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\author{Peter W.\ Graham$^a$, Bart Horn$^b$, Surjeet Rajendran$^a$, Gonzalo Torroba$^a$\\
[7mm]
{\normalsize \it $^a$ Stanford Institute for Theoretical Physics, Department of Physics, }\\
{\normalsize  \it  Stanford University, Stanford, CA 94305, USA }\\
[3mm]
{\normalsize \it $^b$Theory Group and Institute for Strings, Cosmology, and Astroparticle Physics,}\\
{\normalsize  \it Columbia University, New York, NY, 10027, USA}\\
[3mm]}

\bigskip
\title{\bf  Exploring eternal stability with the simple harmonic universe
\vskip 0.5cm}
\maketitle

\begin{abstract}
We construct nonsingular cyclic cosmologies that respect the null energy condition, have a large hierarchy between the minimum and maximum size of the universe, and are stable under linearized fluctuations. The models are supported by a combination of positive curvature, a negative cosmological constant, cosmic strings and matter that at the homogeneous level behaves as a perfect fluid with equation of state $-1<w<-1/3$. We investigate analytically the stability of the perturbation equations and discuss the role of parametric resonances and nonlinear corrections. Finally, we argue that Casimir energy contributions associated to the compact spatial slices can become important at short scales and lift nonperturbative decays towards vanishing size. This class of models (particularly in the static limit) can then provide a useful framework for studying the question of the ultimate (meta)stability of an eternal universe.
\end{abstract}
\bigskip
\newpage

\tableofcontents

\vskip 1cm

\section{Introduction}\label{sec:intro}

The construction of nonsingular cosmological solutions is a central problem in General Relativity (GR), both for its implications on the mathematical structure of the theory and for potential applications to describing our universe. This subject combines deep physical questions --did our Universe have a beginning, or is it eternal?-- with precise theorems regarding general properties of solutions. The singularity theorems in GR\footnote{See e.g.~\cite{Hawking:1973uf} for a detailed discussion.} imply that, under fairly general assumptions, cosmological solutions must be singular. Even inflating solutions have been shown to be geodesically incomplete in the past~\cite{Borde:2001nh}. In this context, it is important to try to extend the validity of these theorems, or to find nonsingular models and understand how the theorems are evaded. Moreover, the existence of such models would have interesting consequences for inflation, by providing possible past-completions or by decaying into realistic cosmologies.

A key assumption in the original singularity theorems is the strong energy condition (SEC) $T_{\mu\nu} v^\mu v^\nu \ge \frac{1}{2}T v_\mu v^\mu$, where $T_{\mu\nu}$ is the energy-momentum tensor, and $v_{\mu}$ is a timelike vector.  The SEC is, however, known to be violated in many consistent systems, and therefore one way to proceed is to relax this condition. It is reasonable to impose instead the (weaker)
null energy condition (NEC) $T_{\mu\nu} v^\mu v^\nu \ge \frac{1}{2}T v_\mu v^\mu$, where this time $v_\mu$ is a null vector. This condition appears to be satisfied macroscopically in our universe. Furthermore, for the purpose of performing a self-consistent GR study, it is important to demand that
the solutions be weakly curved, with all scales much larger than the Planck length. Under these conditions, positive curvature is the main ingredient that allows the circumvention the singularity theorems.\footnote{Another interesting possibility, recently studied in~\cite{Balasubramanian:2014jaa}, is to use translation-breaking effects.}

Ref.~\cite{usfirst} (see also~\cite{Harrison:1967zz, Kardashev:1990, Dabrowski:1995ae} for earlier related work) constructed a family of nonsingular solutions that satisfy the NEC and are everywhere weakly curved. These are bouncing universes that arise from the competition between positive curvature, a negative cosmological constant, and a fluid with equation of state $-1<w<-1/3$. In particular, for $w=-2/3$ the FRW equations can be solved analytically, leading to a scale factor that oscillates harmonically ---the ``simple harmonic universe'' (SHU). Our goal in this work is to continue the analysis of this solution, and improve it in different directions.

First, the model of~\cite{usfirst} is linearly stable only if the ratio $a_-/a_+$ between the smallest and largest sizes of the universe is not too small. In \S \ref{sec:multifluid} we will show that by adding extra matter sources (cosmic strings) it is possible to
find stable solutions with a large hierarchy $a_-/a_+\ll 1$. This is an important step in order to construct realistic bouncing cosmologies.

