text:
\section{Introduction}
During a relativistic heavy ion collision, it is possible to create a
quark-gluon plasma, which will then cool and expand, leading to
hadronization.  At around the same time as hadronization, the system
also undergoes a chiral phase transition which breaks the SU(2)
$\times$ SU(2) symmetry and which then gives physical masses to the
pion and other particles. The out of equilibrium dynamics of this
phase transition can be very interesting if the expansion is fast
enough so that the phase transition resembles a quench. In this case
the effective pion mass can go negative for short periods of proper
time which leads to a temporary exponential growth of low momentum
domains where the isospin can point in a particular direction
(DCC). This is a topic of current interest \cite{ref:dcc}, since DCCs
may provide a signature of the chiral phase transition.  In an earlier
paper \cite{ref:mypaper} the time evolution of this chiral phase
transition, assuming a uniform spherically symmetric expansion into
the vacuum, was considered. The O(4) linear $\sigma$ model in the
large-$N$ approximation, which incorporates both nonequilibrium and
quantum effects, was studied. A range of initial conditions was
analyzed to determine which initial states could lead to the formation
of DCCs.  This radial expansion is interesting because it is expected
at late times in the plasma evolution, and also maximizes the cooling
rate and therefore the possibility of the phase transition resembling
a quench.

In this paper we go beyond previous work \cite{ref:mypaper} and study
the proper time evolutions of the effective hydrodynamic collective
variables of the system, namely the renormalized pressure and energy
density. We want to see to what extent previous intuition coming from
studying classical hydrodynamical models of particle production such
as Landau's model is confirmed.  This requires the solution of a new
technical issue, namely how one obtains a covariantly conserved
energy-momentum tensor when one has an actual cutoff in the theory.
As discussed in earlier work \cite{ref:cooperdcc,ref:phi4}, the linear
$\sigma$ model is an effective theory with a renormalized coupling
constant of order 10, in order to agree with low energy pion
properties. This leads to a maximum cutoff of around 1 GeV in order to
avoid problems at scales of the order of the Landau pole.  It has been
shown earlier \cite{ref:cooperdcc} that as long as the momentum cutoff
determined from the sum over mode functions is below the Landau pole,
there is a regime of cutoffs where the continuum renormalization group
flow is obeyed so that one can safely say that one is in the continuum
regime.  For this class of problems, renormalization methods based on
formal schemes such as dimensional regularization approaches are not
very useful.  Here we have a real cutoff in physical momentum as a
result of using an effective field theory, and we have to be very
careful in order to obtain the correct covariantly conserved
energy-momentum tensor which leads to our definitions of comoving
energy density and pressure variables.  The tools we use to correctly
renormalize the energy-momentum tensor are first to introduce the
physical cutoff $\Lambda$ and then to analyze the divergences in all
the components of the energy-momentum tensor using an adiabatic
expansion of the mode functions.  By comparing our results with a
covariant point-splitting approach \cite{ref:christensen}, we can
identify those terms which should survive in the covariant limit. This
technique allows us to obtain finite energy densities and pressures
which then enable us to study the effective equation of state for the
evolving plasma.

In this paper, we derive the energy-momentum tensor for the O(4)
linear $\sigma$ model, and show how to regularize and renormalize it
in order to obtain the physical energy density and pressure. We study
initial conditions which have instabilities that lead to DCC formation
and also stable initial configurations, and examine the energy density
and pressure in both situations. We also consider the conservation of
the energy-momentum tensor as a check on our numerical methods.

The paper is organized as follows. In Sec.~\ref{sec:tensor} we discuss
the model and coordinate system used, and show how to construct the
energy-momentum tensor. Then in Sec.~\ref{sec:renorm} we explicitly
describe the scheme used to renormalize this tensor in order to
perform a numerical simulation with a physical momentum cutoff.  In
Sec.~\ref{sec:results} we numerically examine the conservation of
energy and the equation of state of the system. Finally in
Sec.~\ref{sec:conc} we discuss the results and provide some concluding
remarks.

\section{Energy-Momentum Tensor}
\label{sec:tensor}
\subsection{Equations of Motion and Coordinate System}
\label{subsec:eom}
We consider the O(4) linear $\sigma$ model, with the classical action
given by \cite{ref:mypaper}
\begin{eqnarray}
   S[\Phi_i,\chi;j_i] = \int d^4 x \sqrt{-g(x)} \left(
      {\cal L}[\Phi_i, \chi] +  j_i \Phi_i \right)
\>,
\end{eqnarray}
where the mesons are in a O(4) vector representation
\begin{equation}
   \Phi_i = (\sigma, \vec \pi)
\>.
\end{equation}
The Lagrangian density is
\begin{eqnarray}
   {\cal L}[\Phi_i, \chi] =
      \frac{1}{2}
      g_{\mu\nu}(x) [ \partial^\mu \Phi_i(x) ][ \partial^\nu \Phi_i(x) ]
      - \frac{1}{2} \chi(x) \Phi_i^2(x) + \frac{v^2}{2} \chi(x)
      + \frac{1}{4 \lambda} \chi^2(x)
\>.
\label{eq:lag}
\end{eqnarray}
In order to quantize the system, we construct the generating
functional of the connected Green's functions and carry out the path
integral to obtain the following quantum effective action (to leading
order in $N$) \cite{ref:mypaper}
\begin{equation}
   \Gamma[\phi_{i},\chi ; j_i]= S[\phi_{i},\chi ; j_i]
      + \frac{iN}{2} {\rm Tr} \ln G_0^{-1}(x,x'; \chi)
\>,
\end{equation}
where
\begin{equation}
   G_0^{-1}(x, x'; \chi) = \left[\Box + \chi(x) \right]
      \delta^4(x,x') / \sqrt{-g(x)}
\>,
\end{equation}
and
\begin{eqnarray}
   \phi_{i}(x) &\equiv& \langle \Phi_{i}(x)\rangle,
   \nonumber \\
   \chi(x) &\equiv& \langle \chi(x)\rangle
\>.
\end{eqnarray}
By varying the effective action with respect to $\phi_i$ and $\chi$, we
obtain the following equations of motion for the mean field $\phi_i$
\begin{equation}
   \left[\Box + \chi(x)\right] \phi_i(x) =
      j_i(x) \equiv H \delta_{i0}\
\>,
\label{eq:eom}
\end{equation}
and the constraint equation for the effective mass squared $\chi$
\begin{equation}
   \chi(x) = \lambda [-v^2 + \phi_i^2(x) - i N G_0(x,x; \chi)]
\>.
\label{eq:chieom}
\end{equation}
Notice that $G_0(x, x'; \chi)$ depends on $\chi$ through its definition,
and it fulfills the following dynamical equation
\begin{equation}
  \left[ \Box + \chi(x) \right] G_0(x, x'; \chi)  =
     \delta^4(x,x') / \sqrt{-g(x)}
\>.
\end{equation}
The current has only a non-vanishing component in the zero ($\sigma$)
direction in order to give mass to the pion \cite{ref:mypaper}.  There
are three parameters in the model: the mass of the pion $m = m_{\pi}$;
the value of $H = f_\pi m_\pi^2$, which gives the vacuum expectation
value $\langle \sigma \rangle = f_\pi$ (using PCAC); and the coupling
constant $\lambda$, which is determined by fitting to low energy
$\pi-\pi$ scattering data \cite{ref:mypaper,ref:cooperdcc}.

The picture one gets from hydrodynamical simulations of heavy ion
collisions is that the energy density is initially in a Lorentz
contracted disk which expands first in the longitudinal direction and
becomes three dimensional at late times.  At early times one expects
the velocity to scale approximately as $z/t$, where $z$ is the
longitudinal direction, since the effective longitudinal size goes to
zero with center of mass energy. This leads to the energy density
approximately becoming a function only of the longitudinal fluid
proper time variable $ \tau_z= (t^2-z^2)^{1/2}$.  At later times when the
expansion is more spherical and the initial distribution looks more
like a ``point'', one expects that the velocity scales as $v=r/t$ and
the energy density then becomes a function of the spherical fluid
variables:
\begin{eqnarray}
   \tau & = & {(t^2 - r^2)}^{1/2},
   \nonumber \\
   \eta & = & {\rm arctanh} (r/t) =
      \frac{1}{2} \ln \left( \frac{t + r}{t - r} \right)
\>,
\end{eqnarray}
where $t = \tau \cosh \eta$ and $r =\tau \sinh \eta$
\cite{ref:cooperlandau}. We restrict the range of these variables to
the future light cone, namely $0 \leq \tau < \infty$ and $0 \leq \eta
< \infty$. The coordinates $(\tau, \eta, \theta, \varphi)$ are useful
to describe a spherically symmetric expansion of a plasma when one is
in a hydrodynamical scaling regime. A spherical expansion provides the
fastest possible expansion rate (and therefore cooling rate) of the
quark-gluon plasma and thus enhances any nonequilibrium effects that
are based on the idea of a rapid quench. Since complete inhomogeneous
evolutions are at the edge of or beyond what is presently numerically
possible, this spherically symmetric expansion provides the other
extreme when compared with the slower, purely longitudinal expansion
studied earlier \cite {ref:cooperdcc}.

In terms of this coordinate system, Minkowski's line element
\begin{equation}
   ds^2 = dt^2 - dr^2 - r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)
\end{equation}
is given by
\begin{equation}
   ds^2 = d\tau^2 - \tau^2(d\eta^2 + \sinh^2 \eta \; d\theta^2 +
      \sinh^2 \eta \sin^2 \theta \; d\varphi^2)
\>,
\end{equation}
from which we can read off the metric tensor
\begin{equation}
   g_{\mu \nu} = {\rm diag} \; (1, -\tau^2, -\tau^2 \sinh^2 \eta,
      -\tau^2 \sinh^2 \eta \sin^2 \theta)
\>.
\end{equation}
The metric in this coordinate system is of the Robertson-Walker form:
\begin{equation}
   g_{\mu \nu} = {\rm diag} \; (1, -{a^2(\tau)},
      -{a^2(\tau)} \sinh^2 \eta,
      -{a^2(\tau)} \sinh^2 \eta \sin^2 \theta)
\>,
\end{equation}
which corresponds to a Ricci flat cosmological model with uniform
expansion $a(\tau) = \tau$, and hyperbolic spatial sections, {\it
i.e.}, with curvature $\kappa = -1$.

It will be convenient to introduce the conformal time $u$ given by
\begin{equation}
   u = \int^{\tau} a^{-1}(\tau') \; d\tau'
\>,
\end{equation}
or equivalently
\begin{equation}
   \tau = \frac{{\rm e}^{u}}{m}
\>,
\end{equation}
where we choose $m = m_\pi$, the only mass scale in the system.  The
metric in the coordinate system with conformal time $(u, \eta, \theta,
\varphi)$ has the form
\begin{equation}
   g_{\mu \nu} = C(u)\; {\rm diag} \; (1, -1, -\sinh^2 \eta,
      -\sinh^2 \eta \sin^2 \theta)
\>,
\end{equation}
with
\begin{eqnarray}
   C(u) & = & a^2(\tau) = \tau^2 = {\rm e}^{2u} / m^2,
   \nonumber \\
   \sqrt{-g} & = & C^2(u) \sinh^2 \eta \sin \theta
\>.
\end{eqnarray}

In a hydrodynamical model, all the expectation values depend only on
the proper time of the system. We therefore will assume that the mean
fields $\phi_i$ and $\chi$ are only functions of $u$, that is $\phi_i
= \phi_i(u)$ and $\chi = \chi(u)$. We then write the full quantum
field $\Phi_i$ in terms of its expansion about its mean value, as
follows:
\begin{equation}
   \Phi_i(u,\eta,\theta,\phi) = \phi_i(u)
      + \hat\phi_i(u,\eta,\theta,\phi),
\end{equation}
where $\hat\phi_i$ are the quantum fluctuations, which include both
vacuum and thermal excitations.  The equations of motion are then
\begin{eqnarray}
   \left[ C^{-1}(u)
      \left( \frac{{\partial}^2}{\partial u^2}
	+ 2 \frac{\partial}{\partial u}
	 \right) + \chi(u) \right]
      \phi_i(u) & = & H \delta_{i0},
   \nonumber \\
   \left[\Box + \chi(u) \right] \hat\phi_i(x) & = & 0
\>,
\end{eqnarray}
where $x$ is the four vector $x = (u, \eta, \theta, \varphi)$.
Then for $G_0(x,x';\chi)$ we find
\begin{displaymath}
   G_0(x,x';\chi) = i \; \langle {\rm T}_c \{
      \hat\phi_i(x), \hat\phi_i(x') \} \rangle
\>,
\end{displaymath}
where T$_c$ corresponds to a $u$-ordered product, following the
closed-time-path formalism of Schwinger \cite{ref:ctp}.

