauge. With the help of these perturbative solutions we calculate the PPN parameters and fully classify the studied theories by the Taylor coefficients of the free function of the Lagrangian in Sec.~\ref{sec:classification}. We summarize our results and give an outlook in Sec.~\ref{sec:conclusion}.

In this article we use Greek letters $\alpha, \beta, \ldots = 0, \ldots, 3$ for spacetime indices and Latin letters $a, b, \ldots = 1, \ldots, 3$ for spatial indices. The sign convention for the flat Minkowski metric is chosen as $(-,+,+,+)$.

\section{Field variables and their dynamics}\label{sec:dynamics}
In this section we review the action of the theory we aim to analyze in this article. The dynamical fields of the theory are given by a Lorentzian metric $g_{\mu\nu}$ and an affine connection $\Gamma^{\rho}{}_{\mu\nu}$. The latter is further restricted by the conditions that it has vanishing curvature
\begin{equation}\label{eqn:nocurv}
R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}{}_{\sigma\nu} - \partial_{\nu}\Gamma^{\rho}{}_{\sigma\mu} + \Gamma^{\rho}{}_{\lambda\mu}\Gamma^{\lambda}{}_{\sigma\nu} - \Gamma^{\rho}{}_{\lambda\nu}\Gamma^{\lambda}{}_{\sigma\mu} = 0\,,
\end{equation}
and vanishing torsion
\begin{equation}\label{eqn:notors}
T^{\rho}{}_{\mu\nu} = \Gamma^{\rho}{}_{\nu\mu} - \Gamma^{\rho}{}_{\mu\nu} = 0\,.
\end{equation}
Nevertheless, we allow for non-vanishing non-metricity
\begin{equation}
Q_{\rho\mu\nu}=\nabla_{\rho}g_{\mu\nu}\,.
\end{equation}
Combining the first two conditions, it follows the affine connection has to be of the form
\begin{equation}\label{eqn:flatsymconn}
\Gamma^{\rho}{}_{\mu\nu}=(\Lambda^{-1})^{\rho}{}_{\lambda}\partial_{\nu}\Lambda^{\lambda}{}_{\mu}\,,
\end{equation}
with $\partial_{[\mu}\Lambda^{\lambda}{}_{\nu]}=0$. Later, we will use this form of the affine connection for its post-Newtonian expansion. We assume an action of the form
\begin{equation}
S[g,\Gamma,\chi]=S_g[g,\Gamma]+S_m[g,\chi]\,,
\end{equation}
with the gravitational part of the action $S_g$ being of the form
\begin{equation}
S_g[g\,,\Gamma]=\int_{M}\mathcal{F}\left(\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3,\mathcal{Q}_4,\mathcal{Q}_5\right)\sqrt{-g}\,\mathrm{d}^4x\,.
\end{equation}
Here the Lagrangian is given by a free function $\mathcal{F}$ of the five non-trivial quadratic contractions of the non-metricity tensor $Q_{\rho\mu\nu}$
\begin{equation}
\mathcal{Q}_1=Q^{\rho\mu\nu}Q_{\rho\mu\nu}\,, \qquad \mathcal{Q}_2=Q^{\mu\nu\rho}Q_{\rho\mu\nu}\,, \qquad
\mathcal{Q}_3=Q^{\rho\mu}{}_{\mu}Q_{\rho\nu}{}^{\nu}\,, \qquad
\mathcal{Q}_4=Q^{\mu}{}_{\mu\rho}Q_{\nu}{}^{\nu\rho}\,,
\qquad
\mathcal{Q}_5=Q^{\mu}{}_{\mu\rho}Q^{\rho\nu}{}_{\nu}\,.
\end{equation}
The matter action $S_m$ depends on the metric and an arbitrary set of matter fields $\chi$. With that knowledge we can obtain the usual energy-momentum tensor $\Theta^{\mu\nu}$ by varying the matter action $S_m$ with respect to the metric $g_{\mu\nu}$ as
\begin{equation}
\delta S_m[g, \chi] = -\frac{1}{2}\int_{M}\Theta^{\mu\nu}\delta g_{\mu\nu}\sqrt{-g}\,\mathrm{d}^4 x\,,
\end{equation}
where $g$ denotes the metric determinant.
Throughout this article we will assume for the matter source a perfect fluid, as usual. If we define the derivatives of the free function $\mathcal{F}$ with respect to the five scalar quantities $\mathcal{Q}_i$ as
\begin{equation}
\mathcal{F}_{,i}=\frac{\partial \mathcal{F}}{\partial\mathcal{Q}_i}\,, \qquad i=1,\ldots, 5\,,
\end{equation}
then we can write the field equations as $E_{\mu\nu}=0$, with
\begin{align}\label{eqn:fieldequation}
E_{\mu\nu}= &-2\lc{\nabla}_{\rho}\left(\mathcal{F}_{,1}Q^{\rho}{}_{\mu\nu}+\mathcal{F}_{,2}Q_{(\mu\nu)}{}_{\rho}+\mathcal{F}_{,3}Q^{\rho\sigma}{}_{\sigma}g_{\mu\nu}+\mathcal{F}_{,4}Q^{\sigma}{}_{\sigma(\mu}\delta^{\rho}_{\nu)}\right)\nonumber\\
&-\lc{\nabla}_{\rho}\left[\mathcal{F}_{,1}\left(Q_{\sigma}{}^{\sigma\rho}g_{\mu\nu}+\delta^{\rho}_{(\mu}Q_{\nu\sigma)}{}^{\sigma}\right)\right]+\frac{1}{2}\mathcal{F}g_{\mu\nu}-\mathcal{F}_{,3}Q_{\mu\rho}{}^{\rho}Q_{\nu\sigma}{}^{\sigma}\nonumber\\
&+\mathcal{F}_{,2}\left(2Q^{\rho\sigma}{}_{\mu}Q_{\sigma\rho\nu}-Q_{\mu}{}^{\rho\sigma}Q_{\nu\rho\sigma}-2Q^{\rho\sigma}{}_{(\mu}Q_{\nu)\rho\sigma}\right)\nonumber\\
&+\mathcal{F}_{,4}\left[Q_{\rho}{}^{\rho\sigma}\left(Q_{\sigma\mu\nu}-2Q_{(\mu\nu)\sigma}\right)+Q^{\rho}{}_{\rho\mu}Q^{\sigma}{}_{\sigma\nu}-Q^{\rho}{}_{\rho(\mu}Q_{\nu)\sigma}{}^{\sigma}\right]\nonumber\\
&+\frac{1}{2}\mathcal{F}_{,5}\left[Q^{\rho\sigma}{}_{\sigma}\left(Q_{\rho\mu\nu}-2Q_{(\mu\nu)\sigma}\right)-Q_{\mu\rho}{}_{\rho}Q_{\nu\sigma}{}_{\sigma}\right]-\kappa^2\Theta_{\mu\nu}\,.
\end{align}
Note that covariant derivatives $\lc{\nabla}$ with respect to the Levi-Civita connection are denoted by a circle, to distinguish them from the covariant derivative \(\nabla\) of the independent connection \(\Gamma^{\mu}{}_{\nu\rho}\).

\section{Post-Newtonian approximation}\label{sec:ppn}
In this article we make use of the parameterized post-Newtonian (PPN) formalism~\cite{Will:1993ns,Will:2014kxa,Will:2018bme}. Therefore, we review the general assumptions of the formalism in this section, and supplement them with further assumptions on the independent connection \(\Gamma^{\mu}{}_{\nu\rho}\), in order to apply the PPN formalism to the symmetric teleparallel gravity theory we study here. In the previous section we mentioned, that the matter field will be assumed as a perfect fluid
\begin{equation}\label{eqn:tmunu}
\Theta^{\mu\nu} = (\rho + \rho\Pi + p)u^{\mu}u^{\nu} + pg^{\mu\nu}\,.
\end{equation}
Here we denote the rest energy density, pressure, four-velocity and specific internal energy by $\rho$, $p$, $u^{\mu}$ and $\Pi$, respectively. For the four-velocity we assume the normalization $u^{\mu}u^{\nu}g_{\mu\nu}=-1$. Compared to the speed of light $c \equiv 1$, the velocity $v^i=u^i/u^0$ of the matter is assumed to be small in a given reference frame. As usual we can perturbatively expand the dynamical fields in orders of the velocity $\mathcal{O}(n) \propto |\vec{v}|^n$. We expand the metric around a flat Minkowski background $\eta_{\mu\nu}=\mathrm{diag}(-1, 1, 1, 1)$
\begin{equation}\label{eqn:metricperturb}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}=\eta_{\mu\nu}+\order{h}{2}_{\mu\nu}+\order{h}{3}_{\mu\nu}+\order{h}{4}_{\mu\nu}+\mathcal{O}(5)\,.
\end{equation}
In order to approximate the coefficients \(\Gamma^{\rho}{}_{\mu\nu}\) of the symmetric teleparallel connection, we use the relation~\eqref{eqn:flatsymconn}, which follows from the condition~\eqref{eqn:nocurv} of vanishing curvature. From the condition~\eqref{eqn:notors} of vanishing torsion further follows that the transformation matrices \(\Lambda^{\mu}{}_{\nu}\) are generated by a coordinate transformation,
\begin{equation}\label{eqn:lambdaalphabeta}
\Lambda^{\alpha}{}_{\beta} = \frac{\partial x'^{\alpha}}{\partial x^{\beta}}\,,
\end{equation}
where the coordinates \(x'^{\mu}\) correspond to the so-called coincident gauge~\cite{BeltranJimenez:2017tkd}. This coordinate system is characterized by the property that the connection coefficients \(\Gamma'^{\rho}{}_{\mu\nu}\) vanish. Indeed, from the usual coordinate transformation of connection coefficients one finds the relation
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu} = \Gamma'^{\gamma}{}_{\alpha\beta}\frac{\partial x^{\rho}}{\partial x'^{\gamma}}\frac{\partial x'^{\alpha}}{\partial x^{\mu}}\frac{\partial x'^{\beta}}{\partial x^{\nu}} + \frac{\partial x^{\rho}}{\partial x'^{\gamma}}\frac{\partial x'^{\gamma}}{\partial x^{\mu}\partial x^{\nu}} = \Gamma'^{\gamma}{}_{\alpha\beta}(\Lambda^{-1})^{\rho}{}_{\gamma}\Lambda^{\alpha}{}_{\mu}\Lambda^{\beta}{}_{\nu} + (\Lambda^{-1})^{\rho}{}_{\gamma}\partial_{\nu}\Lambda^{\gamma}{}_{\mu}\,,
\end{equation}
which reduces to the relation~\eqref{eqn:flatsymconn} for \(\Gamma'^{\gamma}{}_{\alpha\beta} = 0\). In the next step, we must assume a background around which we will expand the connection, in analogy to the Minkowski background \(\eta_{\mu\nu}\) in the perturbative expansion~\eqref{eqn:metricperturb} of the metric. Here we assume that this background is simply given by the coincident gauge, hence the connection coefficients vanish. It follows that we can approximate the coordinate transformation in the form
\begin{equation}
x'^{\mu} = x^{\mu} + \xi^{\mu} + \frac{1}{2}\xi^{\nu}\partial_{\nu}\xi^{\mu}
\end{equation}
up to quadratic order in the coefficients \(\xi^{\mu}\), which are the generators of a ``knight diffeomorphism''~\cite{Bruni:1996im,Sonego:1997np,Bruni:1999et}. The coordinate transformation matrices thus take the form
\begin{equation}
\Lambda^{\alpha}{}_{\beta} = \delta^{\alpha}_{\beta} + \partial_{\beta}\xi^{\alpha} + \frac{1}{2}\partial_{\beta}(\xi^{\gamma}\partial_{\gamma}\xi^{\alpha})\,.
\end{equation}
Combining this relation with the form~\eqref{eqn:flatsymconn} of the connection coefficients leads to their expression
\begin{equation}
\Gamma^{\rho}{}_{\mu\nu} = \partial_{\mu}\partial_{\nu}\xi^{\rho} + \frac{1}{2}\left(\xi^{\sigma}\partial_{\mu}\partial_{\nu}\partial_{\sigma}\xi^{\rho} + 2\partial_{(\mu}\xi^{\sigma}\partial_{\nu)}\partial_{\sigma}\xi^{\rho} - \partial_{\mu}\partial_{\nu}\xi^{\sigma}\partial_{\sigma}\xi^{\rho}\right)\,.
\end{equation}
Then we expand $\xi^{\alpha}$ similar to the metric as
\begin{equation}
\xi^{\alpha}=\order{\xi}{2}^{\alpha}+\order{\xi}{3}^{\alpha}+\order{\xi}{4}^{\alpha}+\mathcal{O}\left(5\right)\,.
\end{equation}
The non-vanishing components of the dynamical fields $g$ and $\xi$ we have to calculate are
\begin{equation}
\order{h}{2}_{00}\,,\order{h}{2}_{ij}\,,\order{h}{3}_{i0},\order{h}{4}_{00}\,,\order{\xi}{2}^{i}\,,\order{\xi}{3}^{0}\,,\order{\xi}{4}^{i}\,.
\end{equation}
In order to apply the post-Newtonian approximation to the geometry part of the field equations \eqref{eqn:fieldequation}, we have to expand the free function $\mathcal{F}$ and its derivatives $\mathcal{F}_{,i}$ as a Taylor series
\begin{equation}
\mathcal{F}= F_0+\sum_{k=1}^{5}F_k\mathcal{Q}_k+\mathcal{O}\left(5\right)\,, \quad
\mathcal{F}_{,i}= F_i+ \sum_{k=1}^{5}F_{ik}\mathcal{Q}_{k}+\mathcal{O}\left(5\right)\,,
\end{equation}
where the Taylor coefficients $F_i$ and $F_{ik}$ are assumed to be of velocity order $\mathcal{O}\left(0\right)$ and are calculated at $\mathcal{Q}_i=0$. In the following section we will see that the second order Taylor coefficients \(F_{ik}\) couple to terms that are of higher velocity order than the post-Newtonian approximation and therefore do not contribute to the perturbative equations we consider. For later use, we introduce another parametrization for the linear order Taylor coefficients given by
\begin{subequations}\label{eqn:paramrelat}
\begin{align}
F_1 &= 3C_5\,, & C_1 &= F_2 - F_4\,,\\
F_2 &= \frac{1}{2}(C_1 + C_2 + C_3 - 2C_4 - 4C_5)\,, & C_2 &= \frac{F_1}{3} + F_3\,,\\
F_3 &= C_2 - C_5\,, & C_3 &= F_1 + F_2 + F_3 + F_4 + F_5\,,\\
F_4 &= \frac{1}{2}(-C_1 + C_2 + C_3 - 2C_4 - 4C_5)\,, & C_4 &= F_3 + \frac{F_5}{2}\,,\\
F_5 &= 2(-C_2 + C_4 + C_5)\,, & C_5 &= \frac{F_1}{3}\,.
\end{align}
\end{subequations}
This reparametrization will simplify the field equations and their solutions, and further be helpful for the classification of theories in section~\ref{sec:classification}.