Secondly, we will perform in \S \ref{sec:stability} a detailed study of the linearized fluctuations, investigating analytically their stability and discussing the role of parametric resonances. This will establish the linearized stability of our solutions over a large range of parameter space. This analysis extends to the quantum level by fixing the size of the initial perturbation by the uncertainty principle.
Nonlinearities could cut off this behavior, leading to a finite but plausibly exponentially long lifetime. It is, however, possible that the nonlinearities do not lead to instabilities -- the nearly static limit is argued to be very different from usual thermodynamic systems, and it is as of yet an open question whether this class of systems may be eternal even after the inclusion of nonlinear effects.

A universe that is stable at the linear and nonlinear level may still be subject to nonperturbative decays, making it metastable (but exponentially long-lived). \S \ref{sec:nonpert} focuses on the possible nonperturbatively suppressed decays of our cyclic cosmologies. The simplest example of an instanton where the universe tunnels to nothing was found in~\cite{Mithani:2011en}. We argue that this instability can be eliminated by introducing additional sources of energy density such as  Casimir energy corrections associated with compact spatial slices.
While we do not provide a definite answer to whether these cosmologies could then be truly eternal, and there are potentially other nonperturbative decay channels that need to be addressed, we believe that this class of models provides a controlled setup where the issue of the beginning of time could be further studied. Our conclusions and future directions are presented in \S \ref{sec:concl}.

\section{Cyclic universes with large bounces}\label{sec:multifluid}

It is interesting to ask if a cyclic cosmology can accommodate a large ratio between the minimum and maximum sizes of the universe which could reproduce, for instance, the hierarchy between the GUT (or Planck) scale, and the size of our Hubble patch. However, it was shown in~\cite{usfirst} that in models with only one type of fluid, the linearized fluctuations with momenta $l \lesssim \sqrt{a_+/a_-}$ grow exponentially. Intuitively, the Hubble antifriction produced by the evolution near the minimum dominates on average over the friction contribution from the maximum, effectively inducing a tachyonic mass. For bounces with  $a_+/a_- \gg 1$, this leads to unacceptable instabilities in a large number of modes.

We will now show that it is possible to construct cyclic models that do not suffer from such instabilities by adding one more matter source -- {e.g.}\ a frustrated network of cosmic strings. The additional source of energy density enables the frequency of oscillation of the universe to be parametrically decoupled from the mass of the linearized fluctuations, which is set by the curvature of the space-time. This freedom was absent in \cite{usfirst} and it permits  the frequency of the oscillations to be small compared to the mass of the fluctuations, enabling stability.

\subsection{General equations and energetic considerations}\label{subsec:general}

Let us begin our analysis by determining how a cyclic solution can arise from the competition of different matter and energy sources.

We consider an FRW metric with positive curvature,
\begin{equation}\label{eq:metric1}
ds^2 = -dt^2 +a(t)^2 \left(\frac{dr^2}{1- K r^2}+ r^2(d\theta^2+ \sin^2 \theta\,d\phi^2) \right)\,.
\end{equation}
Here the curvature $K=1$ and the spatial slices are 3-spheres; we will find it useful to keep the dependence on $K$ explicit, to account more simply below for the effects of cosmic strings. The FRW equations in the presence of a cosmological constant $\Lambda$ and additional matter sources are
\begin{eqnarray}\label{eq:FRW}
\frac{\ddot a}{a} &=& - \frac{4\pi}{3} G_N (\rho + 3 p)+ \frac{\Lambda}{3} \nonumber\\
\left( \frac{\dot a}{a}\right)^2&=& \frac{8\pi}{3} G_N \,\rho - \frac{K}{a^2}+ \frac{\Lambda}{3}\,.
\end{eqnarray}
For simplicity, we will restrict to matter sources that behave macroscopically (but not microscopically, as we discuss in \S \ref{subsec:stability}) as perfect fluids,
\begin{equation}
p = w \rho\;,\;\rho= \frac{c_w}{a^{3+3w}}\,.
\end{equation}