In order to solve the wave equation for the quantum fluctuations, we
follow Parker and Fulling \cite{ref:parker}, and write a mode
expansion for $\hat\phi_i$ as follows
\begin{equation}
   \hat{\phi}_i(u,\vec{x}) = C^{-1/2}(u) \;
      \int_0^{\infty} d s
      \sum_{lm} \bigl[
      \hat{a}_{i,slm} \, g_s(u) {\cal{Y}}_{slm}(\vec{x}) +
      {{\hat{a}}^{\dag}}_{i,slm}
      \, {g_s^*}(u) {\cal{Y}}^\ast_{slm}(\vec{x}) \bigr]
\>,
\label{eq:modes}
\end{equation}
with
\begin{displaymath}
   \vec{x} = (\eta, \theta, \phi),
\end{displaymath}
and
\begin{displaymath}
   \Delta^{(3)}{\cal Y}_{slm}(\vec{x})+ (s^2 + 1)
      {\cal Y}_{slm}({\vec {x}})=0
\>,
\end{displaymath}
where $\Delta^{(3)}$ is the Laplacian of the three dimensional
hyperbolic spatial sections of curvature $\kappa = -1$.  Then the mode
functions $g_s(u)$ satisfy the following differential equation
\begin{equation}
   {\ddot{g}}_{s}(u) + \left[ s^2 +  C(u) \chi (u) \right] g_s(u)=0
\>,
\end{equation}
and for $G_0(x,x;\chi)$ we find \cite{ref:mypaper}
\begin{equation}
   \langle \hat\phi_i^2 \rangle = - i \; G_0(x,x;\chi) =
      C^{-1}(u) \int_{0}^{\infty} \, \frac{s^2 ds}{2{\pi^2}}
      (2n_s+1) |g_s(u)|^2
\>,
\end{equation}
once we have chosen a particular vector state with respect to which we shall
be taking expectation values.  We choose an initial state such that
the pair densities are zero, and the particle number density is
finitely integrable with respect to the corresponding integration
measure, namely
\begin{eqnarray*}
   \langle \hat{a}_{j,s'l'm'}^{\dagger} \hat{a}_{i,slm} \rangle & = &
      n_s \, \delta_{i j} \, \delta(s' - s) \,
      \delta_{l l'} \, \delta_{m m'} \>,
   \\
      \langle \hat{a}_{j,s'l'm'} \hat{a}_{i,slm}^{\dagger}
      \rangle & = & ( n_s + 1 ) \, \delta_{i j} \, \delta(s' - s) \,
      \delta_{l l'} \, \delta_{m m'} \>,
   \\
      \langle  {{\hat{a}}^{\dag}}_{i,slm} {{\hat{a}}^{\dag}}_{j,s'l'm'}
      \rangle & = & p_s \, \delta_{ij} \, \delta(s' - s) \,
      \delta_{ll'} \, \delta_{mm'} = 0 \>,
   \\
      \langle  {{\hat{a}}}_{i,slm} {{\hat{a}}}_{j,s'l'm'} \rangle
      & = & p_s^{\ast} \, \delta_{ij} \, \delta(s' - s) \,
      \delta_{ll'} \, \delta_{mm'} = 0
\>.
\end{eqnarray*}
Here we choose the initial particle number density to be a thermal distribution
\begin{displaymath}
   n_s = [e^{\omega_s(u_0)/k_BT}-1]^{-1}
\>,
\end{displaymath}
where ${{\omega}^2_s(u_0)} = s^2 + C(u_0) \chi (u_0)$.  Notice that we
can choose the pair density to vanish  $(p_s = 0)$,
since one has the freedom to
make a Bogoliubov transformation at $u_0$ so that this always remains
true. The boundary conditions on the mode functions $g_s(u)$ is that
they correspond to the positive frequency adiabatic mode functions
initially. We therefore choose
\begin{eqnarray}
   g_s(u_0) &=& 1 / \sqrt{2 \omega_s(u_0)},
   \nonumber \\
   \dot{g}_s(u_0) &=& - \left\{
      \frac{\dot{\omega}_s(u_0)}{2 \omega_s(u_0)}
      + i \omega_s(u_0) \right\} g_s(u_0)
\>.
\label{eq:initmodes}
\end{eqnarray}
The following rescalings will be useful:
\begin{mathletters}
\label{allrescale}
   \begin{equation}
      \phi_i(u) = C^{-1/2}(u) \rho_i(u),
      \label{eq:rho}
   \end{equation}
   \begin{equation}
      \chi(u) = C^{-1}(u) \tilde\chi(u),
      \label{eq:tildechi}
   \end{equation}
   \begin{equation}
      j_i = C^{-3/2}(u) \tilde j_i(u),
   \label{eq:tildej}
   \end{equation}
   \begin{equation}
      v = C^{-1/2}(u) \tilde v(u)
   \>.
   \label{eq:tildev}
   \end{equation}
\end{mathletters}
In terms of the scaled variables, the equations of motion can be written
\begin{eqnarray}
   \left[ \frac{d^2}{du^2} + s^2 + \tilde\chi(u) \right] g_s(u)
      & = & 0,
   \\
   \left[ \frac{d^2}{du^2} -1 + \tilde\chi(u) \right] \rho_i(u)
      & = & \tilde{j}_i(u)
\>,
\label{eq:motion}
\end{eqnarray}
with the gap equation being
\begin{equation}
   \frac{\tilde\chi(u)}{\lambda} = -{\tilde v}^2(u) + \rho^2_i(u)
      + N \int_{0}^{\infty} \, \frac{s^2 ds}{2{\pi^2}}
      (2n_s+1) |g_s(u)|^2
\>.
\label{eq:gap}
\end{equation}

\subsection{Construction of the Stress-Energy Tensor}
\label{subsec:construct}
The energy-momentum tensor is defined by \cite{ref:BirDav}
\begin{eqnarray}
   T_{\mu\nu} & = & (1 - 2\xi)
      ( \nabla_{\mu} \Phi_i ) \, ( \nabla_{\nu} \Phi_i )
      + \left( 2 \xi - \frac{1}{2} \right) g_{\mu \nu} g^{\alpha \beta}
      ( \nabla_{\alpha} \Phi_i ) \, ( \nabla_{\beta} \Phi_i )
   \nonumber \\
      & - & 2 \xi \Phi_i (\nabla_{\mu} \nabla_{\nu} \Phi_i)
      + \frac{1}{2} \xi g_{\mu \nu} \Phi_i \Box \Phi_i
      + \frac{1}{2}(1 - 3\xi) \chi g_{\mu \nu} \Phi_i^2
   \nonumber \\
      & - & g_{\mu \nu} \frac{\chi v^2}{2} - g_{\mu \nu}
      \frac{\chi^2}{4 \lambda}
      - g_{\mu \nu} \left(1 - \frac{3\xi}{2} \right) j_i \Phi_i
\>.
\end{eqnarray}
Equivalently, we can write this equation as \cite{ref:callan}
\begin{equation}
   T_{\mu\nu} =
      ( \nabla_{\mu} \Phi_i ) \, ( \nabla_{\nu} \Phi_i )
      + \xi (g_{\mu \nu} \Box
      - \nabla_{\mu} \nabla_{\nu}) \Phi_i^2
      - g_{\mu \nu} \left( {\cal L}[\Phi_i,\chi]
      + j_i \Phi_i \right)
\>,
\label{eq:tmunu}
\end{equation}
where ${\cal L}[\Phi_i,\chi]$ is given by Eq.~(\ref{eq:lag}).  We take
the expectation value of the energy-momentum tensor in the thermal
initial state chosen, and make use of the mode expansion given in
Eq.~(\ref{eq:modes}) to examine the components of $T_{\mu\nu}$.

After some algebra, we find for the $T_{uu}$ component
\begin{eqnarray}
   C(u)\langle {T}_{uu} \rangle & =  &
      \frac{1}{2} \left[\dot\rho_i^2 +
      ( \tilde \chi + 1 - 12\xi ) \, \rho_i^2 \right]
   \nonumber \\ &&
   + (6\xi - 1) \rho_i \dot\rho_i
      - \tilde{j}_i \rho_i
      - \frac{\tilde \chi \tilde{v}^2}{2}
      - \frac{\tilde\chi^2}{4 \lambda}
   \nonumber \\ &&
   + \frac{N}{2} \int_0^{\infty} \frac{ s^2 d s }{ 2 \pi^2 }
        ( 2 n_s + 1 )
      \Biggl\{
      \left| \dot{g_s} \right|^2 +
      ( s^2 + \tilde \chi + 2 - 12\xi) \, | g_s |^2
   \nonumber \\ &&
      + (6 \xi -1) (g_s \dot{g_s^\ast}
      + g_s^\ast \dot{g_s}) \Biggr\}
\>,
\label{eq:tuu}
\end{eqnarray}
and similarly for the $T_{\eta \eta}$ component
\begin{eqnarray}
   C(u) \langle {T}_{\eta\eta} \rangle & = &
      \frac{1}{2} \left[ \dot\rho_i^2(1 - 4\xi) +
      ( 1 - \tilde \chi - 8\xi + 4\xi\tilde \chi ) \, \rho_i^2 \right]
   \nonumber \\ &&
      + (6\xi - 1) \rho_i \dot{\rho_i}
      + (1-2\xi) \tilde{j}_i \rho_i
      + \frac{\tilde\chi \tilde{v}^2}{2}
      + \frac{{\tilde\chi}^2}{4 \lambda}
   \nonumber \\ &&
   + \frac{N}{2} \int_0^{\infty} \frac{ s^2 d s }{ 2 \pi^2 }
     ( 2 n_s + 1 )
      \Biggl\{
       (1 - 4\xi)\left| \dot{g_s} \right|^2
   \nonumber \\ &&
    + \left[-\frac{s^2}{3} - \tilde \chi + \frac{2}{3}
      + 4\xi(s^2 + \tilde\chi - 1) \right]
      \, | g_s |^2
   \nonumber \\ &&
      + (6\xi - 1)(g_s \dot{ g_s^\ast }
      + g_s^\ast \dot{ g_s })\Biggr\}
\>.
\label{eq:tetaeta}
\end{eqnarray}

It can be easily shown that the energy-momentum tensor is diagonal and
that the spatial components are all equal, except for a geometrical
factor (see Appendix \ref{app:fluid}).  We then define
\cite{ref:cooperlandau}
\begin{equation}
   \langle \, T_{\mu\nu} \, \rangle \equiv C(u)
      {\rm diag} \, ( \,
         \epsilon, \, p, \,
         p \, \sinh^2 \eta , \,
         p \, \sinh^2 \eta \sin^2 \theta \, )
\>.
\label{eq:diagtmunu}
\end{equation}
\section{Renormalization of $T_{\mu\nu}$}
\label{sec:renorm}
In order to analyze the divergences of $\langle T_{\mu \nu} \rangle$
we make use of the adiabatic expansion of the modes $g_s$, since we
know \cite{ref:bunch} that ${\langle T_{\mu \nu} \rangle}$ and
${\langle T_{\mu \nu} \rangle}_{\rm adiabatic}$ have the same
ultraviolet behaviour. Up to second adiabatic order we have
\cite{ref:bunch}
\begin{mathletters}
\label{alladiabatic}
   \begin{equation}
      g_s^A = \frac{1}{\sqrt{2 \Omega_s}} \;
         {\rm exp} \left[ - i \int^u du' \Omega_s(u') \right] ,
   \label{eq:ga}
   \end{equation}
   \begin{equation}
      |g_s^A|^2 = \frac{1}{2\Omega_s}
         = \frac{1}{2\omega_s} + \frac{\ddot \omega_s}{8\omega_s^4}
         - \frac{3 \dot\omega_s^2}{16 \omega_s^5} + \cdots,
   \label{eq:ga2}
   \end{equation}
   \begin{equation}
      |\dot g_s^A|^2 = \frac{1}{2\Omega_s}
         \left(\frac{\dot\Omega_s^2}{4\Omega_s^2} + \Omega_s^2 \right)
         =  \frac{\omega_s}{2}
         - \frac{\ddot \omega_s}{8\omega_s^2}
         + \frac{5 \dot\omega_s^2}{16 \omega_s^3} + \cdots,
   \label{eq:gadot2}
   \end{equation}
   \begin{equation}
      (g_s^A \dot g_s^{A \ast} + \dot g_s^A g_s^{A \ast}) =
        - \frac{1}{2} \frac{\dot \Omega_s}{\Omega_s^2}
	=    - \frac{1}{2} \frac{\dot \omega_s}{\omega_s^2} + \cdots
   \label{eq:ggdot}
   \>,
   \end{equation}
\end{mathletters}
where $\omega_s^2 (u)= s^2 + \tilde \chi (u)$.

