text:
\section{Introduction}
Ultrarelativistic heavy-ion collisions  are a major experimental tool to study nuclear matter in an extremely hot environment. The energy density in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL and the Large Hadron Collider (LHC) at CERN is so high that partonic degrees of freedom are liberated from nucleons and a deconfined quark-gluon plasma (QGP) is formed.
The QGP then expands hydrodynamically as a fluid with very small shear viscosity over entropy ratio $\eta /s =(1$--$2)/(4\pi)$~\cite{Heinz:2013th,Luzum:2013yya}.
The hydrodynamic paradigm for heavy-ion collisions has been very successful in explaining the various collective flow observables as a dynamical response to event-by-event fluctuations of the initial fireball shape~\cite{Heinz:2013th,Teaney:2009qa,Luzum:2013yya,Gale:2013da,Romatschke:2009im}.

Recently, attention has been paid to another source of fluctuations in the hydrodynamic picture, namely, thermal fluctuations~\cite{Gavin:2006xd,Kapusta:2011gt,Yan:2015lfa,Young:2014pka,Kapusta:2012zb,Murase:2016rhl,Nagai:2016wyx}.
Thermal fluctuations are theoretically required by the fluctuation-dissipation theorem.
Furthermore, thermal fluctuations play an important role in systems with a small number of particles and are essential near the critical point, which is the focus of the ongoing beam energy scan program at RHIC~\cite{Kumar:2013cqa}.

A unique feature of hydrodynamic fluctuations in heavy-ion collisions is the rapidly expanding background flow along the beam direction, which at midrapidity is often modelled as one-dimensional Bjorken flow~\cite{Bjorken:1982qr}.
The distribution of fluctuations around such evolving background is characterized  by a specific wave number scale $k_*$, where the longitudinal expansion and ($k$-dependent) relaxation rates balance, and the distribution function approaches a nonequilibrium steady state.
In the previous publication, we developed an effective kinetic description for \emph{conformal} hydrodynamic fluctuations around the characteristic scale $k_*$ and discussed how to deal with ultraviolet divergences associated with short
wavelength fluctuations~\cite{Akamatsu:2016llw}.
Using the hydrokinetic theory we obtained a universal renormalization of the pressure and shear viscosity in agreement with previous diagrammatic calculations around a nonexpanding background~\cite{Kovtun:2003vj,Kovtun:2011np}.
Furthermore, we applied the hydrokinetic approach to the Bjorken expansion, and found the precise coefficient of the fractional-power-law tail arising from the out-of-equilibrium distribution of hydrodynamic fluctuations.

In this paper, we consider a relativistic \emph{nonconformal} fluid, for which the speed of sound $c_s^2(T)\neq 1/3$ and the bulk viscosity is finite.
The bulk viscosity determines the dissipative correction to the pressure in response to an isotropic expansion or compression and is a measure for scale symmetry breaking.
For example, perturbative calculations in a high-temperature QGP show that it is proportional to the square of the scale symmetry breaking factors (the QCD running coupling and finite quark mass)~\cite{Arnold:2006fz}.
Also, lattice QCD simulations suggest a correlation between the bulk viscosity and the scale symmetry breaking realized in the equation of state~\cite{Meyer:2007dy}.
Spectral sum rules in the bulk channel also indicate some correlation between the bulk viscosity and a nonconformal nature of the equation of state~\cite{Kharzeev:2007wb,Karsch:2007jc,Moore:2008ws,Romatschke:2009ng}.
Finally, near the critical point, the bulk viscosity diverges because of the critical slowing down~\cite{Onuki}.

In the main part of the paper we apply our hydrokinetic theory to a static system perturbed by an isotropic expansion and compute the response function of the energy-momentum tensor in the bulk channel.
We discuss the case of Bjorken expansion in \app{app:bjorken}.
In a nonconformal fluid the two-point correlation function of hydrodynamic fluctuations contributes to the trace of the energy momentum tensor, which gives rise to a renormalization of the bulk viscosity:
\begin{align}
\label{eq:zetaIntr}
\zeta(T)&=
\zeta_0(T;\Lambda) \\
& \quad+\frac{T\Lambda}{18\pi^2}
\left[
\begin{aligned}
&\left(1+\frac{3T}{2}\frac{dc_{s0}^2}{dT} -3c_{s0}^2\right)^2\frac{e_0 + p_0}{\zeta_0+\frac{4}{3}\eta_0} \\
&+4\left(1-3c_{s0}^2\right)^2\frac{e_0 + p_0}{2\eta_0}
\end{aligned}
\right]. \nonumber
\end{align}
Here, $\Lambda$ is a UV cut-off for the hydrodynamic fluctuations and $\zeta_0(T;\Lambda)$ is the bare bulk viscosity.
The fluctuation contribution to the bulk viscosity is positive and proportional to the scale symmetry breaking factors in the equation of state.
It is noteworthy that to arrive at \Eq{eq:zetaIntr}, the temperature of the background fluid must be shifted depending on the cut-off so as to include the energy of the non-equilibrium hydrodynamic modes (see \Sec{sec:tempshift} for details).

The fluctuation-induced renormalization in \Eq{eq:zetaIntr} can be used to estimate a lower bound of the bulk viscosity of QCD --- see \Ref{Kovtun:2011np} for a  similar estimate of the shear viscosity.
Very recently the approach  was also used to estimate the bulk viscosity of a nonrelativistic cold Fermi gas, where the
renormalization was obtained with  diagrammatic methods~\cite{Martinez:2017jjf} (we performed the diagrammatic calculation for the relativistic nonconformal fluid in \app{app:diagrams}).
Using the lattice equation of state for entropy density $s(T)$ and  the speed of sound $c_s^2(T)$~\cite{Borsanyi:2016ksw} in \Eq{eq:zetaIntr}, we calculate the magnitude of bulk viscosity renormalization by setting $\zeta_0=0$, and choosing representative values of the kinematic viscosity, $\eta/s=1/4\pi$, and the temperature-dependent UV
cut-off $\Lambda=2T-4T$ (see \Fig{fig:bulk}).
\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{bulk}
	\caption{A fluctuation-induced bulk viscosity bound as a function of temperature, \Eq{eq:zetaIntr}, for lattice parametrization of the QCD equation of state and shear viscosity over entropy $\eta/s=1/(4\pi)$~\cite{Borsanyi:2016ksw}.
	The UV bound  of hydrodynamic fluctuations $\Lambda$ is varied between $2T$ and $4T$.}
	\label{fig:bulk}
\end{figure}
Because of the small deviation from scale symmetry at high temperatures the bulk viscosity renormalization is vanishing small for $T\gg T_c$.
However, the degree of nonconformality $(c_s^2-\frac{1}{3})^2$ peaks around the pseudocritical temperature where the bulk viscosity reaches $\zeta/s\sim0.03-0.06$ at $T_c\sim150\,\text{MeV}$.

The logic of the estimate in \Fig{fig:bulk} is the following.
The physical bulk viscosity $\zeta(T)$ (which is independent of $\Lambda$) arises from two contributions: the fluctuations above $\Lambda$, which at weak coupling are dominated by single-particle excitations, and the fluctuations below $\Lambda$,  which are described by hydrodynamics.
We have only included the hydrodynamic fluctuations here, and thus we expect the physical bulk viscosity to be larger than the estimate shown in \Fig{fig:bulk}.

The organization of this paper is as follows.
In Sec.~\ref{sec:kinetics}, we derive the kinetic equations for hydrodynamic fluctuations for an isotropically expanding nonconformal fluid.
Then in Sec.~\ref{sec:nonlinear}, we compute the fluctuation contributions to the energy-momentum tensor, and discuss the subtle temperature shift.
After the temperature shift, we renormalize the energy density, the pressure, and the bulk viscosity, and find the finite long-time tails for the weak isotropic expansion.
The summary of the paper is given in Sec.~\ref{sec:summary}. Finally, in  \app{app:bjorken} we repeat the computation of the temperature shift and the renormalization of hydrodynamic fields for Bjorken expansion.
In \app{app:diagrams}, we give a diagrammatic derivation for the bulk viscosity renormalization, which is consistent with our results by the hydrokinetic theory.

\section{Kinetic equations for hydrodynamic fluctuations}
\label{sec:kinetics}
In this section we apply the formalism developed in \Ref{Akamatsu:2016llw} to a nonconformal fluid under isotropic expansion (or compression).
We will follow the same procedure to derive the relaxation type equations for the two-point correlation functions under the presence of background perturbations.

The governing equations for nonconformal hydrodynamics with noise are given by~\cite{LandauStatPart1, LandauStatPart2, [for a recent review: ]Kovtun:2012rj}
\begin{subequations}
\label{eq:hydro+noise}
\begin{align}
d_\mu T^{\mu\nu} &=0\label{eomzero},\quad
T^{\mu\nu}=T^{\mu\nu}_\text{ideal}+T^{\mu\nu}_\text{visc.}+S^{\mu\nu},\\
T^{\mu\nu}_{\rm ideal} &= (e + p) u^{\mu} u^{\nu} + pg^{\mu\nu},\\
T^{\mu\nu}_{\rm visc.} &= - \eta \sigma^{\mu\nu} - \zeta \Delta^{\mu\nu}\Delta^{\alpha\beta}d_{\alpha}u_{\beta},\\
\sigma^{\mu\nu}  &= \Delta^{\mu\rho}\Delta^{\nu\sigma} (d_{\rho} u_{\sigma} + d_{\sigma} u_{\rho} - \frac{2}{3}
g_{\rho\sigma} d_{\gamma} u^{\gamma}), \\
\Delta^{\mu\nu} &= g^{\mu\nu} + u^{\mu} u^{\nu},
\end{align}
\end{subequations}
where $d_{\mu}$ denotes a covariant derivative using the ``mostly-plus" metric convention.
Below we notate the divergence of the flow velocity as $\nabla \cdot u \equiv d_{\mu} u^{\mu}$.
The variance of the stochastic noise is determined by the fluctuation-dissipation theorem:
\begin{align}
&\langle S^{\mu\nu}(x_1)S^{\alpha\beta}(x_2)\rangle \nonumber\\
&= 2T\left[
\begin{aligned}
&\eta \left(\Delta^{\mu\alpha}\Delta^{\nu\beta} + \Delta^{\mu\beta}\Delta^{\nu\alpha}\right) \\
&+\left(\zeta - \frac{2}{3}\eta\right)\Delta^{\mu\nu}\Delta^{\alpha\beta}
\end{aligned}
\right] \frac{\delta(x_1-x_2)}{\sqrt{-\det g_{\mu\nu}}} .
\end{align}
Differently from the conformal case, both shear $\eta$ and bulk $\zeta$ viscosities are now present in the equation of motion and noise correlator.
\subsection{Background fluid}
Dynamics of hydrodynamic fluctuations on a background fluid in a weak isotropic expansion (or compression) is conveniently studied in the reference frame of the fluid.
In the comoving frame for the isotropic expansion, the metric is time dependent,
\begin{align}
ds^2=-dt^2 + (1+h(t))d\vec x^2, \ \ \ (|h(t)|\ll 1)
\end{align}
and the background fluid satisfies
\begin{align}
\label{eq:hydroeq_bkg}
0=\dot e_0(t) + \frac{3\dot h}{2}[e_0(t) + p_0(t)] + \mathcal O(h^2).
\end{align}
The second term on the right-hand side represents the change of energy density from the expansion and the associated work done by the pressure.
Throughout this paper, $X_0$ denotes a quantity $X$ of the background fluid in a perturbed metric ($h\neq 0$).
As discussed previously~\cite{Akamatsu:2016llw}, $e_0(t)$ and $p_0(t)$ denote the background  energy density and pressure from modes with wavenumbers greater than a cut-off $\Lambda$.
In \Sec{sec:tempshift} we detail how $e_0$ and $p_0$ are related to the lattice equation of state.

Solving perturbatively in $h$, the energy density $e_0(t)$ for the background fluid evolves as
\begin{align}
\label{eq:e_bkg}
e_0(t) = \bar e_0 -\frac{3h(t)}{2}(\bar e_0+\bar p_0) + \mathcal O(h^2),
\end{align}
where $\bar e_0$ denotes the energy density of the background fluid in an unperturbed state ($h=0$).
Again, throughout this paper $\bar{X}_0$ denotes a quantity $X$ of the background fluid in an unperturbed state ($h=0$).