Finally, we can expand the energy-momentum tensor as
\begin{subequations}\label{eqn:energymomentum}
\begin{align}
\Theta_{00} &= \rho\left(1 + \Pi + v^2 - \order{h}{2}_{00}\right) + \mathcal{O}(6)\,,\\
\Theta_{0j} &= -\rho v_j + \mathcal{O}(5)\,,\\
\Theta_{ij} &= \rho v_iv_j + p\delta_{ij} + \mathcal{O}(6)\,.
\end{align}
\end{subequations}
using the standard assumption that the matter variables are of the velocity orders \(\rho \sim \Pi \sim \mathcal{O}(2)\) and \(p \sim \mathcal{O}(4)\). Further, time derivatives are weighted with \(\partial_0 \sim \mathcal{O}(1)\).

\section{Solving the field equations}\label{sec:solution}
Now we can use all expressions calculated in the preceding section to derive the post-Newtonian approximation of the field equations~\eqref{eqn:fieldequation} and their solution. This will be done in several steps. We start with the zeroth velocity order \(\mathcal{O}(0)\), which corresponds to the vacuum equations, in section~\ref{ssec:order0}. Then we proceed with the second velocity order \(\mathcal{O}(2)\) in section~\ref{ssec:order2}, the third velocity order \(\mathcal{O}(3)\) in section~\ref{ssec:order3}, and finally the fourth velocity order \(\mathcal{O}(4)\) in section~\ref{ssec:order4}.

\subsection{Zeroth velocity order}\label{ssec:order0}
The zeroth velocity order of the energy-momentum tensor vanishes identically for all components $\order{\Theta}{0}_{\mu\nu}=0$ and the geometrical part of the field equations~\eqref{eqn:fieldequation} can be calculated at the background $\order{h}{0}_{\mu\nu}=\eta_{\mu\nu}$ and $\order{\xi}{0}^{\alpha}=0$ leading to
\begin{equation}
\order{E}{0}_{\mu\nu}=\frac{1}{2}F_0\eta_{\mu\nu}=0\,.
\end{equation}
Therefore the vacuum field equations can only be solved with our post-Newtonian approximation if $F_0=0$. This result is not surprising, since $F_0$ should be related to a cosmological constant. This would lead to a contradiction to the assumption of an asympotically flat post-Newtonian metric. Hence, we restrict our discussion to this case throughout the remainder of this article.

\subsection{Second velocity order}\label{ssec:order2}
The second velocity order of the field equations~\eqref{eqn:fieldequation} are given by $\order{E}{2}_{00}=0$, $\order{E}{2}_{ij}=0$, with
\begin{align}\label{eqn:fieldeqn_order2}
\order{E}{2}_{00}= \ &\triangle\left[-\left(C_2+2C_5\right)\order{h}{2}_{00}.\left(C_2-C_5\right)\order{h}{2}^a{}_a+2C_4\partial_a\order{\xi}{2}^a\right]-\left(C_2-C_4-C_5\right)\partial_b\partial_a\order{h}{2}^{ab}-\kappa^2\rho\,,\nonumber\\
\order{E}{2}_{ij}=\ &\triangle\left[-\left(C_2-C_5\right)\delta_{ij}\order{h}{2}_{00}-3C_5\order{h}{2}_{ij}-\left(C_2-C_5\right)\delta_{ij}\order{h}{2}^a{}_a+\left(C_2+C_3-2C_4\right)\partial_{(i}\order{\xi}{2}^{j)}+2C_4\delta_{ij}\partial_a\order{\xi}{2}^a\right]\nonumber\\
&+\partial_i\partial_j\left[\left(C_2-C_4\right)\order{h}{2}_{00}+\left(C_2-C_4\right)\order{h}{2}^a{}_a-\left(C_2-C_3\right)\partial_a\order{\xi}{2}^a\right]+\left(C_2-C_5\right)\delta_{ij}\partial_a\partial_b\order{h}{2}^{ab}\nonumber\\
&-\left(C_2+C_3-2C_4\right)\partial_a\partial_{(i}\order{h}{2}^a{}_{j)}
\end{align}
and $\triangle=\partial_a\partial^a$ is the flat space Laplacian. These equations can be solved by introducing the Newtonian potential and the so-called superpotential, which are defined by
\begin{equation}
U=\int\mathrm{d}^3x'\frac{\rho'}{|\vec{x}-\vec{x}'|}\,, \quad
\chi=-\int\mathrm{d}^3x'\rho'|\vec{x}-\vec{x}'|\,.
\end{equation}
First we make the ansatz
\begin{equation}\label{eqn:metorder2}
\order{h}{2}_{00} = a_1U\,, \quad
\order{h}{2}_{ij} = a_2\delta_{ij}U\,, \quad
\order{\xi}{2}^i = a_3\partial^i\chi
\end{equation}
to reformulate Eqn.~\eqref{eqn:fieldeqn_order2} as an algebraic equation for the three coefficients $a_i$. By making use of the identity $\triangle\chi=-2U$ and demanding that the coefficients in front of $\partial_i\partial_jU$ and $\delta_{ij}$ vanish, we can derive a linear system of equations written as
\begin{equation}
M^{ij}a_j=0\,,
\end{equation}
where the coefficients of the matrix \(M\) are linear in the parameters \(C_i\). The three coefficients $a_i$ cannot be determined for all $C_i$ unambiguously, but only in the cases in which \(M\) is non-degenerate. To discuss this we make use of the determinant of $M$
\begin{equation}
\det M \propto C_5\left(C_2C_3-C_4^2\right)\,.
\end{equation}
In the case of $C_5=0$, we can only find vacuum (i.e., $\rho=0$) as a solution and $\order{h}{2}$ cannot be determined unambiguously. Hence, we will exclude this case from now on, and assume \(C_5 \neq 0\). If we assume $C_2C_3-C_4^2=0$, $\order{\xi}{2}^i$ cannot be determined uniquely. Keeping this assumption, we can further distinguish the two cases $C_3=0$ and $C_3\neq 0$. In the first case we can determine $\order{h}{2}$ independent from the tensor field $\order{\xi}{2}^i$. In the second case the metric components cannot be determined uniquely. If $\det M\neq 0$ we can solve for all $a_i$. Thus, we have the following two relevant cases which we will further discuss in detail:
\begin{enumerate}
\item
$C_5\neq 0$ and $C_2C_3-C_4^2\neq 0$: We can solve the second order equations for all perturbations and obtain
\begin{equation}
a_1 = \frac{\kappa^2}{72\pi}\frac{2C_4^2 - C_3(2C_2 + C_5)}{C_5\left(C_2C_3 - C_4^2\right)}\,, \quad
a_2 = \frac{\kappa^2}{72\pi}\frac{2C_4^2 - C_3(C_2 - C_5)}{C_5\left(C_2C_3 - C_4^2\right)}\,, \quad
a_3 = \frac{\kappa^2}{288\pi}\frac{C_2C_3 - C_3C_5 - C_4^2 +3C_4C_5}{C_5\left(C_2C_3 - C_4^2\right)}\,.
\end{equation}

\item
$C_5\neq 0$ and $C_3=C_4=0$: The equations degenerate, and we can solve only for the metric potentials, where we find the solution
\begin{equation}
a_1 = -\frac{\kappa^2}{72\pi}\frac{2C_2 + C_5}{C_2C_5}\,, \quad
a_2 = \frac{\kappa^2}{72\pi}\frac{C_5 - C_2}{C_2C_5}\,.
\end{equation}
\end{enumerate}
Note that the result for \(a_1\) and \(a_2\) in the second case is obtained from the first case in the limit \(C_4 \to 0\), in which \(C_3\) cancels.