We regularize our integrals by introducing a non-covariant cutoff
$\Lambda$ in physical momentum, which corresponds to a comoving
momentum cutoff $s_m= C^{1/2}(u) \Lambda$. We shall work with a fixed
physical cutoff $\Lambda$, so that the comoving cutoff $s_m$ depends
on the conformal time $u$.

We have three physical parameters to renormalize: the mass of the pion
$m_\pi$, the self-interaction coupling constant $\lambda$, and the
(dimensionless) coupling constant to gravity $\xi$, necessary in order
for the field theory to be renormalizable, even in Ricci flat
spacetimes, such as the one considered in this paper
\cite{ref:toms}. The quadratic divergences in the gap equation will be
removed by mass renormalization, and the logarithmic divergences will
be subtracted by renormalization of the coupling constants $\lambda$
and $\xi$.  In addition, we shall see that there is an extra quartic
divergence in both the energy density and the pressure, coming from
the mode integrals in ${\langle T_{\mu \nu} \rangle}$ that carry an
extra factor of $k^2$, which can be removed by renormalizing the
cosmological constant.

First we examine the divergences in the gap equation
(\ref{eq:gap}). From the adiabatic expansion (\ref{alladiabatic}), we
know that the mode integral appearing in this equation has a quadratic
divergence. In order to remove this divergence, we perform mass
renormalization by subtracting the regularized gap equation for the
vacuum state:
\begin{equation}
   \frac{C(u) m^2}{\lambda} = -{\tilde v}^2(u) + C(u) f_\pi^2
      + \frac{N}{2} \int_{0}^{s_m} \, \frac{s^2 ds}{2{\pi^2}}
      \frac{1}{{(s^2 + {\rm e}^{2u})}^{1/2}}
\>.
\label{eq:regvacgap}
\end{equation}
This yields for the gap equation
\begin{equation}
   \frac{\tilde \chi (u)}{\lambda} = \frac{C(u) m^2}{\lambda}
      + \rho_i^2 (u) - C(u)  f_\pi^2
      + N \int_{0}^{s_m} \, \frac{s^2 ds}{2{\pi^2}}
      \left[  (2n_s+1)  {|g_s(u)|}^2 -
      \frac{1}{2{(s^2 + {\rm e}^{2u})}^{1/2}} \right]
\>.
\label{eq:reggap}
\end{equation}
Note that the second integral is {\em independent} of time.  Once we
have removed the quadratic divergences in the gap equation, we are
only left with logarithmic divergences, which are subtracted by the
following coupling constant renormalizations
\begin{mathletters}
\label{eq:coupren}
   \begin{equation}
      \frac{1}{\lambda} = \frac{1}{\lambda_{\tiny R}}
         - \frac{N}{4} \int_0^{s_m} \, \frac{ s^2 d s }{ 2 \pi^2 }
         \frac{1}{(s^2 + {\rm e}^{2u})^{3/2}} ,
   \label{eq:lambdaren}
   \end{equation}
   \begin{equation}
      \left( \xi - {1 \over 6} \right) {1 \over \lambda}
         = \left( \xi_{\tiny R}
         - {1 \over 6} \right) {1 \over \lambda_{\tiny R}}
\>.
   \label{eq:xiren}
   \end{equation}
\end{mathletters}
Note that $\xi = 1/6 = \xi_{\tiny R}$ is a fixed point
of the renormalization flow equations \cite{ref:nonminimal}.

The renormalized form of the gap equation is then given by the
following
\begin{eqnarray}
   \frac{\tilde \chi(u)}{\lambda_{\tiny R}}
      & = &
      - {\tilde v}^2_{\tiny R}
      - \frac{NC(u)m^2}{16 \pi^2} + \rho_i^2
      + N \int_0^{s_m} \,
      \frac{ s^2 d s }{ 2 \pi^2 } (2 n_s +1) {|g_s(u)|}^2
   \nonumber \\ &&
      - \frac{NC(u)}{2} \int_0^{\Lambda} \,
      \frac{ k^2 d k }{ 2 \pi^2 } \frac{1}{{(k^2 + m^2)}^{1/2}}
      - \frac{N[\tilde \chi(u) - C(u)m^2]}{4} \int_0^{\Lambda} \,
      \frac{ k^2 d k }{ 2 \pi^2 } \frac{1}{{(k^2 + m^2)}^{3/2}}
\>.
\end{eqnarray}

We now proceed to write the regularized expressions for the energy and
isotropic pressure and analyze the divergences appearing in the mode
integrals.  We shall take $\xi_{\tiny R} = 1/6$ as suggested by
\cite{ref:nonminimal}. The physics behind this choice is clear.  If
one considers an arbitrary composite scalar field (such as the meson
field considered here) and an effective field theory at a scale
$\Lambda$, and one carries out a renormalization group analysis in
the leading large-$N$ approximation (or a fully improved one-loop
renormalization group approximation), $\xi = 1/6 = \xi_{\tiny R}$ is
found to be an attractive renormalization group fixed point in the
infrared limit \cite{ref:nonminimal}.  This means that even if there are
corrections to $\xi = 1/6$ at large scales (of order $\Lambda$), the
observed low energy value of the coupling $\xi$ tends to a physical
value of $\xi_{\tiny R} = 1/6$, as one evolves into the infrared.  The
choice $\xi = 1/6 = \xi_{\tiny R}$ also implies that the divergences
of both the energy and the pressure can be obtained at adiabatic order
zero \cite{ref:bunch}.

Now we examine the divergences present in the mode integrals of
the energy density and pressure. Recall that
\begin{eqnarray}
   {\cal E}(u) & \equiv &   C^2(u) \epsilon(u)
      \equiv  \frac{1}{2} \left[ \dot\rho_i^2
      + \rho_i^2 (\tilde\chi - 1) \right]
      - \tilde{j}_i \rho_i
      - \frac{\tilde\chi \tilde{v}^2}{2}
      - \frac{\tilde\chi^2}{4 \lambda}
   \nonumber \\ &&
      + \frac{N}{2} \int_0^{s_m} \frac{ s^2 d s }{ 2 \pi^2 }
      ( 2 n_s + 1 ) \{
      \left| \dot{g_s} \right|^2
      + ( s^2 + \tilde\chi ) \, | g_s |^2 \}
\label{eq:regepsu}
\>,
\end{eqnarray}
and
\begin{eqnarray}
   {\cal P}(u) & \equiv & C^2(u) p(u)
      \equiv  \frac{1}{6} \left[ \dot\rho_i^2
      - \rho_i^2 (\tilde\chi + 1) \right]
      + \frac{2}{3} \tilde{j}_i \rho_i
      + \frac{\tilde\chi \tilde{v}^2}{2}
      + \frac{\tilde\chi^2}{4 \lambda}
  \nonumber \\ &&
      + \frac{N}{6} \int_0^{s_m} \frac{ s^2 d s }{ 2 \pi^2 }
      ( 2 n_s + 1 ) \{
      \left| \dot{g_s} \right|^2
      + ( s^2 - \tilde\chi ) \, | g_s |^2 \}
  \label{eq:regpu}
\>.
\end{eqnarray}
For the energy the contribution coming from the mode integrals is
\begin{equation}
   \frac{N}{2} \int_0^{s_m} \frac{ s^2 d s }{ 2 \pi^2 }
      ( 2 n_s + 1 ) \left\{
      \left| \dot{g}_s \right|^2 +
      ( s^2 + \tilde\chi ) \, | g_s |^2 \right\}
\>,
\end{equation}
and using the adiabatic expansion given in Eq.~(\ref{alladiabatic}),
the divergent part of this integral is given by
\begin{eqnarray}
   C^2(u) \epsilon_0 & \equiv &
      \frac{N}{2} \int_0^{s_m} \frac{s^2 d s}{2 \pi^2} \, \omega_s =
      \frac{N}{2} \int_0^{s_m} \frac{s^2 d s}{2 \pi^2} \,
      \sqrt{s^2 + \tilde\chi}
   \nonumber \\ &=&
      \frac{N s_m^4}{16\pi^2} + \frac{N s_m^2 \tilde\chi}{16 \pi^2}
      - \frac{N \tilde\chi^2}{64 \pi^2} \ln
      \left( \frac{4s_m^2}{\tilde\chi} \right)
      + \frac{N \tilde\chi^2}{128 \pi^2} + {\cal O}
      \left(\frac{\tilde\chi^3}{s_m^2}\right)
\>.
\label{eq:edivergent}
\end{eqnarray}
For the pressure the contribution coming from the mode integrals is
\begin{equation}
   \frac{N}{6} \int_0^{s_m} \frac{ s^2 d s }{ 2 \pi^2 }
      ( 2 n_s + 1 ) \left\{
      \left| \dot{g}_s \right|^2 +
      (s^2 - \tilde\chi) \, | g_s |^2 \right\}
\>.
\end{equation}
Again using the adiabatic mode expansion, the divergent part of this
integral is given by
\begin{eqnarray}
   C^2(u) p_0 & \equiv &
      \frac{N}{6} \int_0^{s_m} \frac{ s^4 d s }{ 2 \pi^2 }
      \frac{1}{\sqrt{s^2 + \tilde\chi}}
   \nonumber\\
   &=& \frac{N s_m^4}{48 \pi^2} - \frac{N s_m^2 \tilde\chi}{48 \pi^2}
      + \frac{N \tilde\chi^2}{64 \pi^2}
      \ln \left( \frac{4s_m^2}{\tilde\chi} \right)
      - \frac{7N \tilde\chi^2}{384 \pi^2}
      + {\cal O} \left(\frac{\tilde\chi^3}{s_m^2}\right)
\>.
\label{eq:pdivergent}
\end{eqnarray}

Notice that $\epsilon_0 \ne - p_0$, as is required by general
covariance (see Appendix {\ref{app:christensen}}).  We must then
enforce covariance {\em by hand}.  We mention here that the
introduction of a momentum cutoff does not spoil covariance at the
level of the energy density (see Appendix {\ref{app:christensen}}). On
the other hand, we must carefully handle the subtractions that will
lead to the finite isotropic pressure \cite{ref:phi4}.  The quartic
subtraction in the energy density and pressure is a renormalization of
the cosmological constant; in other words, a subtraction of the
cosmological vacuum energy ${N \Lambda^4} / {16 \pi^2}$.

If we now define
\begin{mathletters}
\label{eq:e-prenormalized}
   \begin{equation}
      \epsilon_{\tiny R} = \epsilon
         - \frac{N s_m^4}{16\pi^2 C^2(u)},
      \label{eq:eren}
   \end{equation}
   \begin{equation}
      p_{\tiny R} = p - p_0 - \epsilon_0
      + \frac{N s_m^4}{16\pi^2 C^2(u)}
      \label{eq:pren}
\>,
   \end{equation}
\end{mathletters}
we can easily show that the energy density and pressure are finite, by
making use of Eqs.~(\ref{eq:regvacgap}) and (\ref{eq:coupren}).