To proceed, it will be useful to recast the second FRW equation in (\ref{eq:FRW}) as the vanishing of the GR Hamiltonian\footnote{Recall that the constraint $H=0$ is imposed by the $g_{00}$ Lagrange multiplier, which we have here gauge-fixed to $g_{00}=-1$.}
\begin{equation}\label{eq:GRH}
H =- \frac{G_N}{3\pi a}\left(\pi_a^2 + V(a) \right)
\end{equation}
where $\pi_a$ is the canonical momentum
\begin{equation}
\pi_a = \frac{\partial L}{\partial \dot a} = -\frac{3 \pi}{2 G_N} a \dot a
\end{equation}
and the effective potential
\begin{equation}\label{eq:Va}
V(a) = \left(\frac{3\pi}{2G_N} \right)^2\,a^2 \left(K - \frac{8\pi}{3}G_N c_w \,a^{-3w-1}- \frac{1}{3}\Lambda a^2 \right)\,.
\end{equation}
Given this three-term structure, it is clear what sources are needed for a periodic solution. At the minimum and maximum, $\pi_a=0$, which means $V(a)$ should admit two roots $a_\pm$; furthermore $V(a)<0$ between these roots so that $a$ oscillates between $a_-$ and $a_+$ as determined by (\ref{eq:GRH}). We therefore need
\begin{equation}\label{eq:general-matter}
-1 < w < - \frac{1}{3}\;,\;\Lambda<0\,.
\end{equation}
The evolution near the minimum is dominated by curvature (which tries to make the universe expand) and the fluid (which pushes towards contraction), while near the maximum the negative cosmological constant balances against the fluid. Denoting the period of oscillation by $\tau$ we also obtain, from the integral of the second FRW equation,
\begin{equation}
\frac{\tau}{2} = \int_{a_-}^{a_+} \frac{da}{\left(\frac{8\pi}{3} G_N \,c_w a^{3|w|-1}  + \frac{\Lambda}{3} a^2-K \right)^{1/2}}\,.
\end{equation}

Before discussing models with a large hierarchy $a_- \ll a_+$, we point out that the case $a_- \sim a_+$ is also of conceptual interest. We can tune the parameters $c_w$ and $\Lambda$ in (\ref{eq:Va}) so that the two roots nearly coincide, $a_- \approx a_+$, and this corresponds to the static limit of our cyclic cosmologies. Unlike the Einstein static universe, this class of solutions does not suffer from homogeneous instabilities, partly because it does not require tuning the initial conditions. Therefore, the models with positive curvature and matter content (\ref{eq:general-matter}) provide a framework where many of the questions studied in the Einstein static universe can be addressed without the danger of linearized instabilities.\footnote{For work on the Einstein static universe, its stability and applications to inflationary cosmology see e.g.~\cite{Einstein}.}  In particular, we will return to the question of the ultimate (meta)stability of such static cosmologies in \S \ref{sec:nonpert}.

Let us now find the conditions for a large hierarchy. Assuming $a_- \ll a_+$,  the minimum is approximately determined by balancing in (\ref{eq:Va}) the curvature and fluid contributions,
\begin{equation}\label{eq:amin}
a_- \approx \left(\frac{3}{8\pi} \frac{K}{G_N c_w} \right)^{\frac{1}{3|w|-1}}\,,
\end{equation}
while for the maximum we obtain
\begin{equation}\label{eq:amax}
a_+ \approx \left(8 \pi \frac{G_Nc_w}{|\Lambda|} \right)^{\frac{1}{3-3|w|}}\,.
\end{equation}
Therefore, choosing the cosmological constant and density such that
\begin{equation}\label{eq:cond}
|\Lambda|^{3|w|-1} \ll \frac{(8\pi G_N c_w)^2}{K^{3-3|w|}}\,
\end{equation}
obtains $a_- \ll a_+$. Furthermore, as long as the matter density
\begin{equation}
c_w \ll \frac{K}{(8 \pi G_N)^\frac{1+3|w|}{2}}\,,
\end{equation}
the smallest size of the universe will be much larger than the Planck length, guaranteeing that classical GR remains a good approximation.

We also note that these results generalize easily to multifluid models, replacing $\rho = \sum_i  c_i \,a^{-3(1+w_i)}$ in the expressions before. In particular, let us see what happens when we combine a fluid with equation of state $-1 < w < - 1/3$ with a density of cosmic strings, $w=-1/3$. Since the contribution of strings is $\rho=c_{str}/a^2$, their
effect is simply encoded in a redefinition of the curvature,
\begin{equation}\label{eq:Keff}
 K_{eff} = K - \frac{8\pi}{3} G_N c_{str}\,,
\end{equation}
and the previous equations still apply after replacing $K \to K_{eff}$ (we can now set the background curvature $K=1$ inside $K_{eff}$). Increasing the density of strings decreases the effective curvature, and by tunning $G_N c_{str} \sim 3/8\pi$ it is possible to nearly cancel $K_{eff}$. Now, from (\ref{eq:amin})--(\ref{eq:cond}) we see that taking $K_{eff} \ll 1$ while keeping the other parameters fixed leads to a hierarchy $a_-/a_+ \ll 1$. Moreover, we will show below in \S \ref{subsec:stability} that a smaller $K_{eff}$ gives a larger mass from the angular momentum on the sphere. Therefore, the addition of cosmic strings can simultaneously lead to solutions with large hierarchies between the minimum and maximum size, and to a stabilization of low-lying modes that could otherwise be unstable.