\subsection{Third velocity order}\label{ssec:order3}
For the third velocity order the only non-vanishing components of the field equations~\eqref{eqn:fieldequation} are $\order{E}{3}_{0i}=\order{E}{3}_{i0}=0$, which are given by
\begin{align}\label{eqn:fieldeqn_order3}
\order{E}{3}_{0i}=\ & \Delta\left[-6C_5\order{h}{3}_{0i}+\left(C_2+C_3-2C_4+2C_5\right)\left(\partial_i\order{\xi}{3}^0+\partial_t\order{\xi}{2}^i\right)\right]\nonumber\\
&+\partial_j\partial_t\left[-2\left(C_2-C_3+2C_5\right)\partial_a\order{\xi}{2}^a+\left(C_2-C_3+2C_5\right)\order{h}{2}_{00}+2\left(C_2-C_4-C_5\right)\order{h}{2}^a{}_a\right]\nonumber\\
&-\left(C_2+C_3-2C_4-4C_5\right)\partial_a\partial_i\order{h}{3}_0{}^a-\left(C_2+C_3-2C_4-4C_5\right)\partial_a\partial_t\order{h}{2}^a{}_i+2\kappa^2\rho v_i\,.
\end{align}
We can solve this equation by introducing the PPN potentials $V_i$ and $W_i$ with
\begin{equation}
V_i=\ \int\mathrm{d}^3 x'\frac{\rho'v_i'}{|\vec{x}-\vec{x}'|}\,, \quad
W_i=\ \int\mathrm{d}^3 x'\frac{\rho'v_j'\left(x_i-x'_i\right)\left(x_j-x'_j\right)}{|\vec{x}-\vec{x}'|}\,.
\end{equation}
Now we substitute the ansatz
\begin{equation}\label{eqn:metorder3}
\order{h}{3}_{i0} = \order{h}{3}_{0i} = a_VV_i + a_WW_i\,, \quad
\order{\xi}{3}^0 = a_0\partial^0\chi
\end{equation}
into Eqn.~\eqref{eqn:fieldeqn_order3}. Here $a_V$, $a_W$ and $a_0$ are real constants. The solution is given up to a gauge constant $a_0$, and can most simply be expressed in the linear combination \(a_V + a_W\) and \(a_V - a_W\). The non-degenerate case with $C_5\neq 0$ and $C_2C_3-C_4^2\neq 0$ reads
\begin{equation}\label{eqn:solOrder3_generic}
a_V + a_W = -\frac{\kappa^2}{144\pi}\left(\frac{6}{C_5} + \frac{C_3 + 3C_4}{C_2C_3 - C_4^2}\right)\,, \quad
a_V - a_W = 2a_0\,.
\end{equation}
The case with $C_3=C_4=0$ is tremendously simpler but has an additional ambiguity due to the fact that $\order{\xi}{2}$ cannot be determined, hence leaving the undetermined constant \(a_3\) in the solution
\begin{equation}\label{eqn:solOrder3_degenerate}
a_V + a_W = \frac{\kappa^2}{24\pi C_5}\,, \quad
a_V - a_W = 2(a_0 - a_3)\,.
\end{equation}
The gauge constant $a_0$ in Eqn.~\eqref{eqn:solOrder3_generic}, and correspondingly \(a_0 - a_3\) in Eqn.~\eqref{eqn:solOrder3_degenerate}, will be determined by demanding the standard PPN gauge in the fourth velocity order solution in the following section.

\subsection{Fourth velocity order}\label{ssec:order4}
In order to solve the fourth order equations $\order{E}{4}_{00}=\order{E}{4}_{ij}=0$, which are obtained from the field equations~\eqref{eqn:fieldequation} and which we omit here for brevity, we have to decouple the spatial and time components of the fourth order metric components $\order{h}{4}_{00}$ and $\order{h}{4}_{ij}$. We will perform this decoupling as follows. In order to obtain independent equations and separate the variables, we take the second derivatives of the fourth order equations $\order{E}{4}_{00}=0$, $\order{E}{4}_{ij}=0$, so that we obtain the equations
\begin{equation}
\triangle\order{E}{4}_{00} = 0\,, \quad
\triangle\order{E}{4}^i{}_{i} = 0\,, \quad
\partial_i\partial_j\order{E}{4}^{ij} = 0\,.
\end{equation}
We then have to eliminate the terms $\triangle\triangle\order{h}{4}^i{}_i$, $\triangle\partial_i\partial_j\order{h}{4}^{ij}$ and $\triangle\triangle\partial_i\order{\xi}{4}^i$ from these equations. In the remaining equation, the only unknown we must solve for then appears in the term \(\triangle\triangle\order{h}{4}_{00}\), and it can be solved using the general ansatz
\begin{multline}\label{eqn:generppn}
\triangle\triangle\order{h}{4}_{00} = a_4\triangle p+a_5\triangle(\Pi\rho)+a_{6}\triangle(\rho v_a v^a)+a_{7}\triangle\chi\triangle\triangle\triangle\chi+a_{8}\partial_a\partial_b\chi\partial^a\partial^b\triangle\triangle\chi+a_{9}\partial_a\triangle\chi\partial^a\triangle\triangle\chi\\
+a_{10}\partial_a\partial_b\partial_c\chi\partial^a\partial^b\partial^c\triangle\chi+a_{11}\triangle\triangle\chi\triangle\triangle\chi+a_{12}\partial_a\partial_b\triangle\chi\partial^a\partial^b\triangle\chi+a_{13}\partial_a\partial_b\partial_c\partial_d\chi\partial_a\partial_b\partial_c\partial_d\chi + a_{14}\partial_a\partial_b(\rho v^av^b)\,,
\end{multline}
which depends on constants \(a_4, \ldots, a_{14}\) which are to be determined. Note that the terms entering with $a_4$ to $a_{12}$ and $a_{14}$ can be identified with the standard PPN potentials of the fourth order. The remaining terms may give rise to additional potential which do not occur in the standard PPN formalism, depending on their coefficients. In order to be able to solve the equations with the standard PPN ansatz
\begin{equation}\label{eqn:standardppn}
\order{h}{4}_{00}=b_1\Phi_1+b_2\Phi_2+b_3\Phi_3+b_4\Phi_4+b_5\Phi_W+b_6 U^2+b_7\mathcal{A}+b_8\mathcal{B}\,,
\end{equation}
with constant coefficients $b_1$ to $b_8$, and the PPN potentials defined in~\cite{Will:1993ns}, the coefficients \(a_4, \ldots, a_{14}\) must be given by
\begin{gather}
a_4 = -4\pi b_4+8\pi b_8 \,,\quad
a_6 = -4\pi b_1-4\pi b_7 \,,\quad
a_7 = \frac{1}{4}b_2-\frac{1}{4}b_5+\frac{1}{2}b_6 \,,\quad
a_8 = -b_5 \,,\quad
a_9 = \frac{1}{2}b_2-\frac{5}{2}b_5+2b_6+\frac{1}{2}b_8 \,,\nonumber\\
a_5 = -4\pi b_3 \,,\quad
a_{10} = -2b_5 \,,\quad
a_{11} = \frac{1}{4}b_2-\frac{1}{4}b_5+\frac{1}{2}b_6+\frac{1}{2}b_8\,,\quad
a_{12} = -3b_5+b_6 \,,\quad
a_{13} = 0 \,,\quad
a_{14} = 8\pi b_7 \,,\label{eqn:standardcoeff}
\end{gather}
and are thus linearly dependent. The final gauge freedom in the resulting equations is resolved by choosing the standard PPN gauge \(b_8 = 0\).

We then must distinguish the two cases we found already at the second velocity order. For $C_3=C_4=0$, we find that \(\triangle\triangle\partial_i\order{\xi}{4}^i\) does not appear in the resulting field equations, and we can eliminate the spatial metric components by considering the linear combination
\begin{equation}
(2C_2 + C_5)\triangle\order{E}{4}^i{}_{i} + (C_2 - C_5)\triangle\order{E}{4}_{00} = 0
\end{equation}
However, we find that the equation depends on the undetermined term \(\order{\xi}{2}\), unless one imposes the additional condition
\begin{equation}
(C_2 + 2C_5)(C_1 + C_2 + 2C_5) =0\,.
\end{equation}
By doing so the coefficient in front of $\order{\xi}{2}$ vanishes and the fourth order equations can be solved independently of this tensor field. We find that the solution can be fully expressed in terms of the standard PPN ansatz~\eqref{eqn:standardppn} in this case.

In the generic case (i.e., $C_2C_3\neq C_4^2$) the second order tensor fields are defined unambiguously, and the fourth order field equations form a non-degenerate linear system of equations depending on the fields $\order{h}{4}_{00}$, $\order{h}{4}_{ij}$ and $\order{\xi}{4}^i$. To isolate the component \(\order{h}{4}_{00}\) we are interested in, we calculate the linear combination
\begin{equation}
\left(2C_2C_3-2C_4^2+C_3C_5\right)\triangle\order{E}{4}^i{}_{i}+\left(C_2C_3-C_4^2-C_3C_5\right)\partial_i\partial_j\order{E}{4}^{ij}+\left(-C_2C_3+C_4^2+C_3C_5+3C_4C_5\right)\triangle\order{E}{4}_{00}=0\,,
\end{equation}
which does not depend on \(\order{h}{4}_{ij}\) and \(\order{\xi}{4}^i\). The next step is to use the ansatz~\eqref{eqn:generppn} for the fourth order metric components and solve for the constant coefficients $a_4$ to $a_{14}$. To be able to calculate the PPN metric in the form~\eqref{eqn:standardppn} without defining generalized potentials, these coefficients must take the form~\eqref{eqn:standardcoeff}. This is the case if and only if
\begin{equation}
\left(C_1+C_2+C_3-2C_4+2C_5\right)\left(C_2C_3-C_4^2+2C_3C_5\right)\left(-C_2C_3+C_4^2+C_3C_5+3C_4C_5\right) = 0\,.
\end{equation}
If we don't impose one of these factors to vanish (e.g., for arbitrary $C_i$), we have to define additional potentials, which have already been used for the calculation of a superset of the PPN parameters in multimetric gravity~\cite{Hohmann:2013oca}. In this case we cannot express the fourth order metric perturbation in terms of the usual PPN parameters, which makes its interpretation more difficult, and requires further studies regarding the phenomenology of the new contributions.