In order to numerically evaluate the physical energy density and
pressure, we must also make sure that the vacuum energy that we are
measuring with respect to is zero ({\em i.e.} when we are at the
minimum of the potential). This vacuum energy is calculated in the
``out'' regime of the collision, when $\langle \sigma \rangle =
f_\pi$, $\langle \vec\pi_i \rangle = 0$, $\chi = m_\pi^2$, and the
mode functions are the zeroth order adiabatic ones with $\omega_s^2 =
s^2 + C(u)m^2$.  The renormalized vacuum energy is given by
\begin{equation}
   \epsilon_{\tiny {\rm vac}} =  \frac{m^2 f_\pi^2}{2}
      - \frac{m^2 v^2}{2} - \frac{m^4}{4\lambda}
      - \frac{N \Lambda^4}{16 \pi^2}
      + \frac{N}{2 C^2} \int_0^{s_m} \, \frac{s^2 ds}{2\pi^2}
      \sqrt{s^2 + m^2 C}
\>.
\end{equation}
The vacuum value of the pressure is $p_{\tiny {\rm vac}} = -
\epsilon_{\tiny {\rm vac}}$. These values must be subtracted from
Eqs.~(\ref{eq:e-prenormalized}) so that when we have reached the
``out'' regime, the energy density and pressure go to zero.

We calculate now the trace of the energy-momentum tensor, and show
that these definitions of $\epsilon_{\tiny R}$ and
$p_{\tiny R}$ give a finite trace. We have from its definition
\begin{equation}
   - \langle T_{\mu}^{\mu} \rangle_{\tiny R }
      = 3 p_{\tiny R} - \epsilon_{\tiny R}
\>.
\label{eq:e-mtrace}
\end{equation}
If we make use of Eqs.~(\ref{eq:e-prenormalized}), we can write
\begin{equation}
   - C^2(u) \, \langle T_{\mu}^{\mu} \rangle_{\tiny R} =
      \tilde{v}_{\tiny R}^2 \tilde{\chi}(u)
      + 3 \tilde{j}_i \rho_i(u)
      + \frac{N \tilde{\chi}^2(u)}{32 \pi^2}
\>,
\label{eq:trace}
\end{equation}
where we have defined
\begin{equation}
   \tilde{v}^2 =
      - \frac{m^2 C(u)}{\lambda_{\tiny R}} + f_\pi^2 C(u)
      -\frac{N m^2 C(u)}{16 \pi^2}
      + \frac{N s_m^2}{8 \pi^2} \equiv
      \tilde{v}_R^2 + \frac{N s_m^2}{8 \pi^2}
\>.
\label{eq:v2r}
\end{equation}

The first two terms in Eq.~(\ref{eq:trace}) are the renormalized
classical trace of the energy-momentum tensor, and the second term is
the one-loop quantum trace anomaly.

\section{Numerical Results}
\label{sec:results}
\subsection{Energy-Momentum Tensor Conservation}
\label{sec:engcons}
The renormalized energy-momentum tensor obeys the conservation law
\begin{equation}
   \nabla_{\mu} {\langle T^{\mu\nu} \rangle}_{\tiny R}  = 0
\>.
\label{eq:dmutmunu}
\end{equation}

Using the Christoffel symbols tabulated in Appendix~\ref{app:chris},
we find that the $\nu = u$ component of the conservation equation
takes the form
\begin{equation}
   \dot\epsilon_{\tiny R}(u) + 3 [p_{\tiny R}(u)
      + \epsilon_{\tiny R}(u)] = 0
\>.
\label{eq:uecons}
\end{equation}

In terms of the variables
\begin{mathletters}
\label{eq:e-pcal}
   \begin{equation}
      {\cal E}_{\tiny R} = {\cal E}
         - \frac{N s_m^4}{16\pi^2},
      \label{eq:ecalren}
   \end{equation}
   \begin{equation}
      {\cal P}_{\tiny R} = {\cal P} - C^2(u) ( p_0 + \epsilon_0)
      + \frac{N s_m^4}{16\pi^2}
      \label{eq:pcalren}
\>,
   \end{equation}
\end{mathletters}
we can rewrite the conservation equation as
\begin{equation}
   \dot{\cal E}_{\tiny R}(u) + 3 {\cal P}_{\tiny R}(u)
      - {\cal E}_{\tiny R}(u) = 0
\>.
\end{equation}

Recall that
\begin{eqnarray}
   {\cal E}_{\tiny R} & = & \frac{1}{2} \left[ \dot\rho_i^2
      + \rho_i^2 (\tilde\chi - 1) \right]
      - \tilde{j}_i \rho_i
      - \frac{\tilde\chi \tilde{v}^2}{2}
      - \frac{\tilde\chi^2}{4 \lambda}
   \nonumber \\ &&
      + \frac{N}{2} \int_0^{s_m} \frac{ s^2 d s }{ 2 \pi^2 }
        ( 2 n_s + 1 ) \{
      \left| \dot{g_s} \right|^2
      + ( s^2 + \tilde\chi ) \, | g_s |^2 \}
      - \frac{N s_m^4}{16\pi^2}
\>.
\end{eqnarray}
Therefore
\begin{eqnarray}
   \dot{\cal E}_{\tiny R}
      &=& \dot\rho_i \ddot \rho_i
      + (\tilde\chi - 1)\rho_i \dot\rho_i
      + \frac{1}{2} \dot{\tilde\chi} \rho_i^2
      - \frac{\dot{\tilde\chi} \tilde{v}^2}{2}
      - \tilde\chi \tilde{v} \dot{\tilde{v}}
      - \frac{\tilde\chi \dot{\tilde\chi}}{2\lambda}
      - (\dot{\tilde{j}}_i \rho_i + \tilde{j} \dot\rho_i)
   \nonumber \\
   &+& \frac{N}{2} \int_0^{s_m} \, \frac{s^2 ds}{2\pi^2}
      (2 n_s + 1) \left\{ \dot g_s \ddot g_s^\ast + \ddot g_s \dot g_s^\ast
      + (s^2 + \tilde\chi) (g_s \dot g_s^\ast + \dot g_s g_s^\ast)
      + \dot{\tilde\chi} |g_s|^2 \right\}
   \nonumber \\
   &+& \frac{N}{2} \frac{s_m^3}{2\pi^2}
      (2 n_{s_m} + 1) \left\{ |\dot g_{s_m}|^2
         + (s_m^2 + \tilde\chi)|g_{s_m}|^2 \right\}
         - \frac{N s_m^4}{4\pi^2}
\>,
\label{eq:erdot}
\end{eqnarray}
where we have taken into account the contribution coming from the
upper limit of the mode integral, since $s_m$ depends on $u$.  Making
use of the identities ${\dot{\tilde{v}}}_i = {\tilde{v}}_i$,
${\dot{\tilde{j}}}_i = 3{\tilde{j}}_i$, the equations of motion, and
using the adiabatic approximation for $g_{s_m}$ (we can always choose
$s_m$ to be a high enough comoving momentum so that the adiabatic
approximation is valid in this limit), we then obtain
\begin{equation}
   \dot{\cal E}_{\tiny R} =
      - \tilde\chi \tilde{v}^2
      - 3 \tilde{j}_i \rho_i
      + \frac{N}{4\pi^2} s_m^3 (2 n_{s_m} + 1) \omega_{s_m}
      - \frac{N s_m^4}{4\pi^2}
\>.
\end{equation}
The occupation number $n_{s_m}$ goes to zero for large $s_m$, so we can
neglect it. Expanding out $\omega_{s_m}$ yields
\begin{equation}
   \omega_{s_m} = \sqrt{s_m^2 + \tilde\chi} = s_m
      \left(1 + \frac{\tilde\chi}{2 s_m^2}
      - \frac{\tilde\chi^2}{8 s_m^4} + \cdots \right)
\>.
\end{equation}
So that we have
\begin{equation}
   \dot {\cal E}_{\tiny R} =
      - \tilde\chi \tilde{v}^2 + \frac{N \tilde\chi s_m^2}{8 \pi^2}
	  - 3 \tilde{j}_i \rho_i
	  - \frac{N \tilde\chi^2}{32 \pi^2}
\>,
\end{equation}
or equivalently
\begin{equation}
   \dot {\cal E}_{\tiny R} =
      - \tilde\chi \tilde{v}_{\tiny R}^2
      - 3 \tilde{j}_i \rho_i
      - \frac{N \tilde{\chi}^2}{32 \pi^2}
\>.
\end{equation}
If we compare this expression with Eq.~({\ref{eq:trace}}), we can
easily see that the conservation equation is satisfied.

\subsection{Effective Equation of State}
\label{sec:eos}

Now that we have finite equations for the energy density and pressure,
we can evolve them in proper time and numerically investigate their
behavior.  Since we have a nonequilibrium situation, it does not
really make sense to calculate an actual equation of state, {\em i.e.}
$p = p(\epsilon)$. However, since $\epsilon$ and $p$ are both
functions of $u$, in regions where both are monotonically increasing
or decreasing, one can determine an ``effective'' equation of state $p
= p(\epsilon)$. In this calculation, we do not have two-body
scattering, so that there are many oscillations in the pressure. We
expect at the next order in the $1/N$ expansion, when these effects
are included that one will be able to extract an effective equation of
state.

\subsection{Numerical Simulations}
We choose the initial state at a temperature above the phase
transition in thermal equilibrium, with all particle masses
positive. The equations are solved self-consistently at the starting
time to obtain the values of the mean fields. We fixed the value of
$\chi$ at the initial time as the solution of the gap equation in the
initial thermal state. We also required that the initial expectation
values of the $\sigma$ and $\vec\pi$ fields satisfy ${\vec \pi}^2(u_0) +
\sigma^2(u_0) = \sigma^2_{\tiny T}$, where $\sigma_{\tiny T}$ is the
thermal equilibrium value of $\Phi$ at the initial temperature
$T$. The critical temperature is $T_c \approx$ 160 MeV, so we choose
$T =$ 200 MeV, which gives $\sigma_{\tiny T} = 0.3$ fm $^{-1}$. The
mode functions are chosen as their adiabatic values [see
Eq.~(\ref{eq:initmodes})]. We choose the coupling constant
$\lambda_{\tiny R} = 7.3$ \cite{ref:mypaper}, and take the conformal
value of the coupling to gravity $\xi = \xi_{\tiny R} = 1/6$
\cite{ref:nonminimal}.

In Fig.~\ref{fig:chi}, we show the time evolution of the $\chi$ field,
which is the effective mass squared for the $\phi$ field. When this
field becomes negative, long-wavelength unstable modes begin to grow
exponentially. Therefore this field serves as a measure of instability
in the system. For initial conditions with a negative derivative, we
find that the $\chi$ field is negative for about 3 fm/$c$ of proper
time.  Figure \ref{fig:eandp0} shows the numerical calculation of the
energy density and pressure, in units of fm$^{-4}$, for initial
conditions where no instabilities arose. Note that the system quickly
reaches the ``out'' regime and relaxes to its vacuum values.  Figure
\ref{fig:eandpneg1} shows the same plot for initial conditions with an
instability. For the unstable initial conditions more energy density
was produced initially, but both the energy density and the pressure
drop off very quickly, as is expected (since the system is cooling
very rapidly). It is interesting to note that the pressure becomes
negative when $\chi$ is negative. If we define the speed of sound
$c_0^2(\tau)$ by $p = c_0^2(\tau) \epsilon$, we notice that $c_0^2 \leq 1$.

\begin{figure}
\epsfxsize = 4.0in
\centerline{\epsfbox{chi.eps}}
\caption{Proper time evolution of the $\chi$ field
for the following initial conditions.
Solid line: $\sigma(\tau_0) = \sigma_{\tiny T}$,
$\pi_i(\tau_0) = 0$, and $\dot\sigma(\tau_0) = 0$.
Dashed line: $\sigma(\tau_0) = \sigma_{\tiny T}$,
$\pi_i(\tau_0) = 0$, and $\dot\sigma(\tau_0) = -1$ }
\label{fig:chi}
\end{figure}

\begin{figure}
\epsfxsize = 4.0in
\centerline{\epsfbox{eandp0.eps}}
\caption{Proper time evolution of the energy density and pressure for
the initial conditions $\sigma(\tau_0) = \sigma_{\tiny T}$,
$\pi_i(\tau_0) = 0$, and $\dot\sigma(\tau_0) = 0$.}
\label{fig:eandp0}
\end{figure}

\begin{figure}
\epsfxsize = 4.0in
\centerline{\epsfbox{eandpneg1.eps}}
\caption{Proper time evolution of the energy density and pressure for
the initial conditions $\sigma(\tau_0) = \sigma_{\tiny T}$,
$\pi_i(\tau_0) = 0$, and $\dot\sigma(\tau_0) = -1$.}
\label{fig:eandpneg1}
\end{figure}