\subsection{Evolution of hydrodynamic fluctuations}

For the expanding background described by \Eq{eq:e_bkg}, the hydrodynamic fluctuations excited by thermal noise $\delta e(t,{\bm x})\equiv e(t,{\bm x})-e_0(t)$ and $\vec{g}\equiv(e_0(t)+p_0(t))\vec{v}(t,{\bm x})$ evolve according to the following equations in ${\bm k}$ space:
\begin{subequations}
\label{eq:eom1}
\begin{align}
0&=\partial_t\delta e + ik^i g_i + \frac{3\dot h}{2}(1+c_{s0}^2)\delta e,\\
0&=\partial_t g_i + i c_{s0}^2 k_i \delta e + \frac{3\dot h}{2}g_i \nonumber \\
&\quad +\gamma_{\eta 0} (k^l k_l \delta_i^j-k_ik^j) g_j + \gamma_{\zeta0}k_ik^j g_j+\xi_i,
\end{align}
\end{subequations}
with noise correlation given by
\begin{align}
\langle \xi_i(t, \bm k) \xi_j(t', -\bm k')\rangle &= \frac{2T_0(e_0+p_0)}{\sqrt{-\text{det}\,g_{\mu\nu}}}(2\pi)^3\delta(\bm k-\bm k')\delta(t-t')\nonumber \\
&\quad \times\left[\gamma_{\eta 0} (k^l k_l g_{ij}-k_ik_j) + \gamma_{\zeta 0}k_ik_j\right].
\end{align}
Here $\gamma_{\eta}\equiv \eta/(e+p)$ and $\gamma_{\zeta}\equiv (\zeta+\frac{4}{3}\eta)/(e+p)$ are kinematic viscosities.
Analysis becomes simpler by utilizing a vielbein formalism.
We introduce new variables
\begin{subequations}
\begin{align}
G_{\hat i} &\equiv \left(1+\frac{1}{2}h(t)\right)g^i, \\
K_{\hat i} &\equiv \left(1-\frac{1}{2}h(t)\right)k_i,\\
\Xi_{\hat i} & \equiv \left(1+\frac{1}{2}h(t)\right)\xi^i,
\end{align}
\end{subequations}
which give $G_{\hat i}G_{\hat i}=g_i g^i$, $K_{\hat i}K_{\hat i}=k_i k^i$, and $G_{\hat i}K_{\hat i} = g^i k_i=g_i k^i$.
We define a four-component vector $\phi_a\equiv (c_{s0}\delta e,\vec G)$ of hydrodynamic fluctuations.
The equation of motion for $\phi_a$ is
\begin{subequations}
   \label{eq:matrices}
\begin{align}
\label{eq:phi_evol}
-\dot\phi_a(t,\bm k)
&=i\mathcal{L}_{ab}\phi_b + \mathcal{D}_{ab}\phi_b + \Xi_a + \mathcal{P}_{ab}\phi_b,\\
\label{eq:matrixL}
&\hspace{-4em}\mathcal{L} = \left(
\begin{array}{cc}
0 & c_{s0}\vec K\\
c_{s0}\vec K & 0
\end{array}
\right), \\
\label{eq:matrixD}
&\hspace{-4em}\mathcal{D} = \left(
\begin{array}{cc}
0 & 0\\
0 &
\gamma_{\eta 0}\left(K^2\delta_{\hat i\hat j}-K_{\hat i}K_{\hat j}\right)
+ \gamma_{\zeta 0}K_{\hat i}K_{\hat j}
\end{array}
\right), \\
\label{eq:matrixP}
&\hspace{-4em}\mathcal{P} = \dot h\left(
\begin{array}{cccc}
\frac{3}{2}\left(1+\bar c_{s0}^2 + \frac{\bar T_0}{2}\frac{d\bar c_{s0}^2}{d\bar T_0}\right) & & &\\
 & 2 & & \\
 & & 2 & \\
 & & & 2
\end{array}
\right),
\end{align}
\end{subequations}
with noise correlation given by
\begin{align}
&\langle \Xi_a(t,\bm k)\Xi_b(t', -\bm k') \rangle \nonumber \\
&\quad =\frac{2T_0(e_0+p_0)}{\sqrt{-\text{det}\,g_{\mu\nu}}}\mathcal{D}_{ab} (2\pi)^3\delta(\bm k-\bm k')\delta (t-t').
\end{align}
The matrices $\mathcal L$ and $\mathcal D$ originate from ideal and viscous parts of the hydrodynamic equations respectively, while $\mathcal P$ arises from the remaining interactions between the fluctuations and the background fluid.
Note that the term $\propto \frac{\bar T_0}{2}\frac{d \bar c_{s0}^2}{dT_0}$ in $\mathcal P$ derives from the time dependence of $c_{s0}(T_0(t))\delta e$ in $\phi_1$.
In the kinetic regime, $\mathcal{L}$ drives the evolution of $\phi_a$ so that it will be more convenient to analyze \Eq{eq:phi_evol} in terms of eigenmodes of $\mathcal L$:
\begin{align}
(e_{\pm})_a = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
\pm\hat{K}
\end{pmatrix},
\quad
(e_{T_1})_a =
\begin{pmatrix}
0 \\
\vec{T}_1 \\
\end{pmatrix},
\quad
(e_{T_2})_a =
\begin{pmatrix}
0 \\
\vec{T}_2 \\
\end{pmatrix}.\nonumber \\
\label{eq:vectors}
\end{align}
Here $\hat K \equiv \vec K/|K|$, $\vec T_1$, and $\vec T_2$ form an orthonormal basis.
The subscripts $+,-$  stand for the two sound modes and $T_1,T_2$  for the two transverse diffusive modes.
The corresponding eigenvalues are $\lambda_{\pm}=\pm c_s K$ and $\lambda_{T_1,T_2}=0$.

\subsection{Kinetic equations for hydrodynamic fluctuations}
The two-point correlation functions of $\phi_{A} \equiv \phi_a \left(e_{A}\right)_a$ with $A=+,-,T_1, T_2$ are defined as
\begin{align}\label{eq:NAA}
\left\langle \phi_A(t,\bm k) \phi_B(t,-\bm k')\right\rangle \equiv N_{AB}(t,\bm k)(2\pi)^3\delta(\bm k - \bm k') .
\end{align}
We will determine the equations of motion for $N_{AB}(t,{\bm k})$ using the formalism of \Ref{Akamatsu:2016llw}.
In the rotating wave approximation, the off-diagonal part of the density matrix $N_{AB}$ can be neglected because of its rapid phase rotation\footnote{
$N_{T_1T_2}$ has a stationary phase but vanishes because of the rotational symmetry.
}, while the diagonal part evolves according to
\begin{align}
\label{eq:2pt-func}
\dot N_{AA} = -2\mathcal{D}_{AA}\left[N_{AA} - \frac{T_0(e_0+p_0)}{\sqrt{-\det g_{\mu\nu}}}\right]
+2\mathcal{P}_{AA}N_{AA},\nonumber \\
\end{align}
where we have defined $\mathcal{D}_{AA}\equiv (e_{A})_a \mathcal{D}_{ab}(e_A)_b$ and similarly $\mathcal{P}_{AA}$.
The isotropic system does not distinguish the two transverse modes $T_1$ and $T_2$, and thus we only have two independent kinetic equations: one  for the sound modes ($L=++,--$), and one for the transverse modes ($T=T_1T_1,T_2T_2$).
Using the matrices and eigenvectors of the previous section, \Eq{eq:2pt-func} evaluates to
\begin{subequations}
\begin{align}
\label{eq:hydrokin_L}
\dot N_{L} &= -\gamma_{\zeta 0} K^2 \left[N_{L} - \frac{T_0(e_0+p_0)}{\sqrt{-\det g_{\mu\nu}}}\right] \nonumber \\
& \quad -\frac{\dot h}{2}\left(3\bar c_{s0}^2+\frac{3\bar T_0}{2}\frac{d\bar c_{s0}^2}{d\bar T_0}+7\right)N_{L},\\
\label{eq:hydrokin_T}
\dot N_{T} &= -2\gamma_{\eta 0} K^2 \left[N_{T}- \frac{T_0(e_0+p_0)}{\sqrt{-\det g_{\mu\nu}}}\right]-4\dot h N_{T}.
\end{align}
\end{subequations}
The kinetic equations \eqref{eq:hydrokin_L} and \eqref{eq:hydrokin_T} describe how the distribution of fluctuations $\phi_A$ evolves on the isotropically expanding background.
Perturbative solutions of the kinetic equations for $|h|\ll 1$ take the form,
\begin{align}
\label{eq:kinsol}
N_{L/T}(t,\bm k)&=N_{\rm eq}(t) + \delta N_{L/T}(t,\bm k) + \mathcal O(h^2),
\end{align}
where the equilibrium contribution is
\begin{align}
\label{eq:kinsol_eq}
N_{\rm eq}(t)&=\frac{T_0(e_0+p_0)}{\sqrt{-\det g_{\mu\nu}}} \nonumber \\
&\simeq \left[1-(3+3\bar c_{s0}^2)h(t)\right]\bar T_0(\bar e_0 + \bar p_0),
\end{align}
and the non-equilibrium correction $\delta N_{L/T}$ is
\begin{subequations}
   \label{eq:kinsol_neq}
\begin{align}
\label{eq:kinsol_L}
\delta N_L(\omega,\bm k)&=\frac{\frac{1}{2}i\omega h(\omega)}{-i\omega +\bar\gamma_{\zeta 0} K^2}
\bar C_{\zeta 0} \bar T_0(\bar e_0+\bar p_0),\\
\label{eq:kinsol_T}
\delta N_T(\omega,\bm k)&=\frac{i\omega h(\omega)}{-i\omega + 2\bar\gamma_{\eta 0} K^2}
\bar C_{\eta 0}\bar T_0(\bar e_0+\bar p_0).
\end{align}
\end{subequations}
Here and below we have defined
\begin{subequations}
	\label{eq:C}
\begin{align}
C_{\zeta}(T) &\equiv  1+\frac{3T}{2}\frac{dc_s^2}{dT}-3c_s^2,\\
C_{\eta}(T) &\equiv  1-3c_s^2.
\end{align}
\end{subequations}
Note that when the background fluid is scale invariant $e_0=3p_0$, the corrections $\delta N_{L/T}$ vanish.
Therefore in conformal case, the isotropic expansion or compression does not drive the hydrodynamic fluctuations from the equilibrium distribution $N_\text{eq}(t)$ given by \Eq{eq:kinsol_eq}.

For $k\sim k_{*}$ the distribution of fluctuations in \Eq{eq:kinsol_neq} is not well characterized by the time derivatives of $h(t)$.
However, at large $k$ by the distribution approaches equilibrium with calculable first derivative corrections:\footnote{In the current setup  $\nabla \cdot u=\tfrac{3}{2} \dot h$. }
\begin{subequations}
   \label{eq:nhighk}
   \begin{align}
\delta N_L(t,\bm k)&\simeq - \frac{3
\bar C_{\zeta 0} \bar T_0(\bar e_0+\bar p_0)
      }{\bar\gamma_{\zeta 0} K^2} \,
      \nabla \cdot u,\\
\label{eq:kinsol_T}
\delta N_T(t,\bm k)&\simeq -\frac{3
\bar C_{\eta 0}\bar T_0(\bar e_0+\bar p_0)
      }{2\bar\gamma_{\eta 0} K^2} \,
       \nabla \cdot u.
\end{align}
\end{subequations}
It is these corrections $\propto\nabla\cdot u/K^2$ which are responsible for the renormalization of the bulk viscosity  and the
temperature shift described in \Sect{sec:nonlinear}.