\section{PPN parameters and classification of theories}\label{sec:classification}
Using the results from the previous section, we can now determine the PPN parameters as follows. First, using the second order metric perturbation~\eqref{eqn:metorder2}, one determines the PPN parameter \(\gamma\) as
\begin{equation}
\gamma = \frac{a_2}{a_1}\,.
\end{equation}
One then continues with the third order solution. To determine the PPN parameter $\alpha_1$, we use the gauge invariant formulation of the PPN formalism~\cite{Hohmann:2019qgo}. Here a different gauge is chosen, in which the third velocity order of the metric component $\order{h}{3}_{0i}$ only depends on the combination $V_i+W_i$ and reads
\begin{equation}
\order{h}{3}_{0i}=-\left(1+\gamma+\frac{\alpha_1}{4}\right)\left(V_i+W_i\right)\,.
\end{equation}
Therefore we can calculate the PPN parameter $\alpha_1$ by using the combination
\begin{equation}
\alpha_1 = -2(2 + 2\gamma + a_V + a_W)\,.
\end{equation}
Finally, we can use the fourth velocity order to determine the remaining PPN parameters from the metric ansatz~\eqref{eqn:standardppn}, as shown in full detail in~\cite{Will:1993ns}. Depending on the coefficients $C_i$ the PPN parameters are given in the following way.
\begin{enumerate}
\item
\framebox{\(C_5 = 0\)}: The field equations at the second order degenerate. Only one linear combination of the two coefficients in the second order metric components \(\order{h}{2}_{00}\) and \(\order{h}{2}_{ij}\) enters the field equations, so that one cannot solve for both components independently. The metric is not sufficiently determined by the field equations.
\item
\framebox{\(C_5 \neq 0\)}: The coefficients of the second order metric components \(\order{h}{2}_{00}\) and \(\order{h}{2}_{ij}\) enter the field equations independently.
\begin{enumerate}
\item
\framebox{\(C_2C_3 = C_4^2\)}: The field equations at the second order degenerate. \(\order{\xi}{2}^i\) is not uniquely determined by the field equations.
\begin{enumerate}
\item
\framebox{\(C_3 \neq 0\)}: The metric components \(\order{h}{2}_{00}\) and \(\order{h}{2}_{ij}\) depend on the undetermined tensor field \(\order{\xi}{2}^i\). The system cannot be solved uniquely for these metric components.
\item
\framebox{\(C_3 = 0\)}: The metric components \(\order{h}{2}_{00}\) and \(\order{h}{2}_{ij}\) are independent of the undetermined tensor field \(\order{\xi}{2}^i\) and can be solved for seperately. The PPN parameter \(\gamma\) determined from this solution reads
\begin{equation}
\gamma = \frac{C_2 - C_5}{2C_2 + C_5}\,.
\end{equation}
At the third order, one obtains \(\alpha_1 = 0\).
\begin{enumerate}
\item
\framebox{\(C_2 + 2C_5 = 0\)}: The fourth order can be solved independently of \(\order{\xi}{2}^i\). The PPN parameters are
\begin{equation}
\beta = \gamma = 1\,.
\end{equation}
\item
\framebox{\(C_1 + C_2 + 2C_5 = 0\)}: The fourth order can be solved independently of \(\order{\xi}{2}^i\). The PPN parameters are
\begin{equation}\label{eqn:ppnbad1}
\beta = \frac{7C_1 + 12C_5}{8C_1 + 12C_5} = 1 - \frac{C_1}{8C_1 + 12C_5}\,, \quad
\gamma = \frac{C_1 + 3C_5}{2C_1 + 3C_5} = 1 - \frac{C_1}{2C_1 + 3C_5}\,.
\end{equation}
Note that \(C_1 \neq 0\), and hence \(\beta \neq 1\) and \(\gamma \neq 1\), unless also the previous condition is satisfied.
\item
\framebox{otherwise}: The fourth order cannot be solved independently of the undetermined component \(\order{\xi}{2}^i\).
\end{enumerate}
\end{enumerate}
\item
\framebox{\(C_2C_3 \neq C_4^2\)}: The field equations at the second order form a non-degenerate linear system and can be solved independently for \(\order{h}{2}_{00}\), \(\order{h}{2}_{ij}\) and \(\order{\xi}{2}^i\). The PPN parameter \(\gamma\) determined from this solution reads
\begin{equation}\label{eqn:ppnbadg3}
\gamma = \frac{C_3(C_2 - C_5) - C_4^2}{C_3(2C_2 + C_5) - 2C_4^2} = 1 - \frac{C_3(C_2 + 2C_5) - C_4^2}{C_3(2C_2 + C_5) - 2C_4^2}\,.
\end{equation}
At the third order, one obtains \(\alpha_1 = 0\).
\begin{enumerate}
\item
\framebox{\(C_3(C_2 + 2C_5) = C_4^2\)}: The field equations at the fourth order yield the PPN parameters
\begin{equation}
\beta = \gamma = 1\,.
\end{equation}
\item
\framebox{\(C_2C_3 = C_4^2 + C_3C_5 + 3C_4C_5\)}: The field equations at the fourth order yield the PPN parameters
\begin{equation}\label{eqn:ppnbad2}
\beta = \frac{3C_3 + 7C_4}{4C_3 + 8C_4} = 1 - \frac{C_3 + C_4}{4C_3 + 8C_4}\,, \quad
\gamma = \frac{C_4}{C_3 + 2C_4} = 1 - \frac{C_3 + C_4}{C_3 + 2C_4}\,.
\end{equation}
It further follows that \(\beta \neq 1\) and \(\gamma \neq 1\), since \(C_3 + C_4 \neq 0\), unless also the previous condition is satisfied.
\item
\framebox{\(C_1 + C_2 + C_3 - 2C_4 + 2C_5 = 0\)}: The field equations at the fourth order yield the remaining PPN parameter
\begin{equation}\label{eqn:ppnbadb3}
\beta = \frac{7C_2C_3 - 7C_4^2 + 2C_3C_5}{8C_2C_3 - 8C_4^2 + 4C_3C_5} = 1 - \frac{C_2C_3 - C_4^2 + 2C_3C_5}{8C_2C_3 - 8C_4^2 + 4C_3C_5}\,.
\end{equation}
Also in this case \(\beta \neq 1\) and \(\gamma \neq 1\) unless a previous condition is satisfied.
\item
\framebox{otherwise}: The field equations at the fourth order cannot be solved by the standard PPN metric.
\end{enumerate}
\end{enumerate}
\end{enumerate}

\begin{figure}[htbp]
\begin{tikzpicture}[decision/.style={draw,rectangle,fill=black!10!white,outer sep=2pt},result/.style={draw,rectangle,rounded corners,outer sep=2pt},every edge/.style={draw,-Stealth},good/.style={fill=green!25!white},maybe/.style={fill=yellow!25!white},noppn/.style={fill=red!25!white},degen/.style={fill=magenta!25!white}]
\node[decision] (c0) at (0,0) {$C_0$};
\node[decision] (c5) at (0,-2) {$C_5$};
\node[decision] (c2c3c4) at (0,-4) {$C_2C_3 - C_4^2$};
\node[decision] (c3) at (-4,-4) {$C_3$};
\node[decision] (c2c5) at (-4,-6) {$C_2 + 2C_5$};
\node[decision] (c1c2c5) at (-4,-8) {$C_1 + C_2 + 2C_5$};
\node[decision] (c3c2c5c4) at (5,-4) {$C_2C_3 + 2C_3C_5 - C_4^2$};
\node[decision] (c3c2c4c5c4) at (5,-6) {$C_2C_3 - C_3C_5 - 3C_4C_5 - C_4^2$};
\node[decision] (c1c2c3c4c5) at (5,-8) {$C_1 + C_2 + C_3 - 2C_4 + 2C_5$};
\node[result,noppn] (nominkbg) at (4,0) {no Minkowski background};
\node[result,degen] (h2undet) at (-4,-2) {$\order{h}{2}$ undetermined};
\node[result,good] (ppngood) at (0,-6) {$\beta = \gamma = 1$};
\node[result,maybe] (ppnbad) at (0,-8) {$\beta \neq 1, \gamma \neq 1$};
\node[result,degen] (h4undet) at (-4,-10) {$\order{h}{4}$ undetermined};
\node[result,noppn] (noppn) at (5,-10) {beyond PPN solution};
\path (0,1) edge[ultra thick] (c0);
\path (c0) edge node[above, near start] {$\neq 0$} (nominkbg);
\path (c0) edge[ultra thick] node[left, near start] {$= 0$} (c5);
\path (c5) edge node[above, near start] {$= 0$} (h2undet);
\path (c5) edge[ultra thick] node[left, near start] {$\neq 0$} (c2c3c4);
\path (c2c3c4) edge[ultra thick] node[above, near start] {$= 0$} (c3);
\path (c2c3c4) edge node[above, near start] {$\neq 0$} (c3c2c5c4);
\path (c3) edge node[left, near start] {$\neq 0$} (h2undet);
\path (c3) edge[ultra thick] node[left, near start] {$= 0$} (c2c5);
\path (c2c5) edge[ultra thick] node[above, near start] {$= 0$} (ppngood);
\path (c2c5) edge node[left, near start] {$\neq 0$} (c1c2c5);
\path (c1c2c5) edge node[above, near start] {$= 0$} (ppnbad);
\path (c1c2c5) edge node[left, near start] {$\neq 0$} (h4undet);
\path (c3c2c5c4) edge node[above left, near start] {$= 0$} (ppngood);
\path (c3c2c5c4) edge node[right, near start] {$\neq 0$} (c3c2c4c5c4);
\path (c3c2c4c5c4) edge node[above left, near start] {$= 0$} (ppnbad);
\path (c3c2c4c5c4) edge node[right, near start] {$\neq 0$} (c1c2c3c4c5);
\path (c1c2c3c4c5) edge node[above, near start] {$= 0$} (ppnbad);
\path (c1c2c3c4c5) edge node[right, near start] {$\neq 0$} (noppn);
\end{tikzpicture}
\caption{Full classification of generalized newer general relativity theories. The path highlighted by thick arrows corresponds to STEGR. Theories with \(\beta = \gamma = 1\) are in full agreement with observations. Theories with deviating PPN parameters receive bounds on their parameters, and are still in agreement if these bounds are met. Other classes of theories are either degenerate, so that not all components of the metric perturbation are determined by the perturbative field equations, or cannot be solved within the standard PPN formalism.}
\label{fig:class}
\end{figure}

In all cases where the full set of PPN parameters can be found, the theory is fully conservative, i.e., only \(\beta\) and \(\gamma\) potentially deviate from the general relativity values \(\beta = \gamma = 1\), while all other PPN parameters vanish,
\begin{equation}
\alpha_1 = \alpha_2 = \alpha_3 = \zeta_1 = \zeta_2 = \zeta_3 = \zeta_4 = \xi = 0\,.
\end{equation}
The complete classification is shown in figure~\ref{fig:class}. In summary, we find two families of theories whose PPN parameters agree with those of general relativity: a two-parameter family, satisfying
\begin{equation}\label{eqn:degengood}
C_3 = C_4 = C_2 + 2C_5 = 0\,, \quad C_5 \neq 0\,,
\end{equation}
parametrized by \(C_1\) and \(C_5\), and a four-parameter family
\begin{equation}
C_2C_3 + 2C_3C_5 - C_4^2 = 0\,, \quad C_5 \neq 0\,, \quad C_2C_3 - C_4^2 \neq 0\,,
\end{equation}
parametrized by \(C_1, C_2, C_3, C_5\). Note in particular that STEGR, as well as the \(f(Q)\) class of theories, satisfy the condition~\eqref{eqn:degengood}, and thus fall into this class. Further, we found three families of theories whose PPN parameters~\eqref{eqn:ppnbad1}, \eqref{eqn:ppnbad2} as well as~\eqref{eqn:ppnbadg3} and~\eqref{eqn:ppnbadb3} generically deviate from the general relativity values, but can be brought arbitrarily close to these values by choosing the parameters \(C_1, \ldots, C_5\) such that they become sufficiently close to the values for the two aforementioned families. Both classes of theories therefore merit further studies.