Figure \ref{fig:cons0} shows the conservation of the energy-momentum
tensor. Notice that it
is not conserved for short proper times. Because of the
presence of the Landau pole it is not possible to take the physical
cutoff $\Lambda$ very large (we use 800 MeV). Due to this rather small
cutoff, the occupation number $n_{s_m}$ is not actually zero, which it
was assumed to be for the derivation of the energy-momentum
 conservation [see the
discussion after Eq.~(\ref{eq:erdot})]. As soon as the cutoff $s_m$
becomes large enough, then the occupation number goes to zero and
the energy-momentum tensor is conserved. Figure \ref{fig:consneg1} shows
the energy-momentum tensor
conservation for initial conditions with an instability. We see a
qualitatively similar behavior, regardless of initial conditions.

\begin{figure}
\epsfxsize = 4.0in
\centerline{\epsfbox{cons0.eps}}
\caption{Energy-momentum tensor conservation, for the initial conditions
$\sigma(\tau_0) = \sigma_{\tiny T}$, $\pi_i(\tau_0) = 0$, and
$\dot\sigma(\tau_0) = 0$.}
\label{fig:cons0}
\end{figure}

\begin{figure}
\epsfxsize = 4.0in
\centerline{\epsfbox{consneg1.eps}}
\caption{Energy-momentum tensor conservation, for the initial conditions
$\sigma(\tau_0) = \sigma_{\tiny T}$, $\pi_i(\tau_0) = 0$, and
$\dot\sigma(\tau_0) = -1$.}
\label{fig:consneg1}
\end{figure}

\section{Conclusions}
\label{sec:conc}
We have shown how to regularize and renormalize the energy-momentum
tensor for a spherically symmetric model with a time-dependent
comoving momentum cutoff which breaks covariance. We computed the
finite energy density and pressure, and numerically examined the
equation of state and proved energy conservation for this system. We
found that the energy density and pressure both decrease quickly,
which is expected for a rapidly cooling system. An interesting feature
of the expansion is that when the effective mass squared is negative,
the pressure is also negative. A negative pressure implies cavitation,
which could mean the formation of domains, {\em i.e.} DCCs.

\section*{Acknowledgements}
The authors would like to thank Yuval Kluger, Emil Mottola, and John
Dawson for useful discussions; and give special thanks to Fred Cooper
and Salman Habib for their contributions. UNH gratefully acknowledges
support by the U.S. Department of Energy (DE-FG02-88ER40410). One of
us (MAL) would like to thank Los Alamos National Laboratory for its
hospitality.

redpajama_set_name: RedPajamaArXiv


This week's magical math book is Everyone Can Learn Math by Alice Aspinall. Julie Homenuik, an English teacher from Windsor, Ontario and Alice's colleague wrote about her experience reading this book to her young daughter.
Inspired by the first line of Julie's post (In the world of math, nothing matters more than how we've been taught to feel about math), I wrote Everyone Can Learn Math Part 1 outlining my current thinking about the importance of developing a productive disposition in math class and home. Yesterday mathematics educator and parent David Butler wrote a related thread on twitter. In the thread (click tweet for more), David thoughtfully outlines some possible causes for why his daughter's disposition towards mathematics shifted drastically in a short 8 weeks.
My 10-year-old daughter was rolled in a ball crying yesterday, because of her maths homework. Maths class should not make people feel this way.
This thread is heartbreaking. No child should feel like this. "Math(s) class should not make people feel like this"!
"Well, that is the kind of math smarts school values but there are other kinds of math smarts." (Ilana Horn, tweet from January 2019).
But is is only a beginning, there is much more to figure out regarding supporting children as they struggle with their identities in math classes.
This week's book offers another way for parents and teachers to support young children in developing a productive disposition towards math. A disposition where they see themselves an effective learner and doer of mathematics. Below Julie shares a bit about this book. Thanks to her and her daughter for sharing their #mathbookmagic!
Alice Aspinall is a passionate high school mathematics teacher in Windsor, Ontario. Her book, Everyone Can Learn Math , is born out of her passion for not just math, but a growth mindset.
Published by Friesen Press in fall 2018, this inspiring story follows Amy, a young math student who observes her own achievements and the world around her, and finally makes connections to her personal struggle with a word problem. If others can overcome, if she herself can overcome in dance, then she can do the same in math. She finally recognizes that her ability to tackle this problem has everything to do with her own attitude and perseverance. She can do it; she just has to figure out how.
Alexandria Masse illustrates the book in pastel watercolors that move the reader through Amy's struggles and realizations.
The book is available in hard and soft cover as well as in e-book format at Friesen Press, Amazon, and Barnes and Noble.
The math problem that sets the story in motion.
The fundamental conflict in the book is initially set out as character versus math problem: a word problem in which Amy needs to figure out how many candies each of her friends will receive if she has six friends and three packages of candy with eight candies in each. Truly, though, the problem is character versus self, where Amy needs to resolve her own attitude toward the problem, an issue that is as fundamental to math as the math itself. This book is not just teaching math, but teaching an approach to math. Aspinall recognizes how much our attitudes and preconceptions about mathematical ability impact our students' experience with math. She didn't set out to write a book that incorporated mathematical elements; she wanted to write a book that directly addressed how we can tackle the negative stigma surrounding math, especially for young girls like Amy.
Aspinall builds even more math into other aspects of the story. Take Amy's classroom scenes, for example: The walls of the classroom feature a breakdown of mean, median, and mode, offering up another opportunity to talk math. The other half of the bulletin board displays an array.
Seeing math in all we do.
The front cover provides opportunities for number recognition, and the illustrations within offer opportunities to talk about how we use math in everyday activities—from the shape of our bodies when we dance and the shapes that make up monkey bars, to even the more complex math of sinking a basketball in a free throw. The various situations Amy encounters promotes math talk on many levels.
In fact, the story itself follows a mathematical structure—it is a pattern in which Amy must recognize the constant: perseverance. She perseveres and succeeds. Her peers persevere and succeed. She finally takes this equation—determination plus hard work—and makes it equal success for her in math too.
Of course, as a parent of two young children, Aspinall knows the secret to a child's heart: candy. Combining math and candy creates immediate magic for young learners. My daughter, who is four, already loves math, but when I offered the opportunity to work with math and candy together, her eyes lit up and she couldn't contain her excitement at reorganizing the groups of candy that she would ultimately get to eat. I used three different colors of chocolate eggs for our exploration of the word problem, and although we got stuck a few times on logistics—"Well, there are only three colors" and "Well, there are only four people in our house"—we worked through it. The final result? Six groups of four, a happy dance, and a chocolate feast.
The book is not just magic with the candy, however. After the first time I read the book to my daughter, she said, "I love it. Can we read it again?" An immediate repeat read is a magical book indeed.
Follow Everyone Can Learn Math on Instagram, Facebook, and Twitter.


I enjoyed the general information about climate change and how the planet works to bring itself into equilibrium and the climate systems. The units involving scientific and mathematical formulas were a bit beyond my comprehension.
I wanted information on what is happening in climate change, what we can do locally and what we should be doing globally. This was all covered plus information about the planet and climate systems.
This unit is an introduction to Climate Change. It looks at some of the controversy it has caused and introduces the science and observations which led to the realization that our world is warming and that this is probably caused at least in part by human greenhouse gas emissions. It briefly examines the reaction to this recognition and what is being done on the global stage to mitigate the problem.
This unit looks at meteorology and the distinction between weather and climate. It examines the main drivers of the Earth's weather systems and the way they are influenced by the heat from the Sun. It examines how weather changes throughout the year and depends on location on the Earth's sphere. It also looks at the extreme weather events which affect human activities and examines their cost to society.
Here we examine how fortunate the Earth is in terms of its environment and how this is related to climate. We look at the balance between the incident solar radiation and outgoing infrared energy and relate how the complex of interrelated networks of the Earth's climate system may be viewed in terms of system dynamics.
What was the ancient climate like and how does it compare with current times? What caused changes in past climates and how do we determine their characteristics? Let us find out.
This unit examines the flow of energy to and from the Earth on a global scale. How is the shortwave (visible) energy coming from the Sun balanced by the outflow of infrared energy emanating from the Earth? How is this energy balance affected by greenhouse gases in the atmosphere? How sensitive is the Earth's climate to these energy changes; what global temperature increase or decrease would result from a particular shift in the energy balance?
There are a number of cycles taking place between the atmosphere, biosphere, hydrosphere and geosphere which are critical to Earth's ability to sustain life. The natural carbon cycle exchanges carbon between these four spheres which all contain reservoirs of carbon. The water cycle describes the continuous movement of water above, below and on the Earth's surface. The nitrogen cycle is the process whereby nitrogen is converted between its various chemical forms in the atmosphere and on the surface. The carbonate-silicate cycle exchanges silicates to carbon during weathering and back to silicates during the subduction process and subsequently ejects it in volcanic eruptions.
In the same way that it is not possible to investigate the internal mechanisms of stars or nuclear explosions in a test room, a planet's climate cannot be directly simulated in a laboratory. Crucial tools to investigate these phenomena are computer models. These are built using mathematical equations of physics and Earth processes and applying an initial set of conditions to obtain a result of these processes expressed in a computer model. The model may then be refined in an iterative process by comparing the model output to observed values in the real world. Climate modelling allows scientists to answer "what-if" questions and investigate possible future scenarios.
What are the many impacts of a rise in average temperature of several degrees? Will there be a chain reaction with rising sea levels, ocean acidification, more severe weather systems and changing rain patterns affecting agriculture and biodiversity?
Adaptation seeks to manage the impacts of global warming and reduce the vulnerability of biological and infrastructure impacts such as a policy to build structures on higher ground to circumvent the rise in sea levels.
Mitigation deals with the causes rather than the consequences of climate change. What can be done to minimise the amount of global warming by reducing the emission of greenhouse gasses and enhancing the ability of the Earth to absorb carbon dioxide.
In this final unit, we look at the state of the climate. What is being done in Australia and around the world to combat climate change? What do climate models predict for future years? What can be done if climate change gets "ugly"? Is "climate change" a reality or just a massive hoax?


Crista Johnson, MD, MSc, FACOG, is an Obstetrician/Gynecologist at Maricopa Integrated Health System, Phoenix, AZ, where she is Director of the Refugee Women's Health Clinic. She is also a Clinical Research Affiliate of the Southwest Interdisciplinary Research Center (SIRC), which is a NIH-funded National Center of Excellence in minority health and health disparities at Arizona State University. She received her undergraduate degree from The Johns Hopkins University, medical degree from the Weill Medical College of Cornell University, and completed her residency in Obstetrics & Gynecology at the George Washington University Medical Center. She subsequently completed a fellowship in Female Sexual Medicine at the University of California, Los Angeles and then became a Robert Wood Johnson Foundation Clinical Scholar at the University of Michigan where she obtained her Masters in Health and Health Care Research examining disparities in reproductive health care among refugees/immigrants through mixed-method Community-Based Participatory Research. She has presented nationally and internationally on the challenges faced by health care providers in the care of women with Female Genital Cutting (FGC). Dr. Johnson's current research focuses on investigating strategies to improve health outcomes and psychosexual sequelae for women, particularly those who have undergone FGC; with the aim of improving health care access and utilization, reproductive health education, counseling, community engagement, as well as health care provider cultural competency.