\section{Energy-momentum tensor with nonlinear fluctuations}
\label{sec:nonlinear}
In this section we compute the nonlinear contributions of hydrodynamic fluctuations to the statistically averaged energy momentum tensor $\left<T^{\mu\nu}\right>$.
The main difference from the conformal case~\cite{Akamatsu:2016llw} is additional contributions to the averaged energy
density $\left<T^{tt}\right>$, which are absorbed by a shift in the background temperature $T_0(t,\Lambda)$.
\subsection{Averaged energy-momentum tensor}
The averaged stress tensor consists of contributions from the background fluid and from the two-point functions of the hydrodynamic fluctuations:
\begin{subequations}
\begin{align}
   \label{Tijtotdef}
\langle T^{ij}\rangle&=\left[1-h(t)\right]p_0\delta^{ij} - \frac{3}{2}\dot h(t)\zeta_0\delta^{ij} + T^{ij}_{\rm fluct},\\
T^{ij}_{\rm fluct}
&\simeq \frac{1-h(t)}{e_0+p_0}\left[
\begin{aligned}
&\langle G_{\hat i}(t,\bm x) G_{\hat j}(t,\bm x)\rangle  \\
&+ \delta^{ij}\frac{T_0}{2}\frac{dc_{s0}^2}{dT_0}\langle (c_{s0}\delta e(t,\bm x))^2\rangle
\end{aligned}
\right].
\end{align}
\end{subequations}
The energy density fluctuations $\propto \langle (c_{s0}\delta e(t,\bm x))^2\rangle$ originate from the second-order derivative $\frac{d^2 p_0}{de_0^2}$, which is finite for a nonconformal equation of state.
The trace of the stress tensor from the fluctuations is determined by the two-point functions $N_{L/T}$:
\begin{align}
T^{ii}_{\rm fluct}
&= \frac{1-h(t)}{e_0+p_0}\int\frac{d^3k}{(2\pi)^3}\\
& \quad \times \left[
\left(1+\frac{3T_0}{2}\frac{dc_{s0}^2}{dT_0}\right)N_L(t,\bm k)+ 2N_T(t,\bm k)
\right]. \nonumber
\end{align}
This integral is divergent and is regularized by introducing a cut off $\Lambda$ for $K$ (not $k$).
Substituting the solution \eqref{eq:kinsol}, we write the fluctuating contribution as a sum of two terms,
\begin{equation}
\label{eq:Tii_fluctdef}
  T^{ii}_{\rm fluct}(t,\Lambda) = T^{ii}_{N_{\rm eq}}(t;\Lambda)  +
  T^{ii}_{\delta N}(t;\Lambda).
\end{equation}
The first term arises from equilibrium fluctuations $N_{\rm eq}(t)$ (\Eq{eq:kinsol_eq})
\begin{align}
\label{eq:Tii_cubic}
T^{ii}_{N_{\rm eq}}(t;\Lambda)
\equiv [1-h(t)] \left(1+\frac{T_0}{2}\frac{dc_{s0}^2}{dT_0}\right) \frac{T_0\Lambda^3}{2\pi^2},
\end{align}
while the second term  arises from the nonequilibrium distribution functions, $\delta N_{L/T}$ in \Eq{eq:kinsol_neq}.
In frequency space this  nonequilibrium contribution reads
\begin{align}
\label{eq:Tii_rest}
   T^{ii}_{\delta N}(\omega;\Lambda)  \equiv &
 \frac{h(\omega)\bar T_0}{4\pi^2}\left(1+\frac{3\bar T_0}{2}\frac{d\bar c_{s0}^2}{d\bar T_0}\right)
\bar C_{\zeta 0}f(\omega,\bar\gamma_{\zeta 0},\Lambda) \nonumber \\
& \quad +\frac{h(\omega)\bar T_0}{\pi^2}\bar C_{\eta 0}f(\omega,2\bar\gamma_{\eta 0},\Lambda).
\end{align}
Here we have defined a function,
\begin{align}
\label{eq:freg}
f(\omega,\gamma,\Lambda)
&\equiv
\int_0^{\Lambda\to\infty}p^2dp \frac{i\omega}{-i\omega + \gamma p^2} \\
&=\frac{i\omega}{\gamma}\Lambda
-\left(\frac{|\omega|}{\gamma}\right)^{3/2}\frac{\pi}{2\sqrt{2}}(1+ i {\rm sgn}(\omega)). \nonumber
\end{align}

Next, we calculate the averaged energy density in a similar manner.
It also consists of contributions from the background fluid and from the two-point functions of the fluctuations:
\begin{subequations}
\begin{align}
   \label{eq:defTtt}
\langle T^{tt}\rangle &=e_0 + T^{tt}_{\rm fluct}, \\
T^{tt}_{\rm fluct} &=\frac{\langle \vec G^2 \rangle}{e_0+p_0} \\
&=\frac{1}{e_0+p_0}\int\frac{d^3k}{(2\pi)^3}
\left[N_L(t,\bm k) + 2N_T(t,\bm k)\right], \nonumber
\end{align}
\end{subequations}
The contribution from the fluctuations is again divergent and we regularize with the same cut-off $\Lambda$ on $K$.
Substituting the perturbative solutions \eqref{eq:kinsol}, we find
\begin{align}
   \label{eq:defTttflucts}
   T^{tt}_{\rm fluct}(t;\Lambda) =&  T^{tt}_{N_{\rm eq}}(t;\Lambda) + T^{tt}_{\delta N}(t;\Lambda),
\end{align}
where  the first term arises from the equilibrium distribution $N_{\rm eq}$ (\Eq{eq:kinsol_eq})
\begin{align}
\label{eq:Ttt_cubic}
   T^{tt}_{N_{\rm eq}}(t;\Lambda) \equiv \frac{T_0\Lambda^3}{2\pi^2},
\end{align}
while the second term (in frequency space) arises from $\delta N_{L/T}$,
\begin{align}
\label{eq:Ttt_rest}
T^{tt}_{\delta N} (\omega;\Lambda)
   \equiv&  \frac{h(\omega)\bar T_0}{4\pi^2}
\bar C_{\zeta 0}
f(\omega,\bar\gamma_{\zeta 0},\Lambda) \, \nonumber \\
& \quad +\frac{h(\omega)\bar T_0}{\pi^2}\bar C_{\eta 0}
f(\omega, 2\bar\gamma_{\eta 0},\Lambda).
\end{align}

As will be described in the next section, the divergences in $ T^{ii}_\text{fluct}$  and $T^{tt}_\text{fluct}$ are absorbed by renormalizing the background fields, e.g.,\ $p_0$ and $\zeta_0$.
This renormalization procedure requires a clearer understanding how these bare parameters are defined, and how they depend on the cut-off $\Lambda$.

\subsection{Temperature shift\label{sec:tempshift}}

The bare parameters $e_0, p_0, T_0, \zeta_0, \ldots$ are determined by modes (such as particlelike excitations) with wave numbers above the cut-off,  $k > \Lambda$, which are not explicitly propagated by the statistical hydrodynamic system.
The goal of this section is to carefully explain how these parameters are defined and related to the physical equation of
state $e(T),\,p(T)$ (from lattice QCD) and the cut-off $\Lambda$.

First consider the density matrix for nonhydrodynamic modes with wave numbers above the cut-off $k > \Lambda$.
When the system is driven slightly out of equilibrium by the periodic compression and expansion, the density matrix for these modes $\rho(\Lambda)$ can be decomposed as an equilibrium density matrix $\rho_{\rm eq}(T_0;\Lambda)$ and a nonequilibrium correction which is well characterized by a single gradient $\delta \rho_{\rm neq}(\Lambda)\propto \nabla \cdot u$,
\begin{equation}
 \rho(\Lambda) = \rho_{\rm eq}(T_0;\Lambda) + \delta \rho_{\rm neq}(\Lambda) \, .
\end{equation}
The temperature parameter $T_0$ (which will depend on time and $\Lambda$) is chosen so that the average energy density above the cut-off $e_0(t,\Lambda) \equiv \left\langle T^{tt}(t)\right\rangle_{k>\Lambda}$ equals the energy from the equilibrium density matrix $ \rho_{\rm eq}(T_0;\Lambda)$ alone
\begin{equation}
  \label{eq:landau_constraint}
  e_0(t,\Lambda) \equiv \left\langle T^{tt}(t)\right\rangle_{k>\Lambda} = e_{\rm eq,0}(T_0(t;\Lambda); \Lambda) \, ,
\end{equation}
i.e., $T_0(t;\Lambda)$ is adjusted so that the energy moment associated with $\delta  \rho_{\rm neq}(\Lambda)$ is zero $\delta e_{\rm neq}(t;\Lambda)=0$.
(Otherwise the rhs of \Eq{eq:landau_constraint} would have a correction proportional to $\nabla \cdot u$).
Because of the constraint in \Eq{eq:landau_constraint} we can drop the ``eq" label below, i.e.,
\begin{equation}
e_0(t,\Lambda) = e_{\rm eq,0}(T_0(t;\Lambda);\Lambda)=e_0(T_0(t;\Lambda);\Lambda).
\end{equation}
In kinetic theory a similar constraint is imposed by requiring that the viscous correction to the distribution function $\delta f_{\rm bulk}({\bf p})$ does not change the energy in the system~\cite{Arnold:2006fz,Moore:2008ws}.
Once this prescription for $T_0(t;\Lambda)$ is adopted,
the stress  computed with the density matrix $\rho(\Lambda)$ is given by\footnote{In the current setup $\nabla \cdot u=\tfrac{3}{2} \dot h$.}
\begin{equation}
   \left\langle T^{ij} \right\rangle_{k>\Lambda} =  (1- h) \, p_0(T_0;\Lambda) \, \delta^{ij} -  \zeta_0(T_0;\Lambda) \,  \nabla \cdot u \, \delta^{ij} \, ,
\end{equation}
where the partial pressure $p_0(T_0;\Lambda)$ from modes above $\Lambda$ is determined by the equilibrium density matrix, $\rho_{\rm eq}(T_0;\Lambda)$, while the bulk term comes from the viscous correction, $\delta \rho_{\rm neq}(\Lambda)$.
This is the parametrization of the stress tensor (for $k>\Lambda$) that was used in \Eq{eq:hydro+noise}.
The spatial stress tensor
determines the bulk viscous correction $\zeta_0 \nabla \cdot u$
only after the parameter $T_0(t;\Lambda)$ is defined according to the Landau constraint in \Eq{eq:landau_constraint}~\cite{Arnold:2006fz,Moore:2008ws}.

Later in this section we will define a temperature $T(t)$ by imposing the Landau constraint on the whole system (including the energy of hydrodynamic fluctuations below the cut-off),  and this will lead to a difference between $T_0(t;\Lambda)$ and the cutoff independent temperature $T(t)$.

Now we will relate the partial energy density and pressure, $e_{0}(T_0;\Lambda)$ and $p_{0}(T_0;\Lambda)$, to the equilibrium energy density and pressure, $e(T_0)$ and $p(T_0)$,  as measured by lattice QCD.
Indeed, $e_0$ and $p_0$ are cut-off dependent quantities and are determined by an equilibrium density matrix $\rho_{\rm eq}(T_0;\Lambda)$ which excludes equilibrium hydrodynamic fluctuations below the scale $\Lambda$.
The contribution of such equilibrium hydrodynamic fluctuations to the energy density and pressure are given by \Eq{eq:Ttt_cubic} and \Eq{eq:Tii_cubic}, respectively, and thus the physical energy density and pressure are:
\begin{subequations}
\begin{align}
   \label{eq:erenorm}
   e(T_0) =&  e_0(T_0;\Lambda) + \frac{T_0 \Lambda^3}{2\pi^2}\, , \\
   \label{eq:prenorm}
   p(T_0) =&  p_0(T_0;\Lambda) + \left(1 + \frac{T_0}{2} \frac{dc_{s}^2}{dT_0}\right) \frac{T_0 \Lambda^3}{6\pi^2}  \, .
\end{align}
\end{subequations}
At a practical level these equations serve to define the $e_0$ and $p_0$  parameters that should be used in a stochastic hydrocode with a given  cut-off $\Lambda$ and physical equation of state $e(T_0),\,p(T_0)$.

As discussed above, the temperature $T(t)$ for the complete system (background+fluctuations) is adjusted so that the energy density calculated from the lattice equation of state $e(T(t))$ matches the energy of the partially equilibrated system $\left\langle T^{tt}(t) \right\rangle$,
\begin{equation}
  \left\langle T^{tt}(t) \right\rangle = e(T(t))  \, .
\end{equation}
After imposing this constraint, the time-dependent stress $\left\langle T^{ii}(t) \right\rangle$ of the driven system will deviate from its equilibrium expectation, $3\, p(T(t)) \, (1- h(t))$, and these deviations are described (up to long-time tails) by the bulk viscosity.
Combining Eqs.~(\ref{eq:defTtt}), (\ref{eq:defTttflucts}), (\ref{eq:Ttt_cubic}), and (\ref{eq:erenorm}), the energy of the background+fluctuations is
\begin{align}
   \label{eq:DTdeff}
   e(T(t)) &= e(T_0(t;\Lambda)) +  T^{tt}_{\delta N}(t;\Lambda),
\end{align}
where $T^{tt}_{\delta N}(t;\Lambda)$ was defined in \Eq{eq:Ttt_rest}.
Thus, the temperature for the whole system $T(t)$ (which is
independent of the cut-off) is related to the temperature
parameter of the subsystem $T_0(t;\Lambda)$ by a small shift $\Delta T$
\begin{equation}
 \label{eq:Tshiftdef}
 T_{0}(t;\Lambda) = T(t) + \Delta T(t;\Lambda) \, ,
\end{equation}
so that \Eq{eq:DTdeff} is satisfied.
The temperature shift is given in frequency space by
\begin{align}
\label{eq:T_shift}
   -\frac{de}{dT}\Delta T(\omega;\Lambda)
&=\frac{h(\omega)\bar T}{4\pi^2}
\bar C_{\zeta 0}f(\omega,\bar\gamma_{\zeta 0},\Lambda) \\
& \quad +\frac{h(\omega)\bar T}{\pi^2}\bar C_{\eta 0}f(\omega, 2\bar\gamma_{\eta 0},\Lambda). \nonumber
\end{align}
and clearly depends on the cutoff because the $T_0(t;\Lambda)$ was defined with respect to a specific subsystem labeled by $\Lambda$.
The temperature shift in the time domain takes the form
  \begin{multline}
 \label{eq:Tshiftdiverge}
  -\frac{de}{dT}  \Delta T(t;\Lambda) = -\frac{\bar T\Lambda}{6\pi^2} \left[\frac{\bar C_{\zeta 0} }{\bar\gamma_{\zeta0}} +  4\frac{\bar C_{\eta 0} }{2\bar \gamma_{\eta 0}}\right] \nabla \cdot u \\ + {\rm finite} \, ,
  \end{multline}
where $\nabla \cdot u = \tfrac{3}{2} \dot h$ for this example.
The divergent piece of the temperature shift is universal, but the finite corrections are not.
This is verified by explicit calculation of the temperature shift for the Bjorken background in \app{app:bjorken}.
From practical perspective,  \Eqs{eq:Tshiftdef} and (\ref{eq:Tshiftdiverge}) define how $T_0$ must be chosen for a stochastic hydro code (with a specified cut-off $\Lambda$) to  reproduce the correct physical bulk viscosity for long wavelength hydrodynamic modes and a physical equation of state.
This is detailed in the next section\footnote{In defining $T_0$ from $T$, $\Lambda$, and $\nabla \cdot u$, the finite remainder in \Eq{eq:Tshiftdiverge} can be chosen in any convenient way.}.