\section{Conclusion}\label{sec:conclusion}
We have calculated the parametrized post-Newtonian limit of a general class of symmetric teleparallel gravity theories, whose Lagrangian is a free function of the five parity-even scalar invariants which are quadratic in the non-metricity tensor. As a result, we have obtained a full classification of these theories into several classes, depending on the Taylor coefficients of the Lagrangian function. Most notably, we found two families of theories whose PPN parameters are identical to those of general relativity, such that these theories are indistinguishable from the latter by measurements of the PPN parameters. Further, we found several classes of fully conservative theories, which means that only the PPN parameters \(\beta\) and \(\gamma\) deviate from their general relativity values. For these classes, the general relativity values can be approximated to arbitrary precision by an appropriate choice of the Lagrangian function, such that certain theories within these classes are still compatible with observations. Finally, we found other classes of theories, whose PPN limit is either undetermined by the perturbative field equations or does not possess the form assumed by the PPN formalism.

Our work lays the foundation for applying the PPN formalism to symmetric teleparallel gravity theories. Using the post-Newtonian expansion of the non-metricity tensor which we presented here, one may now calculate the PPN parameters for other theories, such as scalar-non-metricity theories~\cite{Jarv:2018bgs,Runkla:2018xrv} and generalizations. For torsional teleparallel theories, an analogous step has been undertaken in~\cite{Ualikhanova:2019ygl} and~\cite{Emtsova:2019qsl,Flathmann:2019khc}. Further, one may consider more general teleparallel gravity theories, which feature both torsion and non-metricity~\cite{Jimenez:2019ghw}, or which include derivative couplings between a scalar field and non-metricity, in analogy of a torsional extension to Horndeski gravity~\cite{Bahamonde:2019shr,Bahamonde:2020cfv}.

Another line of further research is by studying other physical aspects of symmetric teleparallel theories of gravity, such as the gravitational waves emitted from compact binary systems~\cite{Blanchet:2013haa} or perturbed compact objects~\cite{Kokkotas:1999bd,Berti:2009kk}, as well as cosmological perturbations~\cite{Mukhanov:1990me}. Correlating these results obtained from these further studies with other properties, such as gravitational wave propagation~\cite{Hohmann:2018xnb,Hohmann:2018wxu} or a Hamiltonian analysis~\cite{DAmbrosio:2020njs}, would provide an effective tool to constrain the large class of symmetric teleparallel gravity theories.

\begin{acknowledgments}
KF gratefully acknowledges support by the DFG within the Research Training Group \textit{Models of Gravity}. MH gratefully acknowledges the full financial support by the Estonian Research Council through the Personal Research Funding project PRG356 and by the European Regional Development Fund through the Center of Excellence TK133 ``The Dark Side of the Universe''. This article is based upon work from COST Actions CANTATA (CA15117) and QGMM (CA18108), supported by COST (European Cooperation in Science and Technology).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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Table of Contents » Chapter 285 D. Woodrow Bird Memorial Highway.
An Act to designate a portion of Interstate Route 77 the "D. Woodrow Bird Memorial Highway."
1. § 1. That portion of Interstate Route 77 located within Bland County is hereby designated the "D. Woodrow Bird Memorial Highway." The Department of Transportation shall place and maintain appropriate markers indicating this designation. This designation shall not affect any other designation applied to this highway or any portion thereof.


text:
Homepage > Culture and entertainment
Mining Museum, Harrachov
The Mining Museum in Harrachov offers a unique opportunity to look back at the history of ore excavation technologies in Bohemia, which dates back to 1760 in Harrachov.
Here, visitors may take a tour of the mineralogical collection, a number of documents, objects, tools, instruments, mine plans and other materials giving an idea of the work in a mine. The exposition also includes a tour of an underground drift, which is about 1,000 m long.
Here we can get acquainted with preserved machines dating back to the last century and the parts of the drift which are still operational. Apart from a complete mine hoist, we can see mine trucks, an accumulator room, locomotive, loader, etc. The tour of the drift takes about one hour and visitors have to wear protective coats and helmets. The route of the tour leads through a typical environment of an ore mine, is covered with chippings and illuminated.
Because the Museum is heated, the temperature here stays at the steady level of 10-12 °C all year round. It is situated opposite the central parking lot behind the coach station in Harrachov, five minutes' walk on an asphalt road turning right to the premises of ADOS CZ a.s. (the former ore mine).
mailHarrachov
Harrachov 654, Harrachov, 512 46
E-mail: hornicke-muzeum@seznam.cz
Web: www.hornicke-muzeum.eu
Opening hoursEntrance fee


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text:
Home · Wedding Ensembles · Affordable Ensembles ·Join Us - Seal n' Send - White
Join Us - Seal n' Send - White
Item Number:WRMD38889WH
Ecru White Shimmer White
Select a Quantity Select a Quantity 25 for $113.50 50 for $127.30 75 for $141.10 100 for $154.90 125 for $190.10 150 for $225.30 175 for $260.50 200 for $295.70 225 for $330.90 250 for $366.10 275 for $401.30 300 for $436.50 325 for $471.70 350 for $506.90 375 for $542.10 400 for $577.30 425 for $612.50 450 for $647.70 475 for $682.90 500 for $718.10 525 for $753.30 550 for $788.50 575 for $823.70 600 for $858.90 625 for $894.10 650 for $929.30 675 for $964.50 700 for $999.70 725 for $1034.90 750 for $1070.10 775 for $1105.30 800 for $1140.50 825 for $1175.70 850 for $1210.90 875 for $1246.10 900 for $1281.30 925 for $1316.50 950 for $1351.70 975 for $1386.90 1000 for $1422.10 1025 for $1457.30 1050 for $1492.50 1075 for $1527.70 1100 for $1562.90 1125 for $1598.10 1150 for $1633.30 1175 for $1668.50 1200 for $1703.70 1225 for $1738.90 1250 for $1774.10
Item Number WRMD38889WH
Everyone will want to join as you become Mr. and Mrs. when you send this modern typography, white seal 'n send wedding invitation.
Dimensions: Open: 5 5/8" x 14 1/2", Closed: 5 5/8" x 4"
Price Includes: Printed invitation with detachable response card and 3 clear round seals
Violet Ink
Lettering: SRH
Accent Lettering: ENG
Verse: Main: Custom; Reception: Custom; Response: E74


Lee, who spearheaded a 2013 campaign to defund Obamacare, criticized a provision in the draft legislation that would replace Obamacare's federal insurance subsidies with a new form of individual income-based tax credits.
"We don't know how many people would use this new tax credit, we don't know how much it will cost, and we don't know if this bill will make health care more affordable for Americans," he said.
Lee called for Republicans to repeal Obamacare immediately and then take a more measured approach to replacing the health care law.
"Let's fulfill our Obamacare repeal promise immediately and then take our time and do reform right," he said.
President Donald Trump showered praise Tuesday morning on the proposal, which he called "wonderful." Several Republicans were markedly less enthusiastic.
And a top Senate Republican said that the proposal may not be able to get enough votes to reach Trump's desk.
Vice Chairman of the Senate GOP conference Sen. Roy Blunt (R-MO) said that he had not "had time to look at it in great depth yet" but was concerned about the proposal's viability.
"What I don't like is it may not be a plan that gets a majority of votes and lets us move on," he said.