\subsection{An analytic solution with domain walls and cosmic strings}\label{subsec:DWCS}

We now illustrate our previous general results by studying the special (and solvable) case of domain walls (or any $w = -2/3$ fluid) plus cosmic strings.

The solution to the FRW equations is a simple harmonic universe, albeit now with a nontrivial $K_{eff}$,
\begin{equation}\label{eq:aSHU}
a(t) = \frac{1}{\omega} \frac{\sqrt{K_{eff}}}{\sqrt{\gamma}} \left(1+ \sqrt{1-\gamma} \,\cos\,\omega t \right)\,.
\end{equation}
The frequency of oscillations is
\begin{equation}
\omega = \sqrt{\frac{|\Lambda|}{3}}\;,
\end{equation}
and we have introduced the combination
\begin{equation}
\gamma =  \frac{3 K_{eff}|\Lambda|}{(4\pi G_N c_{dw})^2}\,.
\end{equation}

When $\gamma \sim 1$ the oscillatory term in (\ref{eq:aSHU}) vanishes, and we obtain a static solution with $a_-=a_+=\frac{1}{\omega} \sqrt{K_{eff}}$. On the other hand, the limit $\gamma \ll 1$ gives
\begin{equation}
a_+ \approx \frac{2}{\omega}\frac{\sqrt{K_{eff}}}{\sqrt{\gamma}}\;,\;a_-\approx \frac{1}{2\omega} \sqrt{\gamma K_{eff}}\,,
\end{equation}
in agreement with (\ref{eq:amin}) and (\ref{eq:amax}) for $w=-2/3$ and $K \to K_{eff}$. This leads to a hierarchy
\begin{equation}
\frac{a_-}{a_+} \approx  \frac{\gamma}{4} \ll 1\,.
\end{equation}

In the rest of this work we will often focus on this special case, which allows for analytic results. However, we would like to stress that our results will apply to the more general class of models with fluid $-1<w<-1/3$. For short, we will refer to the general family of solutions as the ``simple harmonic universe'', in a slight abuse of notation.

\subsection{Stability of linearized perturbations}\label{subsec:stability}

As we mentioned before, in the limit $a_- \ll a_+$, cyclic universes supported by curvature, a negative cosmological constant and a matter source with $-1<w<-1/3$ are unstable under linear perturbations. We will now argue that adding an appropriate density of cosmic strings solves this problem.

The key point is that in the presence of cosmic strings the effective curvature becomes an adjustable parameter [see (\ref{eq:Keff})], and using this extra degree of freedom we can keep the ratio $a_-/a_+$ and the period of oscillation $\tau$ fixed, while the overall scale $\sqrt{a_- a_+}$ is decreased.\footnote{For instance, in the analytic solution of \S \ref{subsec:DWCS}, this is achieved by fixing $\Lambda$ and $\gamma$, and decreasing $K_{eff}$ and $c_{dw}$ in a correlated way. For more general $w$, the cosmological constant also needs to be changed together with the other parameters.} Since the masses of linear fluctuations are inversely proportional to this scale, a sufficiently small $K_{eff}$ can overcome the tachyonic contribution from Hubble antifriction, leading to linearly stable perturbations.