\subsection{Renormalized background and long-time tails}

Once the temperature shift $\Delta T(t;\Lambda)$ is obtained, the remaining divergences in $T^{ii}_{\rm fluct}$ can be absorbed by pressure and bulk viscosity renormalization.
Using Eqs.~(\ref{Tijtotdef}), (\ref{eq:Tii_fluctdef}), (\ref{eq:Tii_cubic}), and (\ref{eq:prenorm}), the statistically averaged spatial stress tensor trace reads
\begin{multline}
   \langle T^{ii}\rangle(t)=3\left[1-h(t)\right] p(T_0(t;\Lambda)) \\
 -\frac{9}{2}\dot h(t)\zeta_0 + T^{ii}_{\delta N}(t;\Lambda).
\end{multline}
where $T^{ii}_{\delta N}$ is given in \Eq{eq:Tii_rest}.

Now we will shift the temperature parameter $T_0$ in the pressure to the physical temperature $T(t)$ determined by Landau matching \eq{eq:DTdeff}, $p(T_0) = p(T) + p'(T) \Delta T$.
The fluctuation contribution $T^{ii}_{\delta N}(t;\Lambda)$ and the temperature parameter $\Delta T(t;\Lambda)$ both diverge as $-i\omega h(\omega)\Lambda$.
These two terms gracefully
combine to produce a positive definite renormalization of bulk viscosity $\zeta_0$ in the term $- \frac{9}{2}\dot h(t)\zeta$,
\begin{align}
\label{eq:bulk_ren}
\zeta(T)=\zeta_0(T;\Lambda)+\frac{T\Lambda}{18\pi^2}
\left[
\frac{C_{\zeta 0}^2}{\gamma_{\zeta 0}}+4\frac{C_{\eta 0}^2}{2\gamma_{\eta 0}}
\right] \, .
\end{align}
In this step the coefficients in front of the linear divergences in $\Delta T$ and $T^{ii}_{\delta N}$ have neatly come together to complete the squares of $C_{\zeta 0}$ and $C_{\eta 0}$  defined by \Eq{eq:C}.
Thus the renormalization of the bulk viscosity is positive and only necessary in a system with broken scale symmetry.
We have confirmed that the bulk viscosity renormalization is universal by computing it for a Bjorken expanding background (see Appendix \ref{app:bjorken}).

Once all divergences are absorbed by renormalization, the stress tensor becomes finite and cut-off independent.
In the presence of background expansion, there are remaining finite corrections from the fluctuations in $T^{ii}_{\delta N}$.
The total stress tensor is
\begin{align}
\langle T^{ii}\rangle(t)
&=3\left[1-h(t)\right] p(T(t))
-\frac{9}{2}\dot h(t)\zeta(T(t)) \nonumber \\
& \quad -\int \frac{d\omega}{2\pi}e^{-i\omega t}h(\omega)
|\omega|^{3/2} \frac{\pi}{2\sqrt{2}}(1+i{\rm sgn}(\omega)) \nonumber \\
& \quad\quad \times \frac{\bar T}{4\pi^2}\left[
\bar C_{\zeta 0}^2 \left(\frac{1}{\bar \gamma_{\zeta 0}}\right)^{3/2}
+4\bar C_{\eta 0}^2\left(\frac{1}{2\bar\gamma_{\eta 0}}\right)^{3/2}
\right]\nonumber ,\label{eq:finite}
\end{align}
and has a term with $|\omega|^{3/2}$, which cannot be expressed by local time derivatives.
This term is not analytic at $\omega=0$ and derives from the out-of-equilibrium fluctuations in the kinetic regime $k\sim k_*$.

With $\langle T^{tt}\rangle$ and $\langle T^{ii}\rangle$ known, we can write down the hydrodynamic equations for statistically averaged hydrodynamics with noise
\begin{align}
0=\frac{d}{dt}\langle T^{tt}\rangle + \frac{3}{2}\dot h \langle T^{tt}\rangle + \frac{1}{2}\dot h \langle T^{ii}\rangle .
\end{align}
Because the nonanalytic term in $\langle T^{ii}\rangle$ is of $\mathcal O(h)$, the rest frame energy density $e(t)$ evolves according to
\begin{align}
0=\dot e(t) + \frac{3\dot h}{2}\left[e(t) + p(t)\right] ,
\end{align}
and we obtain the solution:
\begin{align}
e(t) = \bar e - \frac{3h(t)}{2}(\bar e + \bar p),
\end{align}
which will be used to calculate the response function in the next section.

\subsection{Response function in the bulk channel\label{sec:GR}}
The nonanalytic behavior in $\omega$ is also present in the response function in the bulk channel.
In the frequency space, the linear response of stress tensor to the external gravitational field $h(\omega)$ is given by
\begin{align}
\langle T^{ii}\rangle(\omega)
=G_R^{ii,jj}(\omega,\bm k=\bm 0)\frac{1}{2}h(\omega).
\end{align}
The response function $G_R^{ii,jj}$ is defined by
\begin{subequations}
\label{eq:bulk_retarded}
\begin{align}
G_R^{ii,jj}(t,\bm x) &\equiv i\theta(t)
\left\langle\left[ \hat T^{ii}(t,\bm x),\hat T^{jj}(0, \bm 0) \right]\right\rangle, \\
G_R^{ii,jj}(\omega,\bm k)&=\int d^4 x\,G_R^{ii,jj}(t,\bm x) e^{i\omega t -i\bm k\cdot \bm x}.
\end{align}
\end{subequations}
Then from our results, the response function $G_R^{ii,jj}(\omega)\equiv G_R^{ii,jj}(\omega,\bm k = \bm 0)$ is obtained as
\begin{align}
& G_R^{ii,jj}(\omega)
=\frac{\delta}{\delta h(\omega)}\left[2 \langle T^{ii}\rangle(\omega)\right] \Big |_{h=0}\nonumber \\
&= -6\left(\bar p+\frac{3}{2}\bar c_s^2(\bar e + \bar p)\right)
+ 9i\omega \bar \zeta  -\frac{1+i{\rm sgn}(\omega)}{4\sqrt{2}\pi} |\omega|^{3/2}\bar T \nonumber \\
& \quad \times \left[
\bar C_{\zeta}^2\left(\frac{1}{\bar \gamma_{\zeta}}\right)^{3/2}
+4\bar C_{\eta}^2\left(\frac{1}{2\bar\gamma_{\eta}}\right)^{3/2}
\right],\label{eq:GRiijj}
\end{align}
and the spectral function as
\begin{align}
&\rho^{ii,jj}(\omega)=2{\rm Im}\, G_R^{ii,jj}(\omega)\\
&=18\omega \bar \zeta
-\frac{\omega|\omega|^{1/2} \bar T}{2\sqrt{2}\pi}
\left[
\bar C_{\zeta}^2\left(\frac{1}{\bar\gamma_{\zeta}}\right)^{3/2}
+4\bar C_{\eta}^2\left(\frac{1}{2\bar\gamma_{\eta}}\right)^{3/2}
\right]. \nonumber
\end{align}
This spectral function is consistent with a previous diagrammatic computation of the symmetrized correlation function $C^{ii,jj}$ (see the appendix of \Ref{Kovtun:2003vj}) using the fluctuation-dissipation relation:\footnote{
The term $18\omega\bar\zeta$ in $\rho^{ii,jj}(\omega)$ corresponds to a correlation of thermal noise in the stress tensor, which is not explicitly written in the calculation of $C^{ii,jj}$ \cite{Kovtun:2003vj}.
}
\begin{subequations}
\begin{align}
\rho^{ii,jj}(\omega)&=\frac{\omega}{T} C^{ii,jj}(\omega,\bm k = \bm 0), \\
C^{ii,jj}(t,\bm x) &\equiv \frac{1}{2}
\left\langle \left\{\hat T^{ii}(t,\bm x),\hat T^{jj}(0, \bm 0) \right\}\right\rangle_{\rm conn}.
\end{align}
\end{subequations}

We also computed $G_R^{ii,jj}(\omega)$ diagrammatically in \app{app:diagrams} and found identical results to \Eqs{eq:GRiijj} and (\ref{eq:bulk_ren}) up to a contact term\footnote{
Deviation by a contact term is permitted because of different definitions of the two-point functions \cite{Romatschke:2009ng}.
}.

\section{Summary}
\label{sec:summary}

In this paper we applied the kinetic theory of hydrodynamic fluctuations developed in~\Ref{Akamatsu:2016llw} to a relativistic nonconformal fluid.
We calculated the contribution of out-of-equilibrium hydrodynamic fluctuations to the energy momentum tensor, which renormalize the background hydrodynamic fields and the bulk viscosity $\zeta$.
The bulk viscosity renormalization is proportional to the scaling symmetry breaking in the equation of state and can be used to estimate the minimal bulk viscosity value in a hot QCD medium.

In the main body of the paper, we considered a nonconformal charge-neutral fluid, which is driven out of equilibrium by a weak isotropic expansion (or compression).
Analogous calculations for a Bjorken expanding system is summarized in the appendix.
The relaxation of hydrodynamic fluctuations to equilibrium is disturbed by the expansion and the deviation of two-point correlations from equilibrium becomes appreciable for wavelengths $k\alt k_*\sim\sqrt{\omega/\gamma_{\eta,\zeta}}$, where $\omega$ is the frequency of the background expansion and $k_*$ defines the hydrokinetic regime.