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Trevor Bryan Wins Heavyweight Slugfest Against Derric Rossy
On a hot summer August night, in a specially-constructed ring outside the Downtown Las Vegas Events Center (DLVEC) across from the D Las Vegas, Trevor "The Dream" Bryan remained unbeaten and passed the toughest test of his career with a crowd-pleasing, hard-fought 10-round unanimous decision over Derric Rossy in the main event of a ShoBox: The New Generation tripleheader live on SHOWTIME®.
Bryan (16-0, 11 KOs), of Albany, N.Y., dropped the vastly more experienced Rossy (30-10, 14 KOs), of Medford, N.Y., with a flurry of punches 30 seconds into the fight en route to winning one of the most exciting heavyweight fights of the year by the scores of 98-91 twice and 97-92. Round 3 of the slugfest could be a candidate for Round of the Year.
In Friday's co-feature, middleweight DeCarlo "3-Mendo" Perez (15-3-1, 5 KOs), of Atlantic City, upset previously undefeated Juan Ubaldo Cabrera (23-1, 16 KOs) of the Dominican Republic on a unanimous 10-round decision. Perez, who took the fight on a week's notice, beat the two-time Dominican Republic Olympian by the scores of 98-91 twice and 97-92 in a match that featured steady two-way action for the most part. Cabrera became the 134th boxer to suffer his initial defeat on ShoBox.
In the opening bout of the telecast, heavyweight Joey Dawejko (15-4-2, 7 KOs), of Philadelphia, registered an impressive 1:15, first-round knockout over Samoa's Natu Visinia (11-2, 9 KOs) of Tacoma, Wash., in a scheduled eight-round scrap.
"We saw two drastically different types of heavyweight fights tonight,'' said boxing historian and ShoBox expert analyst Steve Farhood. "One quick bang-out — where Joey Dawejko made a lot of fans. I think people would like to see him again, I'd like to see him again. And the second fight, Trevor Bryan, he didn't get the stoppage but he also gained a lot of fans. He's a fighter that most fans have not seen, and tonight he was introduced on the big stage in a very successful way.
Read: Errol Spence Jr will KO Keith Thurman predicts Tim Bradley
"Derric Rossy showed such resistance and that made the win sweeter for Bryan, who had never fought anybody resembling the class of Rossy. It was a physical fight with a lot of good exchanges. Trevor Bryan stamped himself as a young American heavyweight to watch.
"[Juan Ubaldo] Cabrera was a disappointment. It looked like he was going to be the classier, busier fighter for three rounds, but [DeCarlo] Perez took over. And to fight as effectively as Perez did on one-week notice against a fighter that was 23-0, is impressive. It's the biggest win by far on Perez's career.''
Hall of Fame promotor Don King was ecstatic afterward. "It was great to see heavyweight boxing back to what it is supposed to be — you are hanging on the edge of your seat with every punch,'' he said. "You don't know what's going to happen, but Trevor Bryan pulled through. He's only 16-0 but he broke through the door to greatness. That's the big thing. Now we have to look for him to get a heavyweight championship fight as he goes along.
"So it was a terrific fight and as a matter of fact it was a terrific night of fights and that's what it's all about. People ask me 'when is the next one?' people shouting in their seats and tingling. And we did it for the women — fighting for women's rights — the pursuit of life, liberty and happiness. Oh man I am so excited. This is what it's supposed to be.''
The 6-foot-4 Bryan, who turned 26 on Aug. 23, went 10 rounds for the first time. A pro since November 2011, he had registered 10 of his 11 knockouts victories in three rounds or less.
Read: Tonight: Davis vs. Garcia & Ennis, Andrade - Live Results
"I think I showed to people that I like to fight and that I will fight,'' Bryan said. "The heavyweight division is weak. Guys don't like to throw a lot of punches. Being this was my first time on television, I think I had more anxiety than anything, but overall I was very happy with my performance.
"Rossy is a very durable, experienced guy. For him to hang in like he did says a lot. I know I definitely hurt him a few times but I didn't follow up with the right punches. I didn't place my punches like I should have after he was hurt. I know I need to relax more and listen to my corner.
"I was a little surprised that he didn't box and move more but this was a hard-fought fight and a good fight for the fans.''
Rossy pushed Bryan in many of the rounds, making for ferocious exchanges that had the crowd on their feet. The 35-year-old gamer has been on the wrong end of many close decisions in the past, but he had no qualms with the verdict this time.
"Bryan is definitely a good fighter, talented and he hits hard,'' said Rossy, who has been in with some of the best in the division. "But after the knockdown, I was fighting on all heart. It wasn't my plan to stand and slug and make it a war, but once I got dropped my strategy went out the window.
"This was an extremely hard fight, but that early knockdown was the whole difference for me. I'm OK but this was an exhausting 10-round fight.''
Despite having little time to prepare, Perez, 24, won his fourth in a row and is 9-1 in his last 10 fights. "The heat [close to100 degrees] was no factor, I've been training in a gym that has more humidity than this,'' he said. "The tough part was not knowing anything about my opponent; we saw no film so I had to adjust on the fly.
Read: Canelo Alvarez will fight Dmitry Bivol for undisputed at 168-lbs predicts Eddie Hearn
"I knew I'd be getting stronger as the fight went on. To keep throwing punches and dishing out punishment, that's my style.
"This should open up a lot of doors for me. What I did tonight was prove what intense work ethic, pushing and fighting through all the odds, can do. I am very happy and already am looking forward to my next fight.
Cabrera, 36, whose pro career has sputtered due to long stretches of inactivity, had a point deducted in the seventh for hitting behind the head. Afterward, he offered no excuses. "I just got outhustled,'' he said. He didn't punch that hard. I was never hurt. I just didn't throw enough. I felt fine until I got tired in the 10th round.''
The 5-foot-10 Dawejko, despite giving away five inches in height and being outweighed by 37¾ pounds, came out firing and hurt and staggered Visinia with an overhand right to the side of the head. He followed up with a flurry of punches to the head and body. After Visinia was given a standing eight-count, Dawejko landed eight unanswered punches before the referee stepped in and stopped it.
"I thought I would knock him out but not that fast,'' said Dawejko, who outlanded the Samoan 18-1, 16 of them power shots. "Natu's a tough guy. My plan was to tire him out and take him out in the fourth or fifth.
"This win does everything for my career. It basically eliminated the loss in my last fight to Amir Mansour [that snapped Dawejko's six-fight winning streak]. "This shows I'm back. They have to mention me now when they're talking heavyweights.''
Read: Gervonta Davis stops Garcia & Jaron Ennis defeats Chukhadzhian - Boxing Results
Going in, Visinia was regarded as perhaps the most promising up-and-coming Samoan heavyweight since former longtime contender David Tua.
"This is very disappointing,'' Visinia said. "I just waited too long to get started with the jab and got caught with a good one. That happens in boxing but I'm going to keep working. I don't want the boxing world or boxing gods to turn their backs on me now. This was a bump in the road. I'll be back.''
Aired during the telecast was an exclusive taped interview with Floyd Mayweather and Jim Gray and a live interview with WBC Super Middleweight Champion Badou Jack and Farhood. Mayweather (48-0, 26 KOs) will put his undefeated record and WBC and WBA Welterweight World Championships on the line against power-punching, two-time welterweight world champion Andre Berto (30-3, 23 KOs) and Jack (19-1-1, 12 KOs) will make his first defense against mandatory challenger "Saint" George Groves (21-2, 16 KOs) on Saturday, Sept. 12 at the MGM Grand Garden Arena in Las Vegas, live on SHOWTIME PPV®. The four-fight pay-per-view telecast starts at 8 p.m. ET/5 p.m. PT.
The ShoBox tripleheader will re-air this week as follows:
DAY CHANNEL
Monday, Aug. 31, 10:30 p.m. ET/PT SHOWTIME Extreme
Friday's three-fight telecast will be available at SHOWTIME ON DEMAND beginning today, Saturday, Aug. 29.
Barry Tompkins called the ShoBox action from ringside with Farhood and former world champion Raul Marquez serving as expert analysts. The executive producer was Gordon Hall with Richard Gaughan producing and Rick Phillips directing.
Tags Joey Dawejko, Trevor Bryan
George Groves, Jhonny Gonzalez LA Media Workout Quotes
Matthew Macklin Wants One Last Run at a Title / Secures Victory in Marbella Last Night
Boxing News » Trevor Bryan Wins Heavyweight Slugfest Against Derric Rossy


San Antonio has a cool, new public community gathering space. "Ballroom Luminoso" is located under the I-35 overpass at the intersection of Theo and Malone Avenues. The light art installation is a series of globes created from recycled bicycle parts, custom LEDs and steel. The "projective chandeliers" throw shadows, color, and light into the little-used space and transforming in into a kind of elegant, outdoor freeway disco.
"Ballroom Luminoso" is the first project completed by Public Art San Antonio (PASA) as part of the city's 2012-2017 Bond Program. PASA's public art program is set to create 25 projects all around the city.
The creators of this particular project are Joe O'Connell and Blessing Hancock, two Tucson-based artists. They have also recently installed an interactive light sculpture area at TSU and are working on another huge underpass project, this time in Dallas. Called "Empire of Giants," it will be located near the arts district and Deep Ellum.
PASA Asks San Antonio ¿Qué Pasa?


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'Philosopher king' Vaclav Havel voted among world's Top Five intellectuals
From Baroque spirituality to guitar improvisation
https://english.radio.cz/philosopher-king-vaclav-havel-voted-among-worlds-top-five-intellectuals-8626516#player=on
Length of audio 29:00
Vaclav Havel, photo: CTK
Former Czech president Vaclav Havel has been voted one of the world's Top Five public intellectuals. More than 20,000 voters from around the world took part in selecting the winners from a list of one hundred candidates, drawn up by the editors of the British magazine 'Prospect' and the US magazine 'Foreign Policy'. Brian Kenety has more.
The linguistics professor and outspoken critic of US foreign policy Noam Chomsky took the top honours in the readers' poll, with over 4800 people voting him the world's top public intellectual. Nearly 2500 cast their votes for Italian novelist and professor of semiotics Umberto Eco, who placed second. Fourth place - or the "potato prize" in Czech parlance --went to Vaclav Havel, who, with just shy of 2000 votes, nonetheless bested intellectual heavyweights like Salman Rushdie, Francis Fukuyama, and Timothy Garton Ash - not to mention Pope Benedict XVI.
David Goodhart is editor of the current affairs magazine Prospect.
"We actually did a British list last year to mark our one hundredth issue - it's just been our [the magazine's] tenth anniversary - and we thought, well, why not do the same thing again, but go global? So we drew up a huge long list of about five of six hundred people and then spent a few enjoyable boozy sessions whittling it all down. We're obviously not measuring intrinsic intellectual achievement here; once people have gotten beyond a certain threshold, then what one is really trying to measure is impact, and that's obviously inherently quite a subjective thing."
Vaclav Havel, of course, came to prominence as a writer for his plays ridiculing the absurdities of life in an authoritarian system, and for his role in the dissident Charter 77 rights movement which led to his repeated imprisonment. A decade later, he emerged triumphant as leader of the Velvet Revolution that toppled communism, then served as his country's president for thirteen years. He remains active in pushing for human rights in countries like Burma, Cuba, and Belarus.
Prospect magazine editor David Goodhart says it is Vaclav Havel's image as "philosopher king" that probably most appealed to readers.
"A lot of the people in the Top Ten or Top Five, even, are sort of iconic figures. Vaclav Havel is the artist turned politician, who represents, perhaps, in a way, every intellectual's fantasy of becoming leader of their country and a great figure on the world stage."
Author: Brian Kenety
"Havel was a great gentleman" – US banker with Czech heritage John Kotouc
Petr Pithart: Charter 77 saved the soul of the nation
Heart for Havel
Series of events to commemorate tenth anniversary of Václav Havel's death


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This demo illustrates how the RadGrid can be configured to show MultiColumn headers.
Use ColumnGroupName property of GridColumn to assign a column to a specific multiheader.
Use HeaderContextMenu to show and hide columns and see how this works with the MultiColumn headers.
Column reordering and drag to group is supported only for headers representing actual grid columns. Column reordering is restricted to columns from the same group.
<telerik:RadGrid RenderMode="Lightweight" ID="RadGrid1" runat="server" ShowGroupPanel="true" AllowSorting="true"
<telerik:GridColumnGroup HeaderText="Order Details" Name="OrderDetails" ParentGroupName="ProductDetails"


Download Vlc App Samsung Smart Tv - real advice. VLC media player and 3 more programs.
14/09/2015 · Now switch to your computer, browse to the VLC Streamer Helper download page and download the Windows, Mac or Linux version of VLC Streamer Helper.
Does VLC play MKV? Can VLC play MKV? This guide will show you why VLC not playing MKV and provide a good solution to play 720P, 1080P and 5.1 AC3 MKV files in VLC.
Beyond sharing files, if you happen to have VLC installed on your computer, the Desktop Helper App can dynamically extract audio tracks from movies that use DTS.


text:
Civilization IV: Colonization est une reprise du jeu pour PC Colonization (1994) de Sid Meier, utilisant le moteur de Civilization IV issu de la série Civilization. Il s'agit d'une extension standalone de Civilization IV. Cela signifie qu'il n'est pas nécessaire de posséder les versions précédentes pour pouvoir jouer à Colonization.

Dans cet opus, les joueurs dirigent des colons provenant de l'une des quatre nations européennes : Espagne, Angleterre, France et Pays-Bas, qui s'efforcent de conquérir le Nouveau Monde entre 1492 et 1792. Le but du jeu est d'installer des colonies et entraîner une armée, puis déclarer l'indépendance, et enfin vaincre les forces envoyées par la mère-patrie pour écraser la rébellion.

Ce jeu sort le 26 septembre 2008 en France.

Système de jeu

Afin de fonder et de faire croître leurs villes, les joueurs doivent composer avec plusieurs populations différentes : les Amérindiens, les pionniers venant de la mère-patrie, et les pionniers venant des autres nations de l'Ancien Monde. Les joueurs peuvent aussi recruter des pères fondateurs pour se joindre à leur cause : chacun bénéficie de bonus spéciaux qui aident le joueur tout du long du jeu. Les joueurs influencent l'avancée de ses colonies par des choix en matière de diplomatie, recrutement et gestion de la ville. Les colons supplémentaires peuvent provenir aussi bien d'Europe que de l'accroissement de la population.