Let us illustrate this effect of cosmic strings in the analytic SHU model of \S \ref{subsec:DWCS}. It is convenient to change to conformal time,
$d\eta^2 = dt^2/a(t)^2$, which yields a scale factor
\begin{equation}
a(\eta) = \frac{\sqrt{\gamma K_{eff}}}{\omega} \frac{1}{1-\sqrt{1-\gamma}\,\cos (\sqrt{K_{eff}}\eta)}\,.
\end{equation}
As a simple and generic example of linear perturbation, we analyze a probe scalar field $\phi$ in the FRW background,
\begin{equation}\label{eq:massless}
\frac{d^2 \phi}{d\eta^2} + 2 \frac{da/d\eta}{a} \,\frac{d\phi}{d\eta} - \nabla^2_{S^3}\phi=0\,.
\end{equation}
Redefining
\begin{equation}
\hat \eta \equiv \sqrt{K_{eff}}\eta
\end{equation}
obtains
\begin{equation}\label{scalareq}
\frac{d^2 \phi}{d \hat \eta^2} + 2\frac{da/d \hat\eta}{a} \, \frac{d\phi}{d\hat\eta} - \frac{1}{K_{eff}}\,\nabla^2_{S^3}\phi=0\,.
\end{equation}
In this form, the only dependence on $K_{eff}$ appears in the $S^3$ Laplacian term. It is now clear that $K_{eff}\ll 1$ enhances the positive contribution to the mass squared for modes with nonzero momentum.

We can simplify this equation further by expanding in spherical harmonics and redefining the field to cancel the term linear in derivatives. Doing this for the $l^{th}$ harmonic obtains
\begin{equation}\label{eq:psieom}
\psi_l''+ m_l^2 \psi_l = 0\;,\;m_l^2 = \frac{l(l+2)}{K_{eff}}- \frac{a''}{a}\,.
\end{equation}
The largest tachyonic contribution from the time evolution of the scale factor occurs near $\hat \eta \approx \sqrt{\frac{5\gamma}{3}}$ and is of order $\frac{a''}{a}\approx \frac{9}{8\gamma}$. Thus, tuning the density of cosmic strings $c_{str}$ in (\ref{eq:Keff}) such that
\begin{equation}\label{stabilitylargebounce}
K_{eff} \lesssim \gamma
\end{equation}
the linearized perturbations for generic values of the momentum $l$ will be stable even in the regime $\gamma \ll 1$. We show an example of this in Figure \ref{fig:stable}.  We should stress that for special values of $l$ there may exist additional instabilities due to resonances with the oscillations of the scale factor; such effects will be analyzed in \S \ref{sec:stability}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{unstable}
\includegraphics[width=0.4\textwidth]{stable}
\end{center}
\caption{\small{Evolution of the $l=2$ harmonic mode in a SHU with $\gamma=0.225$. In the first plot there are no cosmic strings, and this mode grows exponentially. The second plot shows how the mode becomes stable in the presence of a density of cosmic strings with $K_{eff}=0.2$.}}\label{fig:stable}
\end{figure}

There are some effects, however, which are not captured by (\ref{eq:massless}). In particular, we need to take into account that the fluctuations of the matter sources may have speed of sound $c_s \neq 1$, and that they also source gravitational fields. An important aspect here is that if the equation of state for the perfect fluid were to remain valid at short distances, the speed of sound $c_s^2=w<0$, and there would be instabilities at arbitrarily large momenta. Fortunately, as discussed in~\cite{usfirst}, this can be avoided by having solids that behave, on cosmological scales, as perfect fluids with negative $w$, but which nevertheless have $c_s^2>0$ and are stable. For the models discussed here, we need stable networks of domain walls and cosmic strings. See~\cite{Bucher:1998mh} for more details.

The effects from gravitational backreaction also turn out to be important. Incorporating the scalar metric perturbations and assuming a speed of sound $c^2_s > 0$ one obtains the following equation for the adiabatic mode, for the $l^{th}$ spherical harmonic (see e.g.\ \cite{usfirst}):
\beq
\psi'' + m^2_l \psi = 0 \, , \, m^2_l = \frac{(l+3)(l-1)c^2_s - 1}{K_{eff}}-\frac{7+9c^4_s}{4}\left(\frac{a'}{a}\right)^2 - \frac{3c^2_s - 1}{2}\frac{a''}{a}\,.
\eeq
To avoid introducing instabilities for the low-lying harmonics when $c^2_s(l+3)(l-1) <1$ we wish to keep $c_s$ an $O(1)$ fraction of the speed of light.\footnote{We also need to replace (\ref{stabilitylargebounce}) by the stronger condition $K_{eff} \ll c^2_s \gamma$.}  The $l=1$ spherical harmonic of the scalar metric perturbation is unstable due to a cancellation between terms in the equation of motion~\cite{usfirst} once we include backreaction. When there is only a single adiabatic perturbation sourced by one kind of matter this mode is pure gauge and can be redefined away; however, when multiple sources or fields are present -- as in our case -- this is no longer possible.  We can nevertheless project out this mode (and a few more of the low-lying spherical harmonics, if needs be) by orbifolding along the Hopf fiber of the $S^3$.  This orbifold will also remove Bianchi IX or Mixmaster perturbations, where the scale factor changes in a manner that is homogeneous but not isotropic.