We derive the hydrokinetic equations for the two-point correlation functions $N_{AA}(t,\bm k)$, \Eq{eq:NAA}, of energy $\delta e$ and momentum  $\vec{g}$ density fluctuations in the presence of the expansion.
The nonlinear fluctuations  $N_{AA}(t,\bm k)$ contribute to the statistically averaged energy-momentum tensor $\left<T^{\mu\nu}\right>$.
The divergent part of the fluctuation contributions is regulated by an ultraviolet cut-off $\Lambda$.
The cutoff dependence of $T^{\mu\nu}_{\rm fluct}$ is (partially) absorbed by a universal renormalization of the background energy density $e_0$, the pressure $p_0$, and the bulk viscosity $\zeta_0$ (the same terms are found for the far-from-equilibrium Bjorken expansion\footnote{
Because of the Bjorken expansion is anisotropic, there is an additional linear divergence which renormalizes the background shear viscosity \eqref{eq:shear-ren}:
\begin{align}
\eta(T) &=
\eta_0(T;\Lambda)
+\frac{T\Lambda}{30\pi^2}
\left[
\frac{e_0+p_0}{\zeta_0 + \frac{4}{3}\eta_0}
+\frac{7(e_0+p_0)}{2\eta_0}
\right] . \nonumber
\end{align}
This is a generalization from the conformal case \cite{Akamatsu:2016llw, Kovtun:2011np}.
}; see Appendix \ref{app:bjorken}):
\begin{subequations}
\label{eq:renormalization}
\begin{align}
e(T) &= e_0(T;\Lambda) + \frac{T\Lambda^3}{2\pi^2},\label{eq:eequiv} \\
p(T) &= p_0(T;\Lambda) + \left(1+\frac{T}{2}\frac{dc_{s0}^2}{dT}\right) \frac{T\Lambda^3}{6\pi^2}, \\
\zeta(T)&=
\zeta_0(T;\Lambda) \label{eq:bulkfinal}\\
& \quad+\frac{T\Lambda}{18\pi^2}
\left[
\begin{aligned}
&\left(1+\frac{3T}{2}\frac{dc_{s0}^2}{dT} -3c_{s0}^2\right)^2\frac{e_0 + p_0}{\zeta_0+\frac{4}{3}\eta_0} \\
&+4\left(1-3c_{s0}^2\right)^2\frac{e_0 + p_0}{2\eta_0}
\end{aligned}
\right]. \nonumber
\end{align}
\end{subequations}
The bare unrenormalized background quantities reflect the physical properties of the modes above the cut-off $\Lambda$.
The hydrodynamic fluctuations below the cutoff are dynamical in the hydrodynamics with noise and make an evolving contribution to the energy momentum tensor.
We find that the renormalization of the bulk viscosity is proportional to the nonconformality of the equation of state, e.g.,\ $(1-3c_{s0}^2)^2$, in agreement with other estimates~\cite{Arnold:2006fz,Meyer:2007dy,Kharzeev:2007wb,Karsch:2007jc,Moore:2008ws,Romatschke:2009ng}.
Using the parametrization of the equation of state from the lattice QCD simulations, we find that the fluctuation-induced bulk viscosity is modestly enhanced around the QCD pseudocritical temperature $T_c\sim150\,\text{MeV}$, where deviations from  the conformality are the largest (see \Fig{fig:bulk})~\cite{Borsanyi:2016ksw}.
A diagrammatic derivation of similar bound for bulk viscosity for a nonrelativistic cold Fermi gas was recently presented in \Ref{Martinez:2017jjf} and we performed the calculation for the relativistic nonconformal fluid in \app{app:diagrams} confirming the bulk viscosity renormalization, \Eq{eq:bulkfinal}.

In a nonconformal system, the contribution to the energy density from the hydrodynamic fluctuations  $T^{tt}_{\rm fluct}$ is not completely accounted for by the equilibrium energy density of hydrodynamic modes (the cubic term in \Eq{eq:eequiv}).
The additional cutoff dependent contributions are proportional to the divergence of the flow velocity $\nabla \cdot u$ and are removed by a universal shift in the background temperature $T_0=T(\Lambda)+\Delta T(\Lambda)$, \Eq{eq:Tshiftdiverge}.
Once the cutoff dependence in $T^{\mu\nu}_{\rm fluct}$ is completely absorbed, the remaining finite contribution has a fractional power in the gradient expansion ($\propto \omega^{3/2}$) and makes an essential difference from hydrodynamics without noise (see \Eq{eq:finite}).
In the symmetrized correlation function of the energy-momentum tensor $C^{ii,jj}$, these terms become proportional to $\omega^{1/2}$ and in coordinate space only decay with a power law tail $\propto t^{-3/2}$, and therefore are called the long-time tails.
Comparing the spectral functions $\rho^{ii,jj}$, we find that our computation using the hydrokinetic theory is consistent with the previous diagrammatic calculations~\cite{Kovtun:2003vj}.

In this publication we extended our previous work on hydrokinetic theory to nonconformal systems close to equilibrium and undergoing a Bjorken expansion.
A natural next step is to consider more general background evolution and systems with the net baryon number. It would be particularly rewarding to extend the hydrokinetic theory to critical fluctuations around the critical point, which is the focus of the beam energy scan program at RHIC.

\begin{acknowledgments}
	This work was supported in part by the U.S. Department of Energy, Office of
	Science, Office of Nuclear Physics  under Award Number
	DE\nobreakdash-FG02\nobreakdash-88ER40388 (A.M., D.T.).
	This work was also supported in part by the German Research Foundation (DFG)
	Collaborative Research Centre (SFB) 1225 Isolated quantum systems and universality in extreme conditions (ISOQUANT) (A.M.).
	Y.A. thanks the DFG Collaborative Research Centre 1225 (ISOQUANT) for hospitality during his stay at Heidelberg University.
	\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


January has passed and the 2019 LCK has gone into a brief spring break for the Lunar New Year. With the 10 teams having played 6 matches each, the LCK resumes on the 13th (KST); this Spring Split will now have only 1 and a half weeks left before Round 2 kicks off.
With numerous teams going through vast changes in their starting lineup -- probably the most since 2015, when organizations were prohibited to have multiple rosters -- the 2019 LCK Spring was indeed one of the most anticipated splits. And as expected, this Spring has been packed with surprising spectacles for the fans.
Alongside known veteran players, some new faces are doing pretty well, which is making this LCK even more exciting to watch. Now with the first month of the LCK gone by, we picked the best performing players so far for each role in this split. The following is Inven Global's All-Pro Team of the Month for the 2019 LCK Spring.
Numerous officials and fans did not expect SANDBOX Gaming to do so well this Spring. Their performance during their run in Challengers Korea wasn't as strong compared to Griffin and DAMWON at the time. However, once the split started, SANDBOX surprised everyone. Before going against Griffin, they were undefeated and their prowess undoubtedly came from their top-jungle, Summit and OnFleek.
Summit's individual performance stands out as one of the best in the Spring Split so far. He has been very solid against the best top laners in Korea. This LCK rookie was usually ahead of his opponents in the laning phase and he was a constant threat in teamfights.
Another notable strength Summit has is his vast champion pool. In 2019 Spring, he used a total of 8 champions. He knew how to use meta champions such as Urgot and Aatrox as well as some aggressive split pushers like Jayce.
Every LCK team in this split has a concrete jungler on their side. Even among these prominent junglers, Tarzan seems to be at least a step ahead. His jungle paths were recognized as some of the most creative in the world even since 2018, when Korean junglers were criticized for their passive plays. He knows how to back up his team and at the same time knows how to step up and lead Griffin's early game snowballs.
Tarzan seems more confident than ever. Whether he's on an early-game focused champ such as Xin Zhao, or some late-game teamfight picks like Sejuani, he delivers consistent, top-notch performance. He's top-tier on almost every component in the jungle; it's hard to find a single flaw in his plays this season. That's why he's being called the LCK's "king of the jungle".
Although the mid lane itself still has significant importance in LoL, it is definitely not the main carry role in the current LCK; the role of mid laners are more focused on supporting the other laners. However, that doesn't seem to apply as much to Griffin's mid laner, Chovy.
If you want a glimpse of Chovy's strength in this split, just look at his KDA. Before facing SANDBOX where he 'surprisingly' died twice, he was at an astonishing 104 KDA; he has been only killed 3 times this split. However, this doesn't mean that his performance was passive in any way.
Chovy's individual performance has been dominant on every front. Among the LCK mid laners, he is currently in 1st place for most of the main stats including average kills/deaths, total CS, CS per minute, total gold earned per game, and gold per minute.
He was also the first to confidently use Akali when she was seen as somewhat of a "trap card" in the LCK at the beginning of this Spring. Akali's skill set seemed to suit this young mid laner perfectly. Chovy's Akali was exceptional in skirmishes, especially when he was outnumbered. Watching him go 1 vs 2,3,4, securing a kill, and easily surviving proved just how effective Akali can be in the right hands. His performance contributed greatly to making Akali a meta pick for this LCK split.
Much like the rest of his teammates, SANDBOX's bottom laner Ghost went into this split with very little expectations from viewers. Based on his previous performance in bbq Olivers, he was thought of as a player that is not suited to play for the LCK. However, Ghost proved everyone wrong. His current performance in SANDBOX so far would grant him a guaranteed "MIP (most improved player)" if there was one in the LCK.
Considering how SANDBOX's strategies focus more on their upper lanes, Ghost's in-game stats do not stand out as particularly grand compared to the other top-tier players in the league. Despite this, Ghost's plays - and his positioning in particular - make him such a reliable asset for SANDBOX in teamfights. It seems like he finally found the right balance between his in-game knowledge and the mechanics that he was renowned for during his early days as a pro. If he can successfully maintain his current condition going forward, it won't be that surprising to see him mentioned as an MVP contender.
TusiN was known for his aggressive plays in Afreeca. Although he was sometimes criticized for his shaky performance, his plays on champions such as Rakan and Alistar were second to none in Korea. During this preseason, TusiN teamed up with Deft in Kingzone. Fans and officials were sure that this bottom duo would have good synergy since they are both capable of constantly pressuring opponents in the laning phase.
So far, TusiN seems to be a perfect match for Deft. His aggressive plays on initiators during teamfights actually goes perfectly with Deft's sometimes overaggressive positioning. In this Split, he is also proving that he is capable of carrying his team. During Kingzone's recent match against Gen.G, TusiN went with his signature champ, Alistar, dominating the laning phase, ganking mid, and providing pinpoint initiators during teamfights.
Thanks to TusiN's solid performance, Kingzone recovered from the rather disappointing state they were in earlier in the split. If this prominent support keeps up with the hard work he has shown in recent games, opponents will have quite a hard time taking down the Kingzone bottom duo this Spring.
Q Who's your MVP for this month?


William Blair partner Brian Singer explains how Bitcoin and blockchain encryption has a greater ability to bring more of the world's population out of poverty than anything we've seen in decades.
FORBES: And is bitcoin the currency of the future? Or is it the payment system they're developing?
FORBES: Well, for one thing, it's a cheap payment system.
SINGER: For one thing, it's a cheap payment system. But it's more important than that. It's more important than that. Blockchain cryptography is all about digital transfer of ownership in a completely transparent and public way. Okay, I don't want to go into any of the math and the complexity behind it, but think of what that means.
That means that I can actually record ownership in public, in a digital means. 50% of the world's population has a mobile phone. I read an estimate this month that in 2020, 90% of the world's population will have mobile phones. That's probably a high estimate, but that's okay.
SINGER: We're throwing them away and they can ship them off to emerging markets. But what's powerful about this, now think about Hernando de Soto.
SINGER: Okay. The Institute for Liberty and Democracy in Peru, I really think Hernando de Soto should win the Nobel prize for the work he's done. I hope he does. But he's going around the world and identified one of the most powerful things to the economy and the creation of wealth. And that is the ownership of property.
SINGER: Which you can use as collateral and when you own it, you will develop it. The best thing you can do for the environment is to own it. Because when you own it, you make sure it's sustainable for your future family. And that's powerful. But they don't have the ability to own it. If you live in a shantytown, you operate outside of the legal economy. It's not that you want to operate outside of the legal economy, but you have no ownership of any property.
SINGER: And once you introduce blockchain cryptography, guess what they can do? Have a public ledger of ownership and transfer that cannot be denied because it's absolutely transparent. It's absolutely transparent. It is the means through which these individuals can get exactly what Hernando de Soto thinks they best need to bring themselves out of poverty.
It is so much more profound than that, but at its very basis, it has the ability to give these individuals the right of ownership to property. And that has more than anything else, the ability to bring them out of poverty. And I can't wait. It'll be adopted elsewhere. I mean, blockchain encryption is such a powerful tool. It'll be used by banks, it'll be used by credit card companies, it will become a standard. But what's more interesting is what it can do to poverty around the world to eliminate poverty.
FORBES: Going for a sec on the eliminating poverty, but also, what does that mean, that technology mean for credit card companies and the whole payment system, crazy gobbledygook payment system we have today? Very expensive. Doesn't that blow it away?
SINGER: When there was one manufacturer of phones in the U.S., the U.S. government, what did we have? This big clunky, rotary thing that sat on desks. Now what do we have? Cell phones of all flavors, sizes. I can sit in Africa and on my cell phone, take courses at MIT because they're public record. It's absolutely astounding what happens when you can actually innovate around the world and compete.
SINGER: Right. Well, first the fee is high. And then they reject the transaction because why is this card suddenly showing up in some city, some place, where I don't live? So it gets cut off. Well, with blockchain cryptography, the buyer and the seller automatically trust each other over an untrusted network. That's what's so powerful about it.
A blockchainbased transaction is verified within ten minutes. It's not verified within ten minutes for a credit card, which is why 5% to 10% of all transactions are rejected on the spot. It's not verified until often days later. Even a check is not verified for days. And that is completely taken out of the system. So on day one, on day one, the retailer's sales go up because they don't have to reject the transaction. And they save on the fees. It doesn't really get much better than that if you're looking for a disruptive innovation in any single industry. It's wonderful.