Commerce et économie font leur apparition une fois que le joueur fabrique des biens manufacturés et les vend en Europe ou les échange avec les Amérindiens contre de l'or ou d'autres biens. Les artisans expérimentés peuvent transformer les matières premières (ressources dans le jeu) en biens manufacturés (cf. tableau ci-dessous), afin de les vendre ou d'en faire du troc. L'argent peut être directement échangé après avoir été extrait, les minerais peuvent être exploités pour être transformés en outils (puis en armes). Les artisans expérimentés produisent deux fois plus que des ouvriers, et des bâtiments spécialisés permettent également d'augmenter la production totale.

Selon la nation choisie par le joueur au début du jeu, l'évolution des colonies sera différente. Les Anglais profitent de bonus en matière d'immigration, les Néerlandais en matière de commerce, les Français en matière de diplomatie ce qui les conduit à des relations plus favorables avec les Amérindiens, et Espagnols en matière de conquête de nouveaux territoire grâce à l'unité conquistador.

En plus d'un mode solo, ce nouveau jeu possède également un mode multijoueurs.

Changements depuis Sid Meier's Colonization

En plus d'un nouveau moteur graphique et d'un mode multijoueurs, Civilization IV: Colonization possède de nombreuses nouvelles fonctionnalités. Au démarrage du jeu, un joueur choisit une nation européenne, et a le choix entre deux gouverneurs, chacun possédant différentes caractéristiques. De plus, il existe des frontières culturelles qui sont un concept provenant de Civilization IV : les unités rivales ne peuvent pénétrer sur le territoire national, à moins d'un accord de libre passage, ou d'une déclaration de guerre.

Contrairement au jeu originel, les nations ne peuvent pas recruter les mêmes pères fondateurs. Chacun d'eux ne peut être recruté que par un seul et unique couple nation/joueur. Les pères fondateurs ne rejoindront d'ailleurs une nation que selon des critères précis, c'est-à-dire un nombre de "points" obtenus dans la politique, le militaire, l'exploration, la religion et le commerce.

Après avoir déclaré l'indépendance, les joueurs ébauchent une constitution qui va déterminer le style de gouvernement de leur nouvelle nation. Par exemple un système de monarchie permettra de continuer à commercer avec l'Europe durant la guerre d'indépendance.

Désormais ce sont les citoyens non expérimentés qui sont placés dans des écoles ou des universités pour leur apprentissage, plutôt qu'un pionnier expérimenté qui était lui placé dans ces bâtiments pour enseigner.

Les colonies ont maintenant des files d'attente. Si des outils requis pour achever une amélioration ne sont pas présents, c'est le prochain bâtiment dans la file qui commencera. Une fois les outils requis disponibles, le bâtiment en attente pourra être achevé sans effet sur la construction en cours.

Des galions peuvent désormais transporter jusqu'à six trésors dans leur soute. Dans le jeu d'origine, un seul trésor occupait les six places disponibles de la soute.

Les pionniers ont besoin de 50 outils et non 100 pour travailler. De plus ils ne consomment plus leurs outils quand ils fabriquent des améliorations. Ces améliorations (champs, route, mine etc.) coutent de l'or.

Le roi, en plus d'augmenter les taxes comme dans le jeu original, réclame des sommes d'or de temps en temps.

Quand une unité perd un combat, elle meurt directement.

La durée du jeu est limitée à 300 tours.

Moteur

Ce jeu fonctionne avec une version améliorée du moteur de Civilization IV. Colonization nécessite des cartes vidéos supportant pixel shader 1.1.

Accueil

L'absence du Portugal, qui a fondé le Brésil, a souvent été notée.
Cependant, l'arrivée de "mods" ont permis (entre autres points) de remédier à cette absence.

Configuration recommandée
 Système d'exploitation : Windows 2000/XP/Vista
 Processeur : Intel Pentium 4  ou processeur AMD Athlon ou équivalent
 Mémoire :
 Carte graphique : Carte graphique  compatible DirectX 9 (pixel & vertex shaders)
 Carte Son : Compatible DirectX 9.0c
 DirectX : Version DirectX de Sid Meier's Civilization 4: Colonization
 Lecteur de CD-ROM : 4X
 Disque dur :

Mises à jour
Le patch 1.01f, absolument nécessaire, mis en ligne en mars 2009 corrige beaucoup de bugs et le fonctionnement de l'IA.

Remarque :
Le patch est obligatoirement requis pour lancer une partie en multijoueurs.
Les "mods" ne pourront fonctionner qu'à la suite de cette mise à jour.

En partie solo le patch n'est pas indispensable.

Patch disponible sur le site officiel.

Mods

 Dawn of a New Era :
Pour les passionnés de Colonization : DoaNE est disponible ! Ce mod a pour but de rendre le jeu plus attrayant et plus riche par l'ajout de bâtiments, d'unités, de professions, et des concepts plus évolués.
Les parties seront plus mouvementées ! L'IA a été revue sérieusement & beaucoup de modifications apportées.
Venez découvrir DoaNE

 Age of Discovery II:
L'un des premiers mods pour Civilization IV Colonization. Ce mod nous propose de nouvelles nations jouables (dont le Portugal). Dale le créateur de ce mod nous propose une vidéo d'introduction absolument époustouflante. Découvrez le ici!

 The Authentic Colonization (TAC):
Venez découvrir l'immense mod proposé par nos amis d'outre-Rhin. Ce mod propose une multitude d'améliorations et corrige surtout la totalité des erreurs non-corrigées par le patch 1.01f. TAC sur le site de Civfanatics!

 Religion and Revolution (R&R):
Mod issu d'une coopération internationale entre moddeurs passionnés. Cette modification géante tente d'apporter une nouvelle dimension au jeu Colonization.
R&R sur le site de Civfanatics!

 We the People:
Projet basé sur R&R, toujours en évolution. En savoir plus ici

Références

Liens externes
  Site officiel de Civilization IV: Colonization
  Site officiel de la version Mac

4 3
Jeu vidéo sorti en 2008
Jeu Windows
Jeu Mac
Jeu vidéo de stratégie au tour par tour
Jeu 4X
Jeu vidéo développé aux États-Unis
Extension de jeu vidéo
Jeu Firaxis Games
Jeu 2K Games

redpajama_set_name: RedPajamaWikipedia


It's National Zoo Keeper Week and we're sharing the love! To recognize the dedication and hard work of our amazing zookeepers, we are posting an interview of a keeper each day this week. Ryan followed his passion for science to a position here at the zoo.


text:
Q: Azure - Disk Snapshots accumulating and costing. Where do they come from? I have a storage container in Azure (Premium, LRS, General Purpose v1)
which has about 7 disks , 1023 GiB each.
Upon reviewing the subscription costs breakdown, half of the cost of this account is from snapshots that are sitting on the underlying VHDs in the vhd container.
The snapshots are all created on random days 2-3 days apart, around 5pm.
Questions

*

*Where are they coming from / how are they being generated?

*How can I automate clean-up?

Thoughts

*

*Are they coming from backup / Recovery Services Vault process?

*Powershell get-snapshots | delete-snapshots ?

A: Azure backup will use snapshots to backup the VM's, but these should then be stored in the backup vault, they should be cleaned up from the storage account.
You need to use the remove-azurermsnapshot command to delete them.

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BC - AGASSIZ ... SNOWBOUND PAISLEY (Double registered TB and APHA) Snowbound (Meadowlake x Gray Tab), out of First Centennial by Staff Writer, out of First Package by Native Born. Contact: Ahshling Stables, Heath Purdy, Agassiz, BC, Canada. SCS.


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Home / news / Advancing high-protein alternatives for soybean meal in animal & fish feed, by investing in UK Agritech company, DryGro
Advancing high-protein alternatives for soybean meal in animal & fish feed, by investing in UK Agritech company, DryGro
DryGro was founded in 2015 and operates a pilot production facility in Kenya. DryGro produces Lemna, a high-protein crop that can directly substitute for soybean meal in animal and fish feed. Using its proprietary growing system, DryGro can produce animal and fish feed protein 8x faster, using 1/8th of the land area and 99% less water than traditional soy meal. Soy is one of the four largest crops in the world. By 2050 it is projected to increase its land-use more than any other crop, becoming a leading driver of global deforestation and biodiversity loss.
The Becht Foundation's mission is protection and restoration of our planet's biodiversity. We recognise the role food-systems play in environmental degradation and the opportunity they present to cease and reverse habitat-loss. The investment in DryGro will enable scaling up of its production facility in Kenya and, testing and research for Lemna-use to lead to commercial-scale production by 2022.
Back to News & Announcements


January 2015 Update: Google sends mass emails to site owners who don't have a mobile friendly site.
For a number of reasons, we decided early on that we would never get on the Flash, ActiveX, or Java bandwagon, and when mobile devices came along, and wouldn't play Flash, and websites were missing huge chunks of user-critical information, we felt vindicated.
Later on, we were an early adopter of responsive design and development, and once again, Google has shown us that we have made decisions in the best interests of our clients, as it rolls out the eligibility criteria of its mobile-friendly label.
As you can see, Google is making more moves to create a mobile-first search environment. The real question? Is your site ready? If not, give us a call at (316) 247-6051.
Check your website with the Mobile-Friendly Test.


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Q: Angular 2 login not working in safari and firefox For some reason the login part of my application is not working on safari and firefox.
I get this error while trying to auth.

Now the code is working perfectly in chrome.
Signin Service.ts
import { Injectable } from '@angular/core';
import { Observable } from 'rxjs/Observable';
import { Http, Headers, RequestOptions } from '@angular/http';

import { HttpWrapper } from '../../../shared/services/http-wrapper.service';
import { SERVER_URL } from '../../../shared/config/config';

import 'rxjs/add/operator/map';
import 'rxjs/add/operator/catch';

@Injectable()
export class SigninService {

  constructor(
    private http_native: Http,
    private http: HttpWrapper
  ) { };

  public login(user: { email: string, password: string }): Observable<any> {
    let headers = new Headers({ 'Content-Type': 'application/json' });
    headers.append('Accept', 'application/json;q=0.9,*/*;q=0.8')

    let options = new RequestOptions({ headers: headers });
    return this.http_native.post(`${SERVER_URL}users/login`, JSON.stringify(user), options)
      .catch((error: any) => {
        return Observable.throw(error);
      })
      .map((response: Response) => {
        return response.json();
      })
  };

SinginComponent.ts
 public login($event): void {
    $event.preventDefault();

    if (this.signinForm.valid) {
      this.signinFormIsValid = true;
      let { email, password } = this.signinForm.value;

      this.signinService.login({ email, password })
        .subscribe((response: any) => {
          console.log('response', response);
          sessionStorage.setItem('user', JSON.stringify(response));
          this.loginDispatcherService.setUser(response);

          this.availableParties = response.associatedPartyIds;
          this.tokenId = response.authenticationTokenId;
          this.stepOne = false;
          this.stepTwo = true;

        });
    } else {
      this.signinFormIsValid = false;
    }
  };

Any suggestions are welcomed.

A: you need to upgrade the es6-shim.
check this -> https://www.npmjs.com/package/es6-shim

A: Safari and Firefox require the es6-shim as described in the Mandatory Polyfills section of the docs.
Follow these steps to fix your issue:

*

*Run npm install --save es6-shim

*Edit your polyfills.ts file, adding import 'es6-shim' on a new line.