Lighthouses, these beacons of light have been guiding ships since the 17th century and can be something beauty beyond your imagination. Not surprisingly, lighthouses are most spectacular at night, when their lights beam dutifully for miles out to sea. Here we have 20 Astonishing Lighthouse Photography and hope you like them.


text:
New Suicide Squad Promos Feature New Footage of the Worst Heroes Ever
Zain Charkawi
Saturday, 16 July 2016 - 4:26PM
Suicide Squad is now less than two weeks away from premiering in the U.S., and judging by the projected box office earnings for the opening weekend, it's going to be huge (that is, if it happens to please critics and get good word of mouth from viewers). There's rumored to be one last trailer for the film which will likely debut during Warner Bros. Hall H presentation at San Diego Comic Con. Until then, the marketing team has been putting out a steady stream of TV spots, promo images, and interviews.
Most recently, these brand new promo clips were released, each focusing specifically on a particular character from the movie. Each of them runs around twenty seconds or so, but each of them also seem to contain a tad more footage of the characters. Though it isn't much, viewers really don't need much this close to the release date, and each of the promos still manage to be exciting despite the fact that they aren't jam-packed with fresh footage of the film.
Each of the clips are cut to Sweet's Ballroom Blitz, which was featured in the second trailer for the upcoming film (not counting the Comic Con first look). They also feature the weird and morbid cartoony animations that have been utilized in some of the promotional images and posters for the film.
There are a few interesting moments particularly worthy of note in the clips, such as the bit with Enchantress in a different costume. Could that be a scene of when June Moone becomes possessed? Or is it of the villain achieving one of her dastardly goals, as many have speculated she may be the antagonist of the film?
Hopefully Suicide Squad will be every bit as good as everyone's hoping and brings about a major course correction for the DCEU after the limited success of Batman v Superman.
Suicide Squad premieres on August 5th, 2016.
Joss Whedon's 'Justice League' Reshoots Made Cyborg's Story Less Grim
'Aquaman' Director James Wan Says Jason Momoa Will Make Aquaman Cool Again
'Justice League' Image Shows Off an Enormous New Batplane
'Batman' Voice Actor Kevin Conroy Performs 'The Dark Knight' Monologue
'Shazam!' Director Claims It Will Be a Fun, Lighthearted DCEU Movie
Warner Bros. is Planning a Major Oscars Campaign for 'Wonder Woman'

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Seed fund for African digital start-ups
It's a big boost for small businesses and entrepreneurs. To help support and finance African start-ups, AFD is launching a new €15 million seed fund. It's part of a growing commitment to expanding the continent's digital ecosystem.
Research document
Inequality trends in South Africa: A multidimensional diagnostic of inequality
This report on inequality trends in South Africa is a joint publication produced by Statistics South Africa (Stats SA) in partnership with the Southern Africa Labour and Development Research Unit (SALDRU) based in the School of Economics at the University of Cape Town and the Agence Francaise de
An Application of the Alkire-Foster's Multidimensional Poverty Index to Data from Madagascar: Taking Into Account the Dimensions of Employment and Gender Inequality
generally the sole means of production owned by poor or deprived people. Another shortcoming of the MPI approach is that it does not enable gender inequalities analysis. This is surprising since two out of the three dimensions of the classical MPI are individual attributes. In this study, we also provide a
Published in April 2017
Migration patterns and labor market outcomes in Tunisia
the role of emigration as a security valve for the Tunisian labor market. Moreover, origin households of migrants have a significantly higher wealth index. Remittances play a significant role for the Tunisian economy and at the household level. Our analysis also tends to confirm the effects of
Mining in Africa : Are Local Communities Better Off?
This study focuses on the local and regional impact of large-scale gold mining in Africa in the context of a mineral boom in the region since 2000. It contributes to filling a gap in the literature on the welfare effects of mineral resources, which, until now, has concentrated more on the national
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Reaping Richer Returns: Public Spending Priorities for African Agriculture Productivity Growth
. Choosing to catalyze a shift toward more effective, efficient, and climate-resilient public spending in agriculture can accelerate change and unleash growth. Not only does agricultural public spending in Sub-Saharan Africa lag behind other developing regions but its impact is vitiated by subsidy
Africa's Demographic Transition : Dividend or Disaster?
This book lays out a range of policy actions that are needed at the various phases of the demographic transition and uses global and regional experiences to provide evidence on what has worked and what has not. Countries have a menu of options available to speed up the transition, improve
Nigeria: The Restrained Ambitions of Africa's Largest Economy
the southern part is primarily Christian. Along with such a heterogeneous mix of ethnicities, and a tenuous balance of power, is a wealth of energy resources that are unequally distributed in favour of a few in the south west of the country.
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Highways to Success or Byways to Waste : Estimating the Economic Benefits of Roads in Africa
transportation remains a key strategy for development agencies. Roughly $6.8 billion per year is spent in Sub-Saharan Africa on paving roads, and the World Bank invests more on roads than on education, health, and social services combined. Despite the large sums spent on transportation, there have been no
Youth Employment in Sub-Saharan Africa
Sub-Saharan Africa has just experienced one of the best decades of growth since the 1960s. Between 2000 and 2012, gross domestic product (GDP) grew more than 4.5 percent a year on average, compared to around 2 percent in the prior 20 years (World Bank various years). In 2012, the region's GDP
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Q: How to define a list which evaluates to the sequence [(0,0), (1,0), (2,0), (3,0), etc.] and [(0,0), (0,1), (0,2), (0,3), etc.] I am a beginner in Haskell and i was wondering how can you define two lists one of which which evaluates to the sequence [(0,0), (1,0), (2,0), (3,0), etc.] and the other one to  [(0,0), (0,1), (0,2), (0,3), etc.]? I had multiple tries but every time it just prints out the first sequence and doesn't print the second one. If someone could show me how that would be great. Thanks

A: You can define an infinite list of zeros with:
[0, 0 ..]

and you can define an infinite list of integers starting at 0 in a similar way:
[0, 1 ..]

The function zip :: [a] -> [b] -> [(a, b)] takes two lists and 'zips' them into a list of tuples. So if we combine zip with the above list notation, we can achieve what you ask:
zip [0, 0 ..] [0, 1 ..]

You can try this in ghci:
> take 4 $ zip [0, 0 ..] [0, 1 ..]
[(0, 0), (0, 1), (0, 2), (0, 3)]
> take 4 $ zip [0, 1 ..] [0, 0 ..]
[(0, 0), (1, 0), (2, 0), (3, 0)]

using list-comprehensions
[ (i,0) | i <- [0..] ]

and
[ (0,i) | i <- [0..] ]

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Have you built your website with Jimdo and now do you want to measure your internet audience?
With just three steps you can insert web analytics ShinyStat code in your websites created with Jimdo, a system that allows you to create websites quickly and easily.
This procedure is valid for sites created with JimdoFree, JimdoPro or JimdoBusiness.
Jimdo is an easy-to-use website creator empowering ordinary people to build their own elegant and custom websites. In just a few minutes and without any technical knowledge, everyone can create a place on the Internet – complete with online store, blog, photo galleries, YouTube videos and much more.


Rumble and ramble in blackberry jamble. Yes, we LOVE our Jamberry book. This has been one of the boys' favorites. It is so much fun to read this classic and then take the kids for some good ole berry picking. I was lucky to grow up out in the country where we had blueberries, blackberries and raspberries growing all over. My dad always planted a bunch of strawberries as well. If you aren't lucky enough to have these growing in your back yard then you must take advantage of the "pick your own" farms. I will list several here but please visit the following link for a more detailed list of farms. They are listed by county. I want to focus on ORGANIC "pick your own" but to be honest there aren't really any in Mecklenburg County. So I will post some that are the closest to us. I even think this is worthy of a small road trip for the family. The strawberry farms were a bit further than I wanted to travel. We will have better luck with the organic blueberries in regards to location.
So here I am weighing the benefits of organic versus four hours total driving time (and more pollution). Local Springs Farm won by a close margin. The thought of two grumpy three year olds in the car for four hours that day was not too enticing. It also doesn't hurt that Springs Farm has killer homemade strawberry ice cream. This is a total treat for us. Here it goes and don't forget to check out the tips for making jams on the site. We make jam every year and it is so much better than buying at the store. FYI, you will not find any fruit pectin or jars at EarthFare at Morrison. I asked and the guy said I could find it in a more "rural areas", not too many people in Southpark are doing that sort of thing." I love Earth Fare but I thought that comment was pretty funny…Oh, and yes, you can find it at Harris Teeter.
You might want to check out the farms in SC, since Springs is in Fort Mill. McCleod's in McBee, SC follows organic methods and is only 1 hour 30 minutes from Charlotte.
Remember to always call ahead and make sure that "pick your own" is available. A lot of the farms will close for pick your own if they were over picked the day or weekend before. Strawberries are in season now. Blueberries in June, Blackberries in July and Raspberries in July/August. Time of harvest may vary by region.
– Beans-n-Berries, Conelly Springs, NC 828-403-6989- blackberries, blueberries, veggies.
Remember, this is just a SAMPLE of all the farms listed. If you have a favorite, please comment and tell us what you like about that particular farm. Thanks for reading!
LAST DAY LAST DAY LAST DAY, enter by 5p TODAY!!! Click here to enter the Mother's Day Swagilicious Giveaway! But first be sure you are a newsletter subscriber and facebook fan. This is our small thank you & the break that you rarely get for the demanding role you happily play each and every day. And it's a giveaway just for you gal, so take a deep breath and feast your eyes on the dreamy collection below. Only the Smartiest of mom-centric sponsors would do… and holy pink & green do we have some unbelievable stuff to give to one lucky Smarty! Our winner will be chosen today, Friday, May 7th at 5p.
Thank you so much for posting the organic farms!! We had a great one near our home in Wake Forest, and thought we had found one here. Unfortunately after picking 3 baskets last weekend, I overheard the farmer talking to a customer and said that they sprayed the berries with a pesticide that would go away after two days. He went on to say that they close for those two days. Maybe it's gone in two days, but I don't want to take that risk!!
We picked strawberries at Springs in Fort Mill today. PLENTY of berries still and they are huge! Little disappointed that they are not as sweet this year as in years past.
my double batch jam recipe calls for 12 cups of strawberries and though i love picking I could not do all that concentrated pesticide …very expensive decision was to buy them at Earth Fare which i hated doing. thanks for this list, i'll drive 90 minutes next time.
I just gave McLeod's Farm a call to check on pick-your-own berries for tomorrow. I double-checked the organic status since it wasn't listed on their website. The woman on the phone said she didn't know, so I asked her is they sprayed their crops, and she said they do. 🙁 Just wanted to pass the word along. Thanks so much for this post – it's definitely been a big help!
In 1837, a saddle-making company named hermes bags was established by Thierry Hermes. Having been operated and developed for more than one hundred years, Hermes Kelly becomes a public company. However, Hermes hand bag family still owns thirty percent shares. lida daidaihua proof diet works by changing calories constantly. The secret to daidaihua success is by rotating what you eat to maintain a high metabolism which in the long run creates a quick fat burning procedure. When you buy lida slimming for idiots program, you get some great slimming capsule also. A menu generator, the most valuable tool I have ever seen, allows you to type in your favorite foods and slimming capsules spits out exactly what you should eat that day in a 4 course meal. This lida meal needs to be followed for 11 days. While you look for Tiffany uk designs, the list will keep extending as you can find immense Tiffany pendants collections of designs ranging from elegant to classical designs. If you desire to excite your dear one with an impressive gift, then silver pendants will be right choice. Above all, silver accessories is easily resizable and it of course matches everyone irrespective to age. Discontinued Chanel handbags are unauthorized purses that mimic the styles, sizes, and fashions of well-known designers for a fraction of the cost. While replica Chanel handbags may seem like a great deal because Chanel handbag have only very few differences .


Brighton is a coastal town in the south of England worth a short trip from London any time of year. Here is an unconventional guide to the unconventional beach destination. For such a tiny place in comparison to the capital, Brighton is bursting at the seams both socially and culturally. Good times guaranteed.
Trains run regularly from London Victoria. For bus options see National Express prices and times on their website.
Not much going in the way of hostels. Airbnb (forever and always) is the way to go. A couple of the pubs in town have doubles rooms above the public space - check these out for a cheap place to catch some zzz's.