I discovered I have a gift. Over the years, I've assembled a series of tools and processes that greatly streamlines my business operations, something important for a solopreneur like myself. When I talk to other consultants and busy professionals, they are delighted to learn about these improvements for their own work.
When I can help others focus on what they love to do well, instead of spending time on necessary back-office work, then we all benefit. I am especially motivated to help people doing work to make the world a better place. Through them, I too can make a greater impact.
To that end, I produced my 1st online video on expense filing. I'm really excited about this! Managing expenses used to be such a dreadful task for me, taking up lots of time and filled with errors of omission. I'd forget to file some things or details related to expense items. Now I have a system that allows me to file both paper and digital expenses and real-time, reducing costly delays and the time it takes to submit expenses. It's even fun at times (gasp).
If expense filing is a drag for you, I invite you to take a look. Initial reviews are enthusiastic. All it takes is 30 minutes of your time to watch and you'll benefit immediately from what you learned.
Join now, and you can use coupon "PAINFREE" to get $5 off until Friday, December 9, limited to first 50 subscribers.
Xpenser is the tool that I use for expense management. You will need an account to take this course. Sign up for a free trial. I explain in the 2nd lecture how to configure it and get started.


1. What is Vidyapith? What is its objective and what does it do? See Home.
2. Who can apply for grants and how? See Grants.
3. What is the procedure followed in awarding grants? How are the award-receiving projects evaluated and supervised? See Grants.
4. What are the projects that have already received grants and how much did they receive? See Projects.
5. How can I become a donor? How do I contribute? See Donors.
6. I am a donor. How can I get more involved in Vidyapith's activities? See Donors.
7. I do not want to be a donor right now but I want to help. How can I? You can help spread the word about Vidyapith among your friends and colleagues. You can also help by publicising our work to other interested parties.
8. How much of my donation will actually reach a beneficiary? There are no overhead costs. Excepting some essential administration costs (such as web-hosting fee), the amount we raise is fully used to fund projects. Every service provided by the trustees is done so purely on a voluntary basis. See Reports.


How can something so small make this much poop in one day?!
After all that I've gone through, his first words better be "thank you mama!" or else!
Just for those labor pains alone I should get a medal and a fully paid vacation!
Is there a snooze button on this baby thing?
Just 5 more minutes please.
There are more baby clothes to wash than hair on my head.
And I'm losing both just as fast!
I can't remember what air that doesn't smell like baby powder feels like.
How is this not a paid job?
It's not even 9-5! It's 24/7/365 and I am severely underpaid!
You're telling me changing diapers 10 times day, feeding, cleaning, cooking, and being sleep deprived all at the same time aren't super-human qualities?
Instead of baby gifts, can you watch my baby for 2 hours so I can get some sleep?
What good is this onesie going to do me? Put your money where your mouth is!
If you touch my baby without washing your hands, there will be hell to pay.
WASH YOUR HANDS. IT'S THE POLITE THING TO DO. AND IT TAKES 5 SECONDS.


When your body heats up, or gets red or swollen, that's inflammation at work.
Sometimes you can't even see inflammation happening deep inside your body until you start to feel yourself take a downward turn. But don't worry, there's something you can do to help.
When we eat, the foods we choose to put in our bodies can fight against inflammation or trigger an inflammatory response.
The foundation of an anti-inflammatory diet includes primarily plant-based foods, such as fruits, vegetables, nuts, seeds, beans, and legumes. When eating animal protein sources, make sure to choose wild seafood, organic pasture-raised eggs, and grass-fed land animals.
So think of your next meal as an opportunity to load your body with foods that will make you strong and energized, and also improve your long-term health!
Kale is loaded with anti-inflammatory properties and contains a variety of phytonutrients and antioxidants that help protect our bodies against cellular damage.
Kale helps benefit everything from glowing skin and healthy eyes, to a powerful digestive system and strong bones.
Get it in easily by adding it to your daily smoothie or immune-boosting green juice.
This delicious fruit packs a big punch! Pineapple is loaded with vitamin C and contains an enzyme called bromelain which may help stimulate protein digestion, reduce inflammation of the gut, and boost immune function.
Add pineapple to your fruit plate, smoothies, or juice to help fight inflammation, enhance digestion, and keep your immune system strong.
This cold-water fish is one of the best sources of omega-3 fatty acids, which can help to fight inflammation, lower risk for chronic diseases, and improve mental health.
Salmon is also a great source of protein and is packed with tons of other vitamins and minerals including vitamins B-12, B-3, D, potassium, and selenium.
Cook up salmon any way you like — pan-seared, grilled, or broiled. I like it baked with dill, lemon, and other herbs.
Antimicrobial, antiviral, and , mushrooms contain a variety of compounds that can help to improve immunity and lower inflammation throughout the body.
They consist of long-chain polysaccharides called beta-glucan that promotes a strong immune system, and also houses a powerful antioxidant called ergothioneine that can help fight inflammation.
Mushrooms are a great source of protein, fiber, and various B vitamins, too.
There's so many different types of mushrooms to try, you're bound to find one that fits your taste buds — some of my favorites are shiitake, morel, chanterelle, and porcini.
Packed with vitamins C and K, folate, and fiber, broccoli is an anti-inflammatory powerhouse.
It's especially rich in antioxidants like the flavonoids kaempferol and quercetin, as well as a variety of carotenoids.
Sauté this veggie up with garlic — one of my other favorite anti-inflammatory foods — as a perfect dinner side dish.
Use my recipe, which adds a dash of honey, to make the dish really special.
Dulse is a type of seaweed that contains a unique group of polysaccharides called fucoidans, which work to reduce inflammation within the body.
You can eat dulse fresh or dried. Try adding it to green leafy salads, chopped up with avocado, or blended into dressings.
Low in sugar and high in fiber, these guys are packed with vitamins A, C, and E, and contain a variety of anti-inflammatory properties and antioxidants.
The major antioxidant, anthocyanin, is what gives this berry it's gorgeous deep blue color.
Add organic blueberries to your morning fruit plate or throw them into this green protein smoothie.
Sauerkraut, or fermented cabbage, is loaded with vitamins C and K, iron, and fiber, and naturally contains healthy gut bacteria called probiotics.
By eating foods like sauerkraut, we enhance the health of our digestive system by optimizing our gut flora and balancing our gut microbiome.
We can get probiotics through other fermented foods as well, such as kimchi, miso, and pickles. Try adding sauerkraut to your green salads or using it as a topping on burgers!
Bone broth is a complete serving of the good stuff — minerals like calcium, magnesium, and phosphorus.
A bowl of this may help strengthen the lining of your gut due to the high content of healing compounds including collagen, gelatin, and amino acids like glutamine, arginine, and proline.
Check out my favorite way to eat bone broth in this immunity bone broth veggie soup!
This beautiful yellow-orange spice is often found in curry powders.
Thanks to the active compound curcumin, it has strong anti-inflammatory properties and has been used for thousands of years as a medicinal herb.
Try adding ground turmeric to seasoning on fish and vegetables, or use raw turmeric root chopped up in soups, sauces, or as an addition to your next green juice!
However you take it, remember to add a dash of black pepper to boost absorption.
Most of ginger's anti-inflammatory and medicinal qualities come from its main bioactive compound, gingerol.
Ginger is not only a major immune booster and inflammation fighter, but this plant adds a kick of flavor to smoothies and juices, soups, sauces, and stir-frys. Ginger root can also be used in tea to aid digestion.
Garlic contains sulfur compounds that stimulate our immune system to fight inflammation and illness. It's also antibacterial and antifungal!
This tasty herb is easy to add to any meal and boosts delicious flavor in a variety of dishes. One of my favorite homemade dressings, this creamy tahini dressing, uses garlic as a main ingredient.
Next time you're not feeling like your energetic self, or if you're just ready to take your health to the next level, try incorporating some of these delicious anti-inflammatory foods into your daily routine.
Whether it's experimenting with dulse in your dressings, topping salads with sauerkraut, or adding kale and broccoli to your bone broth soup, these anti-inflammatory foods can benefit your long-term health.
You'll start to see and feel their powerful effects by eating them today!


text:
Ліза «Лефт Ай» Лопес (;  27 травня 1971, Філадельфія, США —  25 квітня 2002, Ла-Сейба, Гондурас) — американська співачка, авторка пісень, танцівниця, реперка, акторка. Колишня учасниця гурту TLC.

Життєпис
Прізвисько «Лефт Ай» () з'явилося, коли співачка носила окуляри з презервативом замість лівого скла в рамках програми популяризації безпечного сексу.

Ліза Лопес страждала від алкогольної залежності (за власними словами, через проблеми в минулому) і намагалася вилікуватися. У 1994 році, запідозривши у зраді свого бойфренда, футболіста Андре Райсона, в сильному сп'янінні Лопес підпалила його будинок. Отримала 5 років позбавлення волі умовно, а також виплатила величезний штраф (який сплачував гурт) і пройшла програму реабілітації.

Вирушаючи у відпустку в Гондурас, 25 квітня 2002 року, щоб уникнути зіткнення з вантажівкою, яка несподівано виїхала їй назустріч, з'їхала з дороги. З семи осіб, що були в автомобілі, загинула лише Лопес. Після смерті співачки її матір подала позов на Mitsubishi Motors, заявивши, що в машині були допущені технологічні помилки, але успіху він не мав.

TLC
Гурт TLC з'явився у 1991 році, названий від перших літер псевдонімів солісток: T-Boz (Тіонн Воткінс), Left Eye (Ліза Лопес) і Chilli (Розонда Томас). Перший альбом 1992 року «Ooooooohhh… On the TLC Tip» розійшовся світом у 6 мільйонів копій. У 1994 році виходить другий альбом «CrazySexyCool», продано понад 15 мільйонів примірників; в 1999 році третій альбом «FanMail» — більше 10 мільйонів екземплярів.

Лопес, перебуваючи в гурті, пише для нього багато пісень, як-от: «ain't 2 Proud 2 Beg», «What About Your Friends», «Hat 2 da Back», «No Scrubs», «Waterfalls», та «Girl Talk».

Сольна кар'єра
Лопес вважалася більш яскравою учасницею в колективі. І тому на початку 2000-х вона починає сольну кар'єру: співпрацює з іншими артистами, бере участь у телевізійних зйомках і випускає в 2001 році свій перший альбом під назвою «Supernova». У 2002 році планувався другий альбом «N. I. N. A.», але відразу після смерті співачки у квітні 2002 року був скасований.

Дискографія
 2001: Supernova
 2002: N. I. N. A. (відмінений; дещо з невиданого альбому з'явилося на альомі «3D» гурту TLC)

 2009: Eye Legacy

Мініальбоми
 2009: Forever... The EP
 2013: Reigndrops in My LeftEye

Фільмографія

Примітка

Посилання
 Lisa Lopes Foundation

Акторки США
Акторки за алфавітом
Музиканти за алфавітом
Персоналії за алфавітом
Загинули в автомобільних аваріях
Народились 27 травня
Народились 1971
Уродженці Філадельфії
Репери США
Афроамериканські співачки
Афроамериканські акторки
Померли 25 квітня
Померли 2002
Поховані в Джорджії
Реперки

redpajama_set_name: RedPajamaWikipedia


text:
Diana Raufelder (* 1978 in Hardheim) ist eine deutsche Erziehungswissenschaftlerin.

Leben
Von 1997 bis 2002 studierte sie Erziehungswissenschaft und Ethnologie an der Universität Heidelberg. Nach der Promotion 2006 an der FU Berlin war sie von 2008 bis 2011 wissenschaftliche Mitarbeiterin an der TU Berlin am Institut für Erziehungswissenschaft, Fachgebiet Pädagogische Psychologie. Nach der Promotion 2014 an der FU Berlin folgte sie 2015 dem Ruf auf die W3-Professur für Schulpädagogik an der Universität Greifswald.

Schriften (Auswahl)
 Von Machtspielen zu Sympathiegesten. Das Verhältnis von Lehrern und Schülern im Bildungsprozess. Marburg 2007, ISBN 3-8288-9471-2.
 mit Angela Ittel: Lehrerrolle – Schülerrolle. Wie Integration gelingen kann. Mit 30 Tabellen. Göttingen 2009, ISBN 978-3-525-61045-9.
 mit Frances Hoferichter: Prüfungsangst und Stress. Ursachen, Wirkung und Hilfe. Stuttgart 2017, ISBN 978-3-17-029391-5.
 Grundlagen schulischer Motivation. Erkenntnisse aus Psychologie, Erziehungswissenschaft und Neurowissenschaften. Opladen 2018, ISBN 3-8252-4961-1.