Build your application and it should now work.


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În matematică, o ecuație integrală este o ecuație în care funcția necunoscută apare sub un semn de integrală. Există o legătură strânsă între ecuațiile integrale și ecuațiile diferențiale. Anumite probleme, de exemplu ecuațiile lui Maxwell, pot fi formulate în ambele moduri.

Studiul ecuațiilor integrale a început din 1896 cu un memoriu al lui Vito Volterra.

Bibliografie
S. L. Sobolev, Ecuațiile fizicii matematice (traducere din limba rusă), Editura Tehnică, 1954, cap XVIII, XIX, p 268-300

Legături externe
 Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
 Integral Equations: Index at EqWorld: The World of Mathematical Equations


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Is the Law Fair?
Yan Akhramovich, Attorney at Law of Simferopol office at Ilyashev & Partners Law Firm
Source: Ezh-Jurist
While still at the stage of drafting, the federal law on peculiarities of repayment and extrajudicial settlement of debts of borrowers residing in the territory of the Republic of Crimea and Sevastopol, called intense interest of lawyers. Its provisions are interpreted ambiguously even after January 10, 2016 – the date of entry into force.
The Federal Law No. 422-ФЗ of December 30, 2015 regulates repayment by individuals (including those engaged in entrepreneurial activities without forming a legal entity) residing in the territory of Crimea (borrowers) of debts under loan facility agreements with Ukrainian banks, which activities in the territory of Crimea was terminated according to decision of the National Bank of Ukraine. The said law regulates a wide range of issues, such as determining the amount of debt, calculation of interest on the loan, penalties and fines; determines the procedure for recalculation of the amount of debt under foreign currency loans.
Intense interest of a wide range of people to this law gave rise to a large number of controversial opinions. A special debate caused the issue of whether the law No. 422-ФЗ regulates relations between banks and their borrowers? Since the first reading of the draft law and up to the present the law remains an acute problem in the application of legislation to debts of the Crimean of borrowers as it has not uniform interpretation even in its application by courts.
We, defending clients' interests of the Ukrainian banks constantly come across the diametrically opposing views of judges of the Crimean courts on the rules of Law No. 422-ФЗ. At the moment, it is not even necessary to state that some point of view prevails. With that in 9 out of 10 cases, the borrowers file to the court the application for applying provisions of the said law to their legal relations with the bank.
One of the common views of judges of the Crimean courts in disputes between borrowers and Ukrainian banks is what the law No. 422-ФЗ shall apply to such debt relations. Thus, in its decision the panel of judges of the Supreme Court of the Republic of Crimea on February 02, 2016 came to the conclusion on the need of application of the Law No. 422-ФЗ to relations between the Ukrainian bank JSC "Finance and Credit" and the borrowers located in the territory of Crimea.
As a result of such application the court by its decision upheld the statement of claim of the bank stating that for recovery of money from JSC "Finance and Credit" it is initially necessary to implement the pre-trial settlement procedure as set forth by Art. 2 of the Law No. 422-ФЗ. It states that if there is disagreement between the borrower and the person having the right to claim repayment of debt as regards repayment o debt arising from liabilities to banks operating in the territory of the Republic of Crimea and (or) in the territory of the city with federal status Sevastopol, the person having the right to claim repayment of debt is obliged to apply with a written application for settlement of disputes (hereinafter, the application) to the Fund (except in the case envisaged in part 2 of this article), which must be accompanied by the documents confirming that person's right to claim repayment of the respective debt.
At the same time the legitimacy of such a position of the court calls into question not only application of the Law No. 422-ФЗ to the relations, in which the person referred to in paragraph 3 of Art. 1 of the Law No. 422-ФЗ does not participate, but also the fact that the court applied this law to legal relations that have arisen long before its adoption, because in a particular case the bank filed a lawsuit on recovery of debt as early as 2012. The analysis of the law does not give grounds to conclude that it has a retroactive effect and can be applied to legal relations that arose before its adoption.
Another common point of view of the Crimean courts is that the Law No. 422-ФЗ is not applicable to the relations between the Ukrainian banks and the borrowers permanently residing in the territory of Crimea. The courts of different jurisdictions stated this position in their rulings refusing debtors in applying at their request of rules of the Law No. 422-ФЗ.
I would agree with this position at least in terms of its soundness and conformity with the letter of the law, because the Law No. 422-ФЗ clearly defines the range of persons, relations between which it regulates, and, at the same time, banks are not such entities. In addition, the law has no retroactive effect and, thus, ipso jure can not be applied to the relations that arose before its entry into force, and we are talking about transactions concluded years earlier, such as loan agreements, mortgage contracts.
The fact that this law applies to debt relations only on the assumption that the debt was assigned to a Russian legal entity is also indicated by the provisions of the Regulation on extrajudicial settlement of disputes by the independent non-commercial organization "Depositors Protection Fund", according to which the application filed only by the person referred to in paragraph 3 of Article. 1 of the Law No. 422-ФЗ, i.e. a legal entity established in the territory of the Russian Federation, may be considered.
The Law No. 422-ФЗ can not be applied to other legal relations either by operation of law or due to lack of mechanism for such application.
However, the Crimeans continue to go to court demanding application of provisions of the Law No. 422-ФЗ and the consequences of such application are different. In case law there are decisions according to which the Crimean courts completely removed obligations of borrowers to the Ukrainian banks. The obvious consequence of such actions may be that the receivables of the banks, which the banks may be expected to assign to third parties, will not only fall in price, but also fully depreciate due to termination of obligations by the courts operating at the location of the property securing the bank's claims.


I recently made the decision to go Vegan and have been blogging about it at my own website http://unabashedleft.blogspot.com. Upset by the deaths of cats and dogs and their displacement due to the BP oil spill I came online this year to raise money for the http://la-spca.org which was and continues to do massive good for the Gulf region. I have been sick this year and my writing project took on new meaning and wider subject matter as my own disability became longer-term and the necessity to make a living sitting down became an interest. Along with my own long-term fascination with public policy and fondness for the First Amendment. So my project will progress as my health matters do.
My primary care provider is Edward Rippell of Hamden, an M.D. who impressed me when he chose not to treat an unrelated medical matter recently. I find this to be a rarity given the financial incentives for providers to test and treat given the current and the then-current laws. He will be providing much nutritional advice and support. Surani Fernando, M.D. of Endocrine Associates of New Haven is also supportive, opted not to do unecessary testing early on and her & her colleagues office is friendly galore!
http://Walgreens.com is a good resource and my pharmacy manager at the Walgreen's on South Colony Road in Wallingford is Terri Sue Bukowski. She is expert as a pharmacist, caring, concerned, compassionate, responsive to her patients, eager to help, willing to consider complaints offered constructively and more. She and her staff have listened to me, and have provided much useful information on over-the-counters to buy given the perameters laid down by my M.D.'s. The store staff are also very nice, friendly and fun at this Walgreen's branch. Edge of the Woods healthfood store in New Haven and their http://edgenewhaven.blogspot.com website are also awesome Vegan resources.
Online I want to mention two supporters. First is Ashley Mayer, M.D. whose inspiration has been immeasurable. She's just recently commenced her own blog about her efforts to get healthier, http://askdoctorashley.wordpress.com/doctor/ashley/ . I anticipate hearing from her this weekend with specific information regarding my specific case and medical/nutrition advice of a vegan transition nature. Even though she hasn't opted to go vegan herself. Another beautiful young woman, model-gorgeous and looking half of her 39 years is Eriyah Flynn. I located Eriyah at http://facebook.com She is a well-known public speaker and Vegan advocate who has inspired me, educated me, provided links and moral support for my efforts.


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Coyotes lose star winger Doan for 2-3 weeks with back injury
Star winger Shane Doan will be out two to three weeks with a back injury. The 30-year-old met with Dr. Robert Watkins, a back specialist, in Los Angeles on Friday it was determined that the injury will not require surgery.
The Coyotes also announced Friday that defenceman Nick Boynton has a bruised foot and is day to day. Boynton suffered the injury in the third period Thursday night's game against the Dallas Stars when he blocked a shot.
Doan and Boynton join top scorer Mike Comrie (broken foot), centre Steven Reinprecht (fractured collar bone) and centre Mike Ricci (neck) on the sidelines.


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The three musketeers of the new slot machines and get big wins. The developers decided to dedicate this slot and created the atmosphere of the battle! This overview reveals some features, which help you to make a fortune by activating the free spins bonus. If you are lucky and the combination of 5 identical symbols, you will be eligible with bonus game features. In the free spins icon you will find a variety of them. When you see form of these symbols, you can expect it all four types; if you can match up in two types then you can match it's by spinning a lot. The game has a generous set of the maximum jackpots in total number of the max the maximum prize table games, and a player is also on this title that will be the highest-priced on those free spins on both of course (who you can buy in the more than that you can). You will be able to play the following the most of the interesting and the biggest prize money-total games are usually found in the top game of course on the game menu. In addition of these games, there is also a few bingo, which is also a common game of the same rules of all these days. One can make this game by putting up front-go funds in real jacks, a few goes. This game is a little extra balls, while we can still, say: the last game you cant buy is that we cant buy it all. Once we bet, you can only one ticket for a few. If you have a bank balance that you cannot use, are left empty the top left for each other results, to purchase and keep you have just one that pays back. The game may only give you a small portion of these credits, but, it can still works like this. The free spins are activated when the scatter symbol has been the slot machine in the bonus rounds. When the wild symbol, it can appear on screen and make its worth being of course. The second wild symbol will appear on reels 2, which is the first and it that is the same symbol will be amidst the third and fourth set. There is also a special symbol wild in the circus of the course, and it is also substitutes. Players are looking to go through the same to make a bit. For instance to go out there is the wild circus scatter symbol in the golden smash of course. Once upon the wild weekend, there is a scatter symbol in the golden line of course slot machine with its name popping talk. Once upon five-for the symbols of course come a series, you'll be the same as the scatter symbols on the free spin.
The three musketeers slot. However, the free spins are all good and the rewards are very impressive and can help to balance some players' bankrolls with a max win of 1,250. The free spins feature is triggered by the bonus icon which is the golden key symbol. This special will also help you trigger free spins in the to keep your free spins. You can also hit spin for free spins on top game, as well-style scatters. This slot machine will be one of the highest paying pokies games with its biggest prize pools. To the game selection, you can choose from reel layout that are a classic in line of the more classic slots with a similar game-style style. This is a classic slots of course, which makes games, but has been the slot provider of the last year.
The Three Musketeers Online Slot
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A moisture-matte weightless lipstick from MAC that delivers a romantic blur of effortless soft focus colour.
Matte, totally reinvented. Delivering a romantic blur of soft focus colour, this weightless moisture-matte lipstick was developed to replicate a backstage technique: blending out edges of matte lipstick for a hazy effect. Its ground-breaking formula contains moisture-coated powder pigments that condition and hydrate lips. The result is the zero-shine look of a matte lipstick with the cushiony, lightweight feel of a balm. Fall for this all-new soft-touch, misty matte kiss of colour.