\section{Stability and analytic considerations}\label{sec:stability}

Cyclic cosmologies have a potentially new type of instability that is not present in expanding universes. Due to the oscillatory behavior of the scale factor, it is possible to have resonances from modes whose momenta are close enough to integer multiples of the frequency of oscillation of the universe.
In a flat infinite model there is a continuum of momentum modes and this always leads to instabilities. However, momenta on $S^3$ are quantized, and so it is possible that none of the modes lie within an instability band. This is the question that we now address.
Specifically, we will show that there exists a set of parameters $K_{eff}, \gamma$ for which there are no parametric resonances, and we discuss how the stability argument may be generalized beyond linear order using KAM theory.

In this section we focus, for simplicity, on the simple harmonic universe solution based on domain walls and cosmic strings; however, our conclusions are expected to be valid for the more general family of cosmologies discussed in \S \ref{subsec:general}.

\subsection{Hill's equation and parametric resonance}

Let us consider the stability of the solutions to the scalar field equation of motion (\ref{eq:psieom}),
\beq
\psi'' + f(\hat{\eta})\psi = 0,
\eeq
where the forcing term for the $l^{th}$ spherical harmonic is
\begin{equation}\label{eq:forcing1}
f(\hat{\eta})= \frac{l(l+2)}{K_{eff}} + \frac{2\sqrt{1-\gamma}\cos\hat{\eta}+(1-\gamma)(-3 + \cos 2\hat{\eta})}{2(1-\sqrt{1-\gamma}\cos\,\hat{\eta})^2}\,.
\end{equation}
Recall that the rescaled conformal time is given by $\hat \eta= \sqrt{K_{eff}} \eta$, in terms of which the oscillating cosmology has period $2\pi$.
The generalization to the full equation for adiabatic perturbations including gravitational backreaction will be discussed later.

This class of equations, with a general periodic forcing term, are known as Hill's equations in mathematics, and are equivalent to a Schr\"odinger/Bloch wave problem in physics. A special and well-known case is the Mathieu equation, corresponding to $f(\hat{\eta}) = \delta + \epsilon \cos{\hat{\eta}}$. Stability of the solutions to Hill's equation has been extensively and rigorously studied; see for instance \cite{diffbook, hillandfourier, asymptoticbook}.  The solutions are known to be of the form
\beq
\psi(\hat \eta)=\psi_1(\hat{\eta})e^{\mu_1 \hat{\eta}} + \psi_2(\hat{\eta})e^{\mu_2 \hat{\eta}},
\eeq
where $\psi_1, \psi_2$ have period $2\pi$ and $\mu_1 \mu_2 = 1$.  More specifically, $\mu_1, \mu_2$ are either real reciprocals, in which case the solution is unstable without fine-tuning the initial conditions, or complex conjugates of unit magnitude, in which case the solution is stable for all time.

Intuition from quantum mechanics suggests that the solutions should be stable in the limit where the forcing term changes adiabatically:
\beq
f'(\hat{\eta})/f(\hat{\eta})^{3/2} \ll 1,
\eeq
which holds in the static limit $(1-\gamma) \ll 1$ and in the large bounce limit as long as $K_{eff} \ll \gamma.$  More generally, when $K_{eff}/\gamma$ is large, the harmonics with $l \gg (K_{eff}/\gamma)^{1/2}$ are in the adiabatic regime.  This intuition is nearly correct; however, there will be `instability bands' in parameter space when the system has a parametric resonance between the free and driven oscillations.  To find a perturbatively stable bouncing universe, these instability bands must be avoided for all harmonics under consideration.