Weblinks
 ews.uni-greifswald.de

Geboren 1978
Schulpädagoge
Frau
Deutscher
Hochschullehrer (Universität Greifswald)


Mark Prisk, MP for Hertford and Stortford, is this week demanding that the Government acts to stop local voluntary groups from facing excessive new water charges as a result of Whitehall blundering.
Under current government rules, East Herts charities, sports clubs, and groups like the Scouts could face large rises in their water bills, due to changes in how surface water is charged.
The Scout Association estimate that the changes could cost £500 a year per Scout hut, and they have organised a campaign against the 'rain tax'.
Mark Prisk met local representatives of the Scouts at a rally last Wednesday at Westminster.
He said: ""This hike in water bills would be devastating for local groups like the Scouts. Ministers and the water regulators have failed to realise how serious this could be. The rules must be changed and I will work with other MPs and the Scout Association to get that change.""
At a rally, Mr Prisk challenged the regulator Ofwat about the problem, and he has signed EDM 1874 which calls for a change in the rules.


It has been proven effective in relieving the clinical signs of pain and inflammation associated with arthritis.
How safe is Rimadyl® for your dog?
is well tolerated by most dogs but, as with any drug, a small number of dogs may develop side effects.
and experiences any of these signs, or anything that is unusual, stop giving the medication and consult your veterinarian immediately.
What results can I expect with Rimadyl®?
during clinical trials. While Rimadyl® is not a cure for arthritis, it can ease the pain and inflammation and improve your dog's mobility and quality of life.
The response to Rimadyl® will vary from dog to dog but a good response should be expected. However, you must remember that, whereas the signs may have improved, the underlying changes to the joint will still be there. Thus, if the medication is discontinued or not given as directed, your dog may experience a return of pain and inflammation.
In many cases, you will have to give Rimadyl® as a lifelong treatment.
You have an important role to play in the successful treatment of your dog. Please follow your veterinarian's instructions carefully.
Give the medication when you should, and give all the medication prescribed.
Finally, if you have any concerns or enquiries, call your veterinarian - they want to provide the best care for your pet.
Is your dog suffering from arthritis ?
The most common form of arthritis (joint disease) that affects animals is osteoarthritis (OA) also known as degenerative joint disease (DJD).
Osteoarthritis can have many causes – previous injuries (i.e. torn cruciate ligaments), obesity, developmental defects (i.e. hip and elbow dysplasia), previous infections, any thing that causes damage to or excessive wear of the joint cartilage. Large breeds of dogs are more likely to develop arthritis. In many cases it is considered as wear and tear and a normal part of the aging process.
Joints that are affected by arthritis have damage to and a loss of joint cartilage. As a result of this damage there is a reduced cushioning, a reduced range of movement of the joints they eventually become thickened and painful.


King County's Brightwater wastewater treatment plant in the USA celebrated a public grand opening on September 24 and officially began treating wastewater. Brightwater employs GE ZeeWeed MBR Technology.
Part of the $1.8 billion Brightwater Treatment System project, the largest expansion of the King County regional treatment system in 50 years, the plant will serve about 189,000 people in north King and south Snohomish County.
Construction of the project started in 2006 in response to increased growth in the region, Treatment plant start-up and operations begins in 2011, with the entire system scheduled to be completed in 2012. The 114-acre treatment plant site includes more than 70 acres of publicly accessible open space, trails and natural habitat designed to fit into and enhance the surrounding neighbourhood.
The plant is already receiving kudos for energy efficiencies. 'Brightwater is achieving the highest possible standards for the treatment of wastewater, which is the main environmental goal. By operating the plant as efficiently as possible, we are further doing our part as environmental stewards,' said Deputy King County Executive Fred Jarrett.
The technologies bringing the savings include installation of several high efficiency micro turbine blowers that will supply air to the plant aeration system − they are expected to operate 30 to 50 per cent more efficiently than traditional blowers − plus modification of the air handling system for the world's largest membrane bioreactor. By optimising the amount of air required for operation of the MBR, Brightwater expects it can reduce the power consumption by around 50 per cent.
It is estimated that the combination of these two systems will save approx 4.5 million kilowatt hours per year.


text:
Q: Does this describe fourier transform correctly? So i saw this paragraph in an article which wasn't really good and was a complete word salad but this one paragraph caught my eye so i asked on Physics Stack Exchange if this paragraph described Fourier Transform accurately and i got an answer saying no. I want to confirm this , so could someone tell me whether this paragraph is wrong?
Fourier showed that any three-dimensional pattern can be analyzed into a set of regular, periodic oscillations that differ only in frequency, amplitude, and phase. Specific waveforms can be exact representations, namely Fourier-transforms, of spatio-temporal objects. When a vessel creates waves on the surface of the sea, it creates Fourier-transforms of its impact on the waters of the sea. It is shown [172] that the surface of the sea is surprisingly information-rich (in-formation, see footnote n˚ 23). When its wave-patterns are subjected to mathematical analysis, it discloses in-formation on the passage of ships, e.g. the direction of wind, the effect of shorelines, and other factors. The interfering wave-patterns may be conserved for hours and sometimes for days, after the vessels that created them have passed. Though ultimately they dissipate, eroded by the combined action of gravity, wind, and shorelines, as long as the wave-patterns persist, they provide in-formation on the events that occurred at the sea's surface. But the waves created by vessels on the surface do more than create in-formation regarding their own motion: they also in-form the motion of other vessels. All vessels that traverse the wake that spreads out behind a given vessel are rocked by those waves; in this sense the motion of the wave-creating vessel is translated into the motion of the wave-rocked vessels. The medium that transmits the effects is the surface of the sea: it interconnects the wave-creating with the wave-rocked vessels. And, as all vessels both create waves and are rocked by them, the sea interconnects the motion of all vessels on its surface.
I feel that it is wrong but please let me know

redpajama_set_name: RedPajamaStackExchange


text:
package com.godaddy.sonar.ruby.resources;

import static org.junit.Assert.*;

import org.junit.After;
import org.junit.Before;
import org.junit.Test;

import com.godaddy.sonar.ruby.core.Ruby;

public class RubyFileTest {
	protected Ruby ruby;

	@Before
	public void setUp()
	{
		ruby = new Ruby();
	}

	@After
	public void tearDown()
	{

	}

	@Test
	public void test()
	{
		assertEquals("ruby", ruby.getKey());
		assertEquals("Ruby", ruby.getName());

		String[] expected = new String[] {".rb"};
		assertArrayEquals(expected, ruby.getFileSuffixes());
	}