Use this form to request a book be purchased for the libraries' collections, or contact the academic liaison in your subject area.
A service for UK faculty, staff, and students that allows for the delivery of books from any campus library to another campus library for pick-up. You may also request a book from a library to be picked up at that library. For example, you request a collection of Shakespeare's works from Young library and wish to pick it up at Young Library. Please keep in mind that it may take 24 - 48 hours to process your request, pull the book(s) and get it to the desired pick up location. You will receive an email when the book is ready for pick-up. See Document Delivery for more information.
UK students, faculty and staff may request a book that is currently checked out to another borrower. Once a request is placed, the borrower who currently has the book is sent a system-generated letter to let him/her know that the due date has been changed. At this time the current borrower is given a two week due date (unless the original due date is sooner than 2 weeks) to allow him/her to finish using the book.
Once the book is returned, the person who requested the book is notified that the item is available for pickup.
Any current UK Library patron may request a book from Storage. Please keep in mind that it may take 24 - 48 hours to process your request, pull the book(s) and get it to the desired pick up location. One of our Storage facilities is located off campus. You will receive an email when the book is ready for pick-up.
To request a book: Search for the book in InfoKat Discovery. Click on the title of the item you wish to request. Click on 'Sign-in for more options'. Under 'Get It' click on 'Request'. Fill out out 'Details of title you requested:' only * (starred) fields are required. Chose PickUp location. Click green 'Request' to complete the request. You will be notified via email once the book is available for pick-up.
Any current UK library patron may request a book that is currently checked out to another patron. The patron with the book will be notified that the book as been recalled and give a new due date. That patron has 2 weeks to finish using the book (unless the origin due date is earlier than 2 weeks). The UK patron requesting the book will be notified once the book has been returned and is ready for pickup.
NOTE: Recalled (requested) items not returned by the new due date will accrue a recall overdue fine of $2.00 per day. Please respond to these notices promptly.
Items that have been recalled may not be renewed.
If you are unable to locate an item that should be on the shelf but is not, inquire at your Circulation Desk. Library personnel will search for the item and notify you if the item is found.
UK faculty and staff may authorize one or more persons to borrow and request library materials under the authorizing person's name by submitting this form. A verification email will be sent to the requesting faculty/staff member. Other services and equipment may vary from library to library. Please consult individual campus libraries for your specific needs.


text:
// ==========================================================================
//                                DG.Case
//
//  Copyright (c) 2014 by The Concord Consortium, Inc. All rights reserved.
//
//  Licensed under the Apache License, Version 2.0 (the "License");
//  you may not use this file except in compliance with the License.
//  You may obtain a copy of the License at
//
//    http://www.apache.org/licenses/LICENSE-2.0
//
//  Unless required by applicable law or agreed to in writing, software
//  distributed under the License is distributed on an "AS IS" BASIS,
//  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
//  See the License for the specific language governing permissions and
//  limitations under the License.
// ==========================================================================

sc_require('models/model_store');
sc_require('models/base_model');

/** @class
 *
 * Represents an individual case in the collection. For the base collection (the
 * rightmost collection, the one with no child collection) there is one case
 * for each item in the data set. For all upper level collections a case is a unique
 * combination of attribute values among the data items with the same parentage.
 * Each collection manages an attribute group. An individual case is the set of
 * DataItems in the DataSet for which the attributes in the attribute group all
 * have the same values.
 *
 * @extends SC.Object
 */
DG.Case = DG.BaseModel.extend((function() {
  /** @scope DG.Case.prototype */

  /**
    Utility function to convert dates to strings while leaving
    all other values alone.
   */
  function convertValue(x, iAttr) {
    if (iAttr && (iAttr.type === 'date' && (DG.isDate(x)))) {
      return DG.formatDate(x, iAttr.precision);
    }
    else {
      return (DG.isDate(x)) ? DG.formatDate(x) : x;
    }
  }

  return {

    /**
     * Revision counter for the case.
     *  Incrementing the revision property causes change notifications to be sent.
     *  Calling beginCaseValueChanges()/endCaseValueChanges() will automatically
     *  increment the revision counter when endCaseValueChanges() is called.
     * @property {Number}
     */
    revision: 0,

    /**
     * A relational link back to the parent collection.
     * @property {DG.Collection}
     */
    collection: null,

    /**
     * A relational link back to the parent case (if any).
     * @property {DG.Case}
     */
    parent: null,

    /**
     * A relational link to the child cases of this case.
     * @property {[DG.Case]}
     */
    children: null,

      /**
       * Every case gets its attribute values from an item.
       * @type {DG.DataItem}
       */
    item: null,

    _cachedDate: null,

    /**
     * Associative array of {AttrID:value} pairs for quicker lookups.
     */
    _valuesMap: function () {
      return this.get('item').values;
    }.property(),

    /**
     * Initialization function for the case.
     */
    init: function() {
      sc_super();

      this._cachedDate = {};
      DG.Case._addCaseToItemMap(this);

      this.children = [];
      if (this.parent) {
        this.parent.children.pushObject(this);
      }
    },

    verify: function () {
      if (SC.empty(this.collection)) {
        DG.logWarn('Unattached case: ' + this.id);
      }
      if (typeof this.collection === 'number') {
        DG.logWarn('Unresolved reference to collection id, ' + this.collection +
          ', in case: ' + this.id);
      }
      if (typeof this.get('parent') === 'number') {
        DG.logWarn('Unresolved reference to parent, ' + this.get('parent') +
          ', in case: ' + this.id);
      }
    },

    /**
     * Destruction function for the case.
     */
    destroy: function() {

      DG.Case._removeCaseFromItemMap(this);

      sc_super();
    },

    beginCaseValueChanges: function() {
      this.beginPropertyChanges();
    },

    /**
     * Ends a set of property changes and increments the revision property.
     */
    endCaseValueChanges: function() {
      this.incrementProperty('revision');
      this.endPropertyChanges();
    },

    /**
     * Returns the raw value of the specified attribute for this case.
     * Dates are returned as Dates rather than converted to strings.
     * @param {Number} iAttrID ID of the attribute for which the value should be returned.
     * @returns {Number | String | null}
     */
    getRawValue: function( iAttrID) {

      if( !SC.none( iAttrID)) {

        // If we have an attribute formula, we must evaluate it.
        var tAttr = DG.Attribute.getAttributeByID( iAttrID);
        var valuesMap = this.get('_valuesMap');
        if (tAttr && tAttr.get('hasFormula') &&
          // we only do the evaluation if it's one of this case's attributes
          (this.getPath('collection.id') === tAttr.getPath('collection.id'))) {
          return tAttr.evalFormula( this);
        }

        // If we have a cached value, simply return it
        if( tAttr && valuesMap && (!SC.none(valuesMap[iAttrID])) &&
            // we only do the evaluation if it's one of this case's attributes. This is important!
            (this.getPath('collection.id') === tAttr.getPath('collection.id'))) {
          return valuesMap[iAttrID];
        }

        // one last chance if we've got a parent and it has 'getValue'
        // Need for this seems to have been brought about by a particular import of boundary file data. Can't hurt.
        var tParent = this.get('parent');
        if( tParent )
          return tParent.getRawValue( iAttrID);
      }
      // if we get here, return value is undefined
    },

    /**
     * Returns the value of the specified attribute for this case.
     * Dates are converted to strings.
     * @param {Number} iAttrID ID of the attribute for which the value should be returned.
     * @returns {Number | String | null}
     */
    getValue: function( iAttrID) {
      var rawValue = this.getRawValue(iAttrID);
      var attr = DG.Attribute.getAttributeByID( iAttrID);
      return rawValue != null ? convertValue(rawValue, attr) : rawValue;
    },

    /**
     * Returns the numeric value of the specified attribute for this case.
     * Note that the built-in isFinite() can be used with the result (rather than
     * DG.isFinite()) because nulls/undefineds/etc. have already been filtered out.
     * @param {Number} iAttrID -- ID of the attribute for which the value should be returned.
     * @param {Object} oOptInfo -- Additional information about the value can be returned to
     *                              the client via this object.
     *                  .type -- The type of the value, e.g. "string", "number", etc.
     *                  .isNominal -- True if the value is non-numeric and can be
     *                                converted to a non-empty string.
     * @returns {Number}
     */
    getNumValue: function( iAttrID, oOptInfo) {
      var value = this.getRawValue( iAttrID),
          valType = typeof value,
          tAttr = DG.Attribute.getAttributeByID( iAttrID),
          attrType = tAttr && tAttr.type;
      if (DG.isDate(value)) {
        // treat dates numerically
        value = Number(value);
        valType = "number";
      } else if (attrType === 'date') {
        if (!this._cachedDate[iAttrID]) {
          this._cachedDate[iAttrID] = DG.parseDate(value, true);
        }
        value = Number(this._cachedDate[iAttrID]);
        valType = "number";
      } else if (DG.isDateString(value)) {
        this._cachedDate[iAttrID] = DG.parseDate(value);
        value = Number(this._cachedDate[iAttrID]);
        valType = "number";
      }
      else if (attrType === 'numeric') {
        value = DG.MathUtilities.extractNumeric(value);
        valType = "number";
      }
      if( oOptInfo) {
        oOptInfo.type = valType;
        // Note: we don't consider non-primitive types (e.g. objects, arrays, etc.)
        oOptInfo.isNominal = (valType === "boolean") ||
          ((valType === "string") && !SC.empty(String(value)));
      }
      switch( valType) {
        case "number":
          return value;
        case "string":
          return (value !== "" ? Number(value) : NaN);
      }
      return NaN;
    },

   /**
     * Returns a numeric value of the specified attribute for this case where the raw value has been stripped
    *   of non-numeric characters and the first valid number.
     * @param {Number} iAttrID -- ID of the attribute for which the value should be returned.
   */
    getForcedNumericValue: function( iAttrID) {
      var tInfo = {},
          tValue = this.getNumValue( iAttrID, tInfo);
      if( tInfo.type === 'number')
        return tValue;
      else
        return DG.MathUtilities.extractNumeric( this.getRawValue( iAttrID));
    },

    /**
     * Returns the string value of the specified attribute for this case.
     * @param {Number} iAttrID ID of the attribute for which the value should be returned.
     * @param {Object} oOptInfo -- Additional information about the value can be returned to
     *                              the client via this object.
     *                  .type -- The type of the value, e.g. "string", "number", etc.
     * @returns {String}
     */
    getStrValue: function( iAttrID, oOptInfo) {
      var value = this.getValue( iAttrID),
          valType = typeof value;
      if( oOptInfo) {
        oOptInfo.type = valType;
      }
      return DG.DataUtilities.toString(value);
    },

    /**
     * Return the value that fits the attribute's type
     * @param iAttrID {Number}
     * @return {number|string|null}
     */
    getTypedValue: function( iAttrID) {
      var tAttribute = DG.Attribute.getAttributeByID( iAttrID);
      if( tAttribute) {
        switch (DG.Attribute.getAttributeByID(iAttrID).get('type')) {
          case 'numeric':
            return this.getForcedNumericValue(iAttrID);
          case 'date':
            return this.getValue(iAttrID);
          case 'nominal':
            return this.getStrValue(iAttrID);
          default:
            return this.getRawValue(iAttrID);
        }
      }
      else
        return this.getRawValue( iAttrID);
    },

    /**
     * Is there a value of the specified attribute for this case.
     * Note that a formula can return a null value, so some cases will be thought
     * to have a value for the given attribute ID, when actually they don't.
     * @param {Number} iAttrID ID of the attribute for which the value should be returned.
     * @returns {Boolean}
     */
    hasValue: function( iAttrID) {
      if (!SC.none( iAttrID)) {
        var tAttr = this.get('collection').getAttributeByID( iAttrID);
        if(tAttr && tAttr.get('hasFormula'))
          return !SC.empty( this.getValue( iAttrID));

        if( tAttr)
          return this.get('_valuesMap') && !SC.empty( this.get('_valuesMap')[iAttrID]);

        var tParent = this.get('parent');
        return tParent && tParent.hasValue( iAttrID);
      }
    },

    /**
     * Sets the value of the specified attribute for this case.
     * @param {Number} iAttrID ID of the attribute for which the value should be set.
     * @param {String} iValue Value of the newly created case value.
     */
    setValue: function(iAttrID, iValue) {
      var children = this.get('children');
      if( !this.get('_valuesMap')) return; // Looks like case has been destroyed

      this._cachedDate[iAttrID] = null;
      if (children.length === 0) {
        this.item.setValue(iAttrID, iValue);
      } else {
        children.forEach(function (child) { child.setValue(iAttrID, iValue); });
      }
    },