For a forcing term of the form
\beq\label{eq:generalpsi}
\psi'' + \omega(\omega - \frac{\epsilon}{\pi} p(\hat{\eta}))\psi = 0,
\eeq
where $ \int_0^{2\pi} p(x) dx = 1$, $\epsilon \ll 1$, as $\omega \to \infty$ the solution will pass through an infinite number of instability bands where the system approaches a parametric resonance point and becomes unstable.  Many asymptotic estimates exist for the width $\Delta \omega_n$ of the $n^{th}$ instability band -- in \cite{hillandfourier} it was shown that the width of the band is closely related to the $n^{th}$ Fourier coefficient of the forcing term $p(\hat{\eta})$.  Specifically, the solution will be unstable within a range
\beq
\label{intervals}
\omega_n = \frac{n}{2} + \frac{\epsilon}{2 \pi}(1 \pm |c_n|) + O(\epsilon^2)
\eeq
as long as the $n^{th}$ Fourier coefficient
$$
c_n = \frac{1}{2\pi}\int^{2\pi}_0 p(\hat{\eta}) e^{-i n \hat{\eta}}d\hat{\eta}
$$
is nonzero.  For the forcing term in the simple harmonic universe, $c_n \to 0$ exponentially in $n$ as $n \to \infty$, as we will check below, and at first glance we might worry that the linear estimate for the size of the gap will eventually become smaller than the quadratic remainder term in \ref{intervals}.  However, it is easy to confirm using the proof in \cite{hillandfourier} that the $O(\epsilon^2)$ correction to the width of the instability interval remains subleading at large $n$.  More generally, when the forcing term is infinitely differentiable $\Delta \omega_n\,\to\,\frac{\epsilon}{\pi}|c_n|$ faster than any inverse power of $n$~\cite{asymptoticbook}.  The position of the instability band will also receive corrections order by order in powers of $\epsilon$ -- these, however, have a power-law falloff in $1/n$.

We will therefore use the estimate $\Delta \omega_n \approx \frac{\epsilon}{\pi}|c_n|$ to show that there exists a set of values for the parameters $K_{eff}, \gamma$ such that we can avoid the instability bands for all values of $n$ and $l$.   For the simple case of the Mathieu equation it is possible to give a constructive proof of stability, showing explicitly how to choose the values of $K_{eff}$ that avoid all parametric resonances --see e.g.~\cite{Auzzi:2012ca}. However, our forcing term is not just a cosine function and this complicates an explicit calculation of the stable values of $K_{eff}$.\footnote{One source of difficulties, absent in the Matthieu equation, is that the positions of the instability bands (\ref{intervals}) receive corrections in $1/n$ that cannot be neglected when estimating the closeness of $\omega$ to parametric resonances.} We will instead follow a different approach:
starting with a continuous set of $K_{eff}$ we discard the values that hit an instability band for the first spherical harmonic, then the ones that hit an instability interval for the second band, and so on, until we are left with the values that never hit an instability for any harmonic. We will show below that this process converges, and that the fraction of values of $K_{eff}$ that lead to instabilities is bounded and exponentially small both in the large bounce and quivering limits.

Before proceeding to our analysis of the large bounce and static limits, let us collect a few useful expressions.
For the system at hand, the free oscillation is given by
\beq
\omega^2 = \frac{l(l+2)}{K_{eff}}\,,
\eeq
the parameter $\epsilon$ is equal to
\beq
\epsilon = \sqrt{\frac{K_{eff}}{l(l+2)}}\left(\frac{2\pi^2}{\sqrt{\gamma}} - 2\pi^2 \right)\,,
\eeq
and the normalized forcing term is given by
\beq
p(\hat \eta)=-\frac{1}{2\pi} \frac{\sqrt \gamma}{1- \sqrt \gamma}\frac{2\sqrt{1-\gamma}\cos\hat{\eta}+(1-\gamma)(-3 + \cos 2\hat{\eta})}{2(1-\sqrt{1-\gamma}\cos\,\hat{\eta})^2}\,.
\eeq
The $n^{th}$ Fourier coefficient is given by
\beq
c_{n} = -\frac{3(1 - \gamma)}{2}a_{n} + \frac{\sqrt{1-\gamma}}{2}(a_{n-1} + a_{n+1}) + \frac{1 - \gamma}{4}(a_{n-2} + a_{n+2})\, ,
\eeq
where
\beq
a_{n} = \frac{1}{\sqrt{\gamma} - 1}\left(\frac{\sqrt{1 - \gamma}}{1 + \sqrt{\gamma}}\right)^{|n|}\frac{(1 + |n|\sqrt{\gamma})}{\gamma}\,.
\eeq
For $|n|\ge 2$ this expression simplifies to
\beq\label{eq:cnapprox}
c_{n} = -\frac{2 |n| \sqrt{\gamma } -1 }{1-\sqrt{\gamma }}\left(\frac{\sqrt{1-\gamma }}{1+\sqrt{\gamma }}\right)^{|n|}\,.
\eeq