}

redpajama_set_name: RedPajamaGithub


text:
St. John Church, Middletown, CT
← Pastoral Sharings: "Easter Sunday"
Pastoral Sharings: "Third Sunday of Easter" →
Pastoral Sharings: "Second Sunday of Easter"
Posted on April 26, 2014	by bobdraz
Father Alex McAllister SDS
Second Sunday of Easter
Very often you see pictures of St Thomas touching the wound in Christ?s side but in actual fact the Gospel does not record this event. Christ certainly showed him his wounds but it is never mentioned that Thomas reached out his hand to actually touch them.
Interestingly, apart from this incident, Thomas is portrayed in the Gospels as being very brave. In the account of the raising of Lazarus that we heard on the last Sunday of Lent when Jesus gets the message of Lazarus? illness and he decides to go up to Jerusalem we find Thomas saying, ?Let us go too and die with him.?
These are not the words of a timid and fearful man; a man beset by doubts. And yet when the other Apostles tell him of their meeting with the Risen Lord, which for some unknown reason he had missed, Thomas flatly refuses to believe them.
What Thomas had missed out on was an encounter with the Risen Christ. And, no matter what the other Apostles said, he refused to believe. He wasn?t open to persuasion or reasoning.
And I think we have to say, ?Rightly so!? After all, faith does not come from reasoning or from what anyone else tells us. Faith is a gift of God and it principally comes though an encounter with the Lord.
In Thomas? case this was the actual presence of the Risen Jesus who showed him the wounds of his crucifixion. For St Paul it was his Damascus experience.
In every case, let me suggest, faith comes through an encounter with the Lord. Mostly these are not physical encounters like that of Thomas, but they are just as real nonetheless.
Each one of us comes to faith by a different route. Things happen to us on life?s journey that help us to see the hand of God at work in our lives.
As a child we might be brought up by our parents to believe in God and we grow up accustomed to pray each day. In this way prayer becomes a natural and even essential part of our lives.
But this is not merely the saying of prayers. What our parents have initiated us into is a dialogue with the Lord ?with a person, with God himself. Each time we pray we are entering into an encounter with God.
At some point or other the young person faces the criticism of others and they question whether this is a real dialogue or whether they are just talking to themselves. If their prayers are more than merely superficial then they may well come to the realisation that this is no empty dialogue but a real and meaningful conversation with the Lord. And through this insight their faith is strengthened and moves to a new and deeper level. As life goes on our faith is validated by all sorts of events and occurrences. I clearly remember talking to a group of secondary school pupils. We were discussing prayer and I asked them if they ever felt that their prayers were answered.
One girl said that together with her whole family she had prayed very hard for her grandmother who had cancer. She explained that although they had prayed for a cure the grandmother actually got worse and eventually died.
Nevertheless she felt that her prayers had been answered because her grandmother had died peacefully and was happy to go to God. It was also clear that her own faith had been strengthened by this and that she felt closer to God and to her family as a result. Thomas said that he wanted proof. He said that unless he could put his hand in the wounds he would not believe. But when Christ appeared to him that was enough, he never reached his hand out to touch the wounds.
Instead he fell to the ground with the great words ?My Lord and my God? on his lips. Throughout his life Thomas never lacked courage. Tradition has it that he preached the Gospel in many different countries ultimately travelling as far as India where he was martyred. As we have heard together with Jesus he set out on the road to Jerusalem saying to his fellow Apostles, ?Let us go too, and die with him.? Well, he certainly got his wish even if he had to wait a few years for it!
His faith was surely tested more in that final moment of his death than it ever was before. But in the end he remained resolute. His words to Jesus, ?My Lord and my God? or something very like them were surely on his lips as he gave up his spirit.
Thomas had the extraordinary privilege of knowing Jesus in the flesh and also of meeting him in his risen form. But the greatest encounter of all was at the moment of his own martyrdom when he was drawn into the presence of God in heaven.
It is this final and ultimate encounter that we are all preparing for. And the best preparation of all is for us to open our eyes and see the hand of God in our lives and for us to spend time in prayer and dialogue with him, but most of all by sharing his body and blood in the Eucharist.
It is in these ways that our faith is fed and strengthened. It is by doing these things that at that final moment, with God?s grace, we will make that great and wonderful prayer of Thomas our own.
http://www.catholicwealdstone.org/wordpress/?p=1645
SaintJohnChurchMiddletown.com
Digest of Articles from Catholic Blogs and Websites
2nd Sunday of Easter (Divine Mercy Sunday)
Gospel John 20:19-31
The Gospel begins with Easter Sunday and ends with the Sunday after Easter. On Easter Sunday the disciples were still mourning the death of Jesus, and even though they had heard reports that the tomb was empty and that Jesus is Risen, this reality was too much for them to accept and they remained in fear hidden away behind the locked doors of the upper room. This was same upper room where they had only days earlier celebrated the Passover with Jesus and were probably already referring to it as the "Last Supper." While Jesus had been buried in a tomb of rock, the upper room was becoming their tomb.
Second Sunday of Easter: Doubts, Divine Mercy and St. John Paul II
The Sunday after Easter always presents the event that took place in the Upper Room one week after Jesus rose from the dead. Pope John Paul II also designated this Sunday as Divine Mercy Sunday. And today the world celebrated the Canonization of this pope whom so many refer to as John Paul the Great. I believe I can tie all three of these themes together.
The Gospels tell an incredible story. A virginal conception. Miraculous healings. Even people coming back from the dead. How are we to know that it's not all just a fanciful fabrication?
There is much evidence for the reliability of the Gospels, but here is one of the strongest bits of evidence I know. Think for a minute. If you were part of a group who decided to perpetrate an elaborate hoax, what would be your motive? Wouldn't you want to gain some significant benefits from such a risky business? Maybe fortune, fame, and privilege? And if you were to be prominent figures in this tall tale, wouldn't you at least want the story to make you look good?
Second Sunday of Easter, Year A (Divine Mercy Sunday)
Gospel (Read Jn 20:19-31)
The celebration of our Lord's Resurrection on Easter Sunday usually focuses on the sheer ecstasy of His victory over death. All during Holy Week, we are absorbed with the details of His horrific Passion. When we reach Easter, our hearts nearly burst with joy that Jesus is alive and vindicated as God's Son. In other words, it's easy to dwell on the fact of the Resurrection and be so dazzled by it that we do not think much beyond that. The mercy of Divine Mercy Sunday (yes, intended pun) is that now we begin to meditate on the meaning of the Resurrection. Today's Gospel gets us started.
Jesus Died to Save You – What Does THAT Mean???
You see the signs beside the highway blazoned "Jesus Saves". You hear the religious slogans from Catholics and Protestants, "Jesus died to save you from your sins." or "Jesus blood was shed to wash away your sins."
I'm sympathetic when I hear modern secularists say with frustration, "What on earth does that mean??!! How can the death of a criminal two thousand years ago cleanse away my 'sins'?"
The Five-Fold Argument for the Resurrection
The New Testament consists of first-century, seemingly first-hand accounts of the life, death, and resurrection of Jesus Christ. It also claims to be an eyewitness account and makes some mighty bold claims about what the authors saw and heard. Yet a first-century audience (that is, an audience which would have been able to call "shenanigans" if the authors were just making things up) believed these accounts, copied them, and spread them both textually and orally throughout the entire Roman Empire and to faraway places like India.
Live Out Divine Mercy
According to both Blessed John Paul II and Saint Faustina Maria Kowlasksa, the virtue of mercy is the greatest attribute in the Heart of Jesus. This being the case we should strive to understand this virtue and even more important try to live it out to the full.
Honoring Two Beloved Popes
Hundreds of thousands of pilgrims are expected to flock to Rome on Divine Mercy Sunday, April 27, for the canonizations of John XXIII and John Paul II.
But for those Americans who can't make the trans-Atlantic trip, there will be plenty of fanfare and celebration stateside, as two of the 20th century's most beloved popes become saints.
I Can Tell He Loves Me
The old saying is that all roads lead to Rome.
And that has been my experience being here over the last two days.The faithful are flocking into Rome from all corners of the globe, with various countries waving their flags, and every language imaginable being spoken.
Two Modern Saints for the Modern World
Pope Francis shook the Catholic world on July 5th, 2013, with the announcement that Pope John XXIII and Pope John Paul II will be canonized together on April 27th, 2014.
Pope Francis approved the healing of Floribeth Mora of Costa Rica from a brain aneurysm as miraculous, making it the second approved miracle attributed to the beloved Polish pope who passed away in 2005. The first miracle was the healing of a French nun, Sister Marie Simon-Pierre, who had been afflicted with Parkinson's disease—the very ailment that John Paul II suffered from. Both of these women prayed for John Paul's intercession and both of their prayers have been answered.
3 Things Catholics Need to Confidently Reclaim and Own Again
I'm thinking of a person who describes himself as a Christian, gets his beliefs from the Bible, and has a passion for sharing Jesus with others. What kind of person comes to mind?
I bet the first thought most people have is an evangelical Protestant.
But not a Catholic.
And that's a problem.
Seek the Lord's Will, Even in Uncertainty
Wouldn't it be great if we had a direct connection to Heaven — a phone line by which we could call God Himself and ask questions such as "Lord, what's the easiest way for me to solve this problem?"; or "Father, how much longer do I have to put up with this situation?"; or "God, I really don't know what it is that You want me to do; will You please show me?" There have been such cases recorded; the Old Testament hero Moses, for instance, used to talk to God face-to-face,123 (123 Exod. 33:11) and the Apostles were able to question Jesus whenever they wanted. Many mystics and visionaries throughout the centuries have allegedly held conversations with Jesus or one of the saints, and there are hundreds (perhaps thousands) of persons today who claim to be in contact with Heaven.
At the Gate Called "Beautiful." What the Miracle by Peter and John Teaches us about our Spiritual Journey
At the daily masses of the Easter Octave, we have been reading about, among other things, the story of a paralyzed man whom Peter and John encounter just outside the Temple at the Gate called "Beautiful." This paralyzed man's story is our story and as we read it we learn something of our own spiritual journey to the Lord and to heaven, symbolized here by the Temple. Let's look at this moving story, as it is not merely the recounting of an event taking place 2000 years ago; rather it is our story.
God is Not a Vending Machine
"If God is so good and loving, if He's really there at all, why didn't He answer my prayer?" This question has been asked by atheists and agnostics and doubters and believers and skeptics and observers and disinterested third-parties alike. For some it presents a serious challenge to their faith, or to the possibility of their having any faith at all.
The Dignity of Dying
Within the last month we've marked the anniversaries of two important deaths that, while linked inextricably because of when they occurred, could not have been more different.
Does Faith Need Culture? Answers from Scripture & the Church
"I came that they may have life, and have it abundantly"
(John 10:10 ).
Is culture something necessary for the life of faith? Or, is it rather a distraction? Does it pull us further away from a focus on the next life, by rooting us in the things of the earth? Is it a temptation to try to build a lasting city, when Hebrews says "here we have no lasting city, but we seek the city which is to come" (13:14).
The Meal at Emmaus – Jesus's Todah
The story of Jesus's journey to Emmaus has to be one of favorite stories in the whole of Scripture. Cleophas and another disciple were traveling from Jerusalem to Emmaus, devastated by their Master's crucifixion only days before , and utterly bewildered by the women's report of the empty tomb, when Jesus sidled up alongside them and struck up a conversation. "What are you discussing?" Then, while preventing their eyes from recognizing him, Jesus "explained to them what was said in all the Scriptures concerning himself" (Lk. 24:27). And as if that wasn't enough, after he accepted their invitation to dinner, "he took the bread and blessed, and broke it, and gave it to them. And their eyes were opened and they recognized him; and he vanished out of their sight." (Lk. 24:30-31).
Are There Souls in Hell Right Now?
It has become fashionable in some Catholic quarters these days to question where there are now or will ever be any souls populating hell. Hell, it is taught, is a "real possibility," but whether there are any souls actually there, or whether there will ever be any souls there, is unknown to us.
It is, of course, true that hell is a "real possibility" for each of us. And that is a sobering thought. But it is also true that souls are actually in hell now, and will be for all eternity. This is a teaching of our Catholic Faith.
Origins of the Rosary
The rosary is one of the most cherished prayers of our Catholic Church. Archbishop Fulton Sheen said, "The rosary is the book of the blind, where souls see and there enact the greatest drama of love the world has ever known; it is the book of the simple, which initiates them into mysteries and knowledge more satisfying than the education of other men; it is the book of the aged, whose eyes close upon the shadow of this world, and open on the substance of the next. The power of the rosary is beyond description."
Knights' success built on founder's desire for charity
In many ways, Father Michael J. McGivney was just one more of that band of hardworking Irish-American priests who spent themselves building up the Church in America in the latter years of the 19th century. But in one truly extraordinary respect, he was unique: Before he was 30, Michael McGivney had founded what was to become the largest Catholic men's organization in the world: the Knights of Columbus.
Why Catholics Must Reject Elite Culture
It seems that Catholics have been getting nowhere in the public square lately. The problem is not just losing ground on this issue or that, but an increasing inability to get our issues recognized as real and legitimate. That's true not only with moral issues, but also with more basic ones like the rationality of religion and the very existence of human nature.
Why Catholicism Is Preferable to Protestantism
My new book, The Protestant's Dilemma, shows in a myriad of ways why Protestantism is implausible. We sifted through many arguments to boil the book down to the most essential. A few chapters didn't make the cut but are still good enough to share.
Here's one of them.
Mao Thought He Could Eliminate Religion, He Was Wrong
China, according to the Telegraph, may very well be home to more Christians than any other country in the world by 2030. I know that seems like some futuristic date but hey, that's only 15 years.
Officially, the People's Republic of China is an atheist country but that is changing fast as many of its 1.3 billion citizens seek meaning and spiritual comfort that neither communism nor capitalism seem to have supplied.
Christian congregations in particular have skyrocketed since churches began reopening when Chairman Mao's death in 1976 signalled the end of the Cultural Revolution.
Less than four decades later, some believe China is now poised to become not just the world's number one economy but also its most numerous Christian nation.
How To Pass A Personhood Amendment
Everybody knows that Roe v. Wade created a national right to kill your unborn children, and everybody knows that the #1 item on the pro-lifer To-Do list is "Overturn Roe v. Wade." However, once Roe is overturned, we'll still have a lot of work to do. Abortion will be a state issue. Unfortunately, most state courts would gladly recreate Roe v. Wade at the state level, using the state constitution to magic abortion rights back into existence. Some have already done this. Meanwhile, although many states provide some legal protections to unborn children, these protections are not considered inalienable human rights; they could be completely repealed with a single legislative act.
Palm Sunday of the Lord's Passion
Pastoral Sharings: Solemnity of the Epiphany
Pastoral Sharings: Feast of the Holy Family
"Are You Missing Out? Week 1: Jesus' Warning
Pastoral Sharings: "Feast of Christ the King"
Franklyn on Twenty-Eighth Sunday in Ordina…
Joel Schmidt on Pastoral Sharings: Fr. Mike…
Jeanette Marie on Time To Clean House
Dale Turenne on Saved by Faith or Works?
Popular Piety and Ma… on Open The Door
Father Jim
Father Mike

redpajama_set_name: RedPajamaCommonCrawl


This works when first adding a site to the search console, but after adding a site use the following instructions to add a sitemap.
Go to the search console dashboard https://www.google.com/webmasters/tools/home. Is your site listed? Click its listing, click the Sitemaps area.
If it says "No sitemap found" click the "add a sitemap" button.


text:
Courteney Cox Arquette - Details
Voted one of the worlds most sexy women, Courteney Cox has risen to stardom through her many achievements in the showbiz industry. Originating from Birmingham AL, Cox migrated to New York following the completion of high school and commenced her career as a model.
In 1984, Cox made her acting debut , in the daytime drama "As The World Turns", from this platform Cox was launched into the world of acting, staring in Hollywood movies,and TV shows, but gaining notoriety as the infamous Monica in the hit TV sitcom 'Friends'
Sometimes Credited as Courteney Cox, or Courteney Cox Arquette
Married David Arquette 12/6/1999
15th June 1964 - Birth


At First glance of Eric Orchard's work, several names and titles spring to mind. Edward Gory, Tim Burton, Gloom Cookie, Emily the Strange - but that's where the similarities end.
Orchard has echoes (or even homages, if you will) of what are clearly his influences. Everything from the gothic fairy tales and old 40's and 50's subtle horror movies to the whimsical comedies of the 70's and 80's (Dough Zombies?? Okay!) But he separates all of these influences to create his own world populated with his own characters. The world of Marrow Bones.
Marrow Bones is, for all it's gloom and muck, a world of dreamy magic. A place where all things are possible and anything can (and does) happen. For Nora, the books protagonist, it is her home. Her safe haven from a world of loneliness and cruel people. She is employed, as a hand, at the Ravensbeard Inn, her Uncle Barnaby's place of business.
The first issue involves Nora's escape from where her parents abandoned her in the prison for strange kids. Her escape is aided by some of her Uncle's friends known only as the Gord Brothers (I Can't wait to learn more about these guys!) We're introduced to a lovely cast of characters including the Narrator who reminds me somewhat of the crypt keeper but I feel like I could be this guys pal (he's also known as the Librarian). Mrs. Strumm, a ghost who has lived at the Ravensbeard Inn for as long as anyone can remember. Ollie the Vampire (Easily my favorite character) who's a nervous little wreck. Uncle Barnaby (he's a werewolf, permanently. Though I did mistake him for a gigantic rat in one of the panels). Nora is the center point, and like most stories, she's the catalyst that keeps it all moving forward. Among her many other accomplishments, she's also very skilled at fencing!
I loved this issue. It was my first time reading a completely digital book which presented some difficulties for me. However, I understand the seas of change and I can tell which way the wind is blowing. I'll still have bookshelves full of thousands of pounds worth of cherished paper stories bound in leather - but, perhaps I'll start creating a virtual library as well. I'd be proud to add this first issue of Marrow Bones to the collection.
Eric Orchard clearly puts a lot of passion into his work. His characters are alive and there's a real sense of intimacy between them. After seeing the map at the end of the book, the only thought I had was "I can't wait to see what issue two holds!"
The digital copy rings in at 47 pages for a measly two bucks. You can check out Eric Orchards Site here and pick up a copy of the first issue while you're there too!
Eric Orchard has a way with words- and drawing, that never fails to tweak my sense of humor.
I agree with you completely! It also has this longing nostalgia feel for wanting to be a child, yet retaining the social skill to converse as an adult. Very effective.


Black Friday saw increased activity on the UK's Top100 online retailers. We've graphed the average page load time of the Top100 and measured how many sites were unresponsive at some point.
*Edit at 12.30am on Monday 30 November, 2015: this article previously stated that websites were 'down'. This ambiguous terminology was queried by some readers and so 'unresponsiveness' is referred to and briefly defined in the updated version. All tests were conducted from a London-based server using a Chrome desktop browser. Our testing via GTmetrix.com does not take into account local caching or ISP caching, nor the customer's willingness to wait. For more information on what constitutes a timed-out 'unresponsive' test, please see GTmetrix' FAQ.