    /**
     Returns a copy of the case's values.
     */
    copyValues: function() {
      var valuesMap = this.get('_valuesMap'),
          attrs = this.collection.attrs,
          values = {};
      attrs.forEach(function (attr) {
        values[attr.name] = valuesMap[attr.id];
      });
      return values;
    },

    /**
     Override to handle 'values' specially.
     During runtime, the _valuesMap, which maps from attrID to value,
     is the definitive contents of the case. In this method, we copy the
     contents of the _valuesMap to the 'archivableValues' property, which
     maps from attrName to value and converts values for archival purposes.
     */
    archivableValues: function() {
      var values = {};
      var valuesMap = this.get('_valuesMap');
      var attrs = this.collection.attrs;
      attrs.forEach(function (attr, ix) {
        values[attr.name] = convertValue(valuesMap[attr.id]);
      });
      return values;
    }.property(),

    debugLog: function(iPrompt) {
      DG.log('Case ' + this.get('id'));
    },

    toArchive: function () {
      var result;
      result = {
        parent: (this.get('parent') && this.get('parent').id)  || undefined,
        guid: this.id,
        id: this.id,
        itemID: this.item.id,
        values: this.get('archivableValues')
      };
      return result;
    }
  };
})());

/*
  Map from { collectionID: { itemID: DG.Case }}
  Used for looking up cases from items.
 */
DG.Case._itemCaseMaps = {};

/*
  Adds a (new) case to the item case map.
 */
DG.Case._addCaseToItemMap = function(iCase) {
  var collectionID = iCase.getPath('collection.id'),
      itemID = iCase.getPath('item.id'),
      itemCaseMap = collectionID && DG.Case._itemCaseMaps[collectionID];
  if (!SC.none(itemID) && collectionID) {
    if (!itemCaseMap)
      DG.Case._itemCaseMaps[collectionID] = itemCaseMap = {};
    itemCaseMap[itemID] = iCase;
  }
};

/*
  Removes a case from the item case map.
 */
DG.Case._removeCaseFromItemMap = function(iCase) {
  var collectionID = iCase.getPath('collection.id'),
      itemID = iCase.getPath('item.id'),
      itemCaseMap = collectionID && DG.Case._itemCaseMaps[collectionID];
  if (!SC.none(itemID) && itemCaseMap) {
    delete itemCaseMap[itemID];
  }
};

/*
  Find a case (if any) that corresponds to the specified item.
 */
DG.Case.findCase = function(iCollectionID, iItemID) {
  var itemCaseMap = iCollectionID && DG.Case._itemCaseMaps[iCollectionID];
  return itemCaseMap && !SC.none(iItemID) && itemCaseMap[String(iItemID)];
};

/**
 * Creates a new case with the specified properties.
 * @param iProperties
 *    {{collection:DG.Collection|id, parent:DG.Case|id, extentStart: number}}
 *    Properties to apply to the newly-created DG.Case
 *
 * @return {DG.Case}  The newly-created DG.Case
 */
DG.Case.createCase = function( iProperties) {
  var collection = iProperties.collection,
    parent = iProperties.parent,
    newCase;
  iProperties.collection = DG.store.resolve(collection);
  if (!SC.empty(parent)) {
    iProperties.parent = DG.store.resolve(parent);
  }
  newCase = DG.Case.create(iProperties || {});
  return newCase;
};

/**
 * Destroys the specified DG.Case.
 * @param {DG.Case} iCase The case to destroy
 */
DG.Case.destroyCase = function( iCase) {
  iCase.destroy();
};

redpajama_set_name: RedPajamaGithub


text:
Wilma Elizabeth McDaniel (December 22, 1918 – April 13, 2007) was a long-time resident of California's Central Valley. Wilma was one of thousands who emigrated from Oklahoma to California during the Dust Bowl years of the mid-1930s. Named the Tulare County poet laureate in the 1970s, McDaniel was dubbed "The Okie Poet" because of her writings about Oklahoma throughout her lifetime.

McDaniel published more than fifty books of poetry and was the subject of a film documentary by Chris Simon, Down an Old Road: The Poetic Life of Wilma Elizabeth McDaniel.

Biography
Wilma Elizabeth McDaniel was born December 22, 1918, in Lincoln County, Oklahoma. Her parents were Benjamin and Anna (Finster) McDaniel. She was the fourth born out of eight children. McDaniel studied at Green Ridge School, and another in Depew, Oklahoma. When Wilma was 17 years old, in 1936, her father removed the family to California in the wake of the Great Depression and Dustbowl. Fortunately the McDaniels had family in California that helped to make the transition easier. They were able to secure employment through these family members. Wilma picked grapes at a vineyard for her Uncle Jeremiah Finster. Wilma also worked for a time as a housekeeper and maid. Her father died 10 years after moving to California and Wilma lived with her mother until her death in 1983.

Poetic career
McDaniel's love of poetry began at an early age. At age eight, she began to write on scraps of paper, envelopes, and grocery bags, storing them away for later publication. In the 1970s -when Wilma was in her 50s- she took some of her poems to the Tulare Advance-Register newspaper editor, Tom Hennion. Seeing her inborn talent, Hennion continued to publish her work in the newspaper which lead to her wider publication later on in life. McDaniel wrote with simple language and harsh imagery and was referred to as the "California Walt Whitman" by Gerald Haslam.

References

External links
 Origination page for Haslam article reproduced above.
 Highlights this late Okie Folk poet in images and poetry; also includes a fairly complete bibliography of her work.
 Wilma Elizabeth McDaniel Papers at the UC Merced Library and Special Collections
 Wilma Elizabeth McDaniel Collection at the Oklahoma State University Library

1918 births
2007 deaths
Poets from California
American women poets
20th-century American poets
20th-century American women writers
21st-century American women

redpajama_set_name: RedPajamaWikipedia


Walking the streets of Dublin a couple of months ago, I came across this message for U2's Bono.
Any guesses as to what the heck staks are?
STAXS is a dynamic, leading clothing retailer – successfully providing today's women with sophisticated fashion, styled for today's world!
He's a short man, staks are probably shoes or inserts in shoes to make you appear taller. We know them as "lifts"
Agree with Pete. Ever notice his footwear on stage? The soles of his boots/shoes are usually about 3" high. He's but a wee leprechaun.
That makes sense. That's got to be it.
Bono is a great person... Talented too..
U2 base all their crews and operations out of Ireland.
The main reason they moved their business aspect was because the Irish taxation on their international operations was extremely high. Any smart business would do that same.
He wants governments to commit more tax money to Africa but doesnt want to pay his own.
Bono does grat charity work, no doubt about that. However, the guy is still a tool, his ego is unmatched (other than our own Irv prehaps)and the guy wants everyone to know how great he is and how much charity work he does, he gets off on it.
I actually really enjoyed Rattle and Hum, but that's the last U2 album I loved.
@ Mike - Not familiar with Rattle and Hum, could you name a single I might recognize off it.
That being said, U2 is kinda like Pearl Jam for me. I completly recognize that both are hugely successfull bands, and I quite enjoy a number of their songs, I just never got what the big deal about them was...I know some people that put them in the same leauge as the Beatles and Zepplin...personally I just don't see it in terms of quality of music....longevity for sure, but in my view, not quality.
But, as I've said before, that's the beauty of music, it means different things to different people.
What I liked about Rattle and Hum was that it's part great new songs, part live versions of great U2 songs and part cool covers.
U2 is the musical equivalent of Walmart but they sell organic carrots as a feel good move then tell the world how in touch with the planet they are.
Rattle and hum actually seems pretty intersting, might have to check that one out.
Achtung Baby is their greatest album and widely considered by most music critics one of the top 10 rock albums of all time.
Achtung Baby was the beginning of the end for U2.
I'll take Boy, or Joshua Tree, or Rattle and Hum.
And I can't believe this is happening AGAIn, but I have to agree with Irvine here too. Bono (I won't throw all of U2 under the bus, since the other 3 stay pretty quiet), is a two-faced db. When he started crticizing Harper and Canada about what we do to help 3rd world countries, Bono lost all credibility he had.
519 Rob - He had every right to rag on Harper. Harper cut federal help to a number of releif organizations world wide - after he said he wouldn't!
These are the same projects that Harper spoke and promoted himself only a year before. It essence Harper said one thing and then did the exactly opposite.
Rattle & Hum is great too but everything else is so-so. But in concert, they're more than worth the price of admission.
Argie - 100% Agreed... Live No One Comes Close... Their Simply Amazing!
@ MrFascination - I've seen U2 live, and it was a great show.
That being said, Dave Matthews Band blows them out of the water for live shows. If you've never been, I highly reccomend it!
Mississauga Phil - We have different views of Dave Mathews Band.. I consider them one of the most over rated bands ever.. I've seen them live (friend dragged me to it back June 2010).. I thought it was an awful concert - BUT i'd say that crowd loved it... Almost like a cult following.. Really strange.. Just not for me.
@ MrFascination - No doubt that DMB are one of those groups that you either love or hate, there is usually no middle ground. If that 2010 show was at the amphetheatre downtown, then I was at that show too...loved it!!
I despise U2 now as well but for me they were at their best when War came out.
Hey Toronto Mike, thanks for taking a pic of this. My wife and I were in Dublin last month and I noticed this message (It's a bit faded now and a few scribbles over it). I mentioned it to her this evening and she said she didn't remember seeing it. I was a bit amazed I was able to Google it and find it here!
I was kicking myself for not taking a pic, but glad someone caught it for posterity. LOL Cheers!
Hey, I took a photo of that back in Sept last year. I was searching the web looking for the meaning of staks. Thanks for clearing this up.


If you've been a developer on the App Store for any length of time, you've probably faced the difficult decision of killing an app. Maybe it's just not making as much money as you'd hoped. Maybe its codebase has, through neglect, gotten too old to bother updating. Whatever the reason, choosing to end-of-life an app can be a painful choice to make. Today we talk about removing apps from the App Store, and all the business emotional factors that guide our decision.
Every month there are more great articles and content about the business of Mac and iOS development than we have time to discuss on the show. If you'd like to receive our monthly newsletter with the best of these links, as well as a summary of past episodes, and early access to conference tickets when they become available, be sure to sign up for our mailing list at http://releasenotes.tv/mailinglist.
This episode is sponsored NSNorth. NSNorth is Canada's independent conference crafted specifically for developers and designers who specialize in Apple's platforms of iOS and Mac. Charles and Joe have both attended in years past, and we both highly recommend it. So check your calendars and grab your wallets; tickets go on sale Jan 20, 2015.
This entry was posted in Release Notes on January 19, 2015 by admin.


To say that our senses were dazzled during our 'Melian Restaurant' dinner would definitely be an understatement. An all-round sensory experience never to be forgotten.
Not able to look away from the magnificent sunset, we 'blindly' sipped our cocktails.
Only once the sun had set and our eyes were finally able to avert from the horizon, could we ask the waiter what our divinely tasting cocktails were (when asked what we would like to drink, our answer had been "Surprise us!"). Apparently, a Negroni & a Greek Mojito is what we had drunk.
We ordered another round, waved back to a yacht full of singing French people, and started dipping our fresh bread into the little dish of olives, capers, and kritamo which were bathing in, wonderfully scented, extra virgin olive oil.
Again not being able to decide what to choose from the menu, (we nailed it down to the bewitching sunset) and being the 'culinary risk-takers' that we are, we simply stated, "We'll try a few dishes. Bring us what you think we'd like." Not having to decide actually felt quite liberating and we leisurely started with the Feta Saganaki (lightly fried, breaded feta with ouzo & sunflower seeds), served with the most delicious, slightly spicy, tomato jam and watermelon balls. Fava (yellow-split peas) and grilled octopus followed, as did Salmon with Carrot Cream and the evening's protagonist Sea Bass Ceviche.
An unparalleled sunset combined with gastronomic artistry. An experience we will never forget. You won't either. Trust us.
Melian Restaurant is located on the premises of the luxurious Boutique Hotel 'Melian Hotel & Spa' in the Pollonia area of Milos Island.