\subsection{Large bounce limit}

In the large bounce limit $\gamma \ll 1,$ we consider the regime $K_{eff} \ll \gamma$.  Since the instability intervals we have to avoid obey $n \gtrsim 1/\sqrt{K_{eff}}$, $n\sqrt{\gamma}$ is large.  Using (\ref{eq:cnapprox}), the leading term in the Fourier coefficient in this limit can be checked to be
\beq
c_n \rightarrow -2n \sqrt{\gamma}\,e^{-n\sqrt{\gamma}} + \,...
\eeq
The size of the $n^{th}$ instability interval is then
\beq
\Delta \omega_n \approx \frac{\epsilon}{\pi}|c_n| \sim \sqrt{\frac{K_{eff}}{l(l+2)}}\,n\,e^{-n\sqrt{\gamma}}\,.
\eeq
We see then that the lengths of the instability bands are exponentially small in the regime of interest $K_{eff} \ll \gamma$. We will next use this to argue that most values of $K_{eff}$ in such a range avoid all the resonant instabilities.

Consider $K_{eff}$ to be a continuous parameter varying within a narrow interval $I = \left[K^0_{eff}, K^0_{eff} + \delta K_{eff}\right]$ around a small value  $K^0_{eff}$.  By narrow we mean that $\delta K_{eff}/K^0_{eff} \ll 1$.  For a particular spherical harmonic $l = l_{*}$, as $K_{eff}$ varies over $I$, the function $\omega_{l_{*}}(K_{eff}) = \sqrt{\frac{l_{*}(l_{*}+2)}{K_{eff}}}$ will vary over a corresponding range $\left[ \omega_0, \omega_0 + \delta \omega \right]$ that includes potentially many instability bands.  The region $\delta_{l_{*}}K_{eff}$ of $I$ such that $\omega$ lies in an instability interval can be estimated using
\beq
\frac{\delta_{l_{*}}K_{eff}}{\delta K_{eff}} \sim \Delta \omega_n \,,
\eeq
which implies that the fraction of $I$ such that $\omega_{l_{*}}$ lies in an instability band is approximately
\beq
\frac{\delta_{l_{*}}K_{eff}}{\delta K_{eff}} \sim \Delta \omega_n \sim e^{-\sqrt{l_{*}(l_{*}+2)}\sqrt{\gamma/K^{0}_{eff}}}\,.
\eeq

The total fraction of the $K_{eff}$ that hit an instability band for some value of $l$ as $l$ ranges from $1$ to $\infty$ is therefore bounded by the expression
\beq
\sum_{l=1}^{\infty}\,e^{-\sqrt{l(l+2)}\sqrt{\gamma/K^{0}_{eff}}}
\eeq
which is exponentially small as long as $K^{0}_{eff} \ll \gamma$.  Therefore in the parametric region $K_{eff} \ll \gamma$, most values of $K_{eff}$ will avoid instability bands for all $n$ and $l$, and the proof is done.

Note that while the fraction of $K_{eff}$ that avoid all parametric resonances is nearly one in a measure theoretic sense, the set of $K_{eff}$ that will hit an instability band eventually for some large but finite $l$ is dense, meaning that any neighborhood around a `good' choice will contain infinitely many bad choices.  Finding an explicit working set of values and assessing how `naturally' they are tuned depends on the context of the specific model used, while our aim here has been to show that there is a set of parameters that are free of resonant instabilities.  In practice, for the purposes of perturbative stability it suffices only to show stability for a finite (though potentially very large) set of modes, since for $l$ above a certain cutoff value $l_c$ the modes have Planck scale masses, in which case the classical instability will be replaced by nonperturbative particle or black hole production.  Under these relaxed criteria it is possible to find (exponentially small) open neighborhoods of the parameter $K_{eff}$ that avoid the finite collection of instabilities for $l < l_c$.

\subsection{Static limit}

As we discussed before, the static, or `quivering' limit $(1-\gamma) \ll 1$ is also conceptually interesting, especially for the possibility of constructing an eternal universe. Linearized fluctuations were found to be numerically stable in~\cite{usfirst} already with $K_{eff}=1$. We now show that it is also possible to avoid parametric resonances for most choices of parameters.

The Fourier coefficients of the normalized forcing term $p(\hat{\eta})$ in the static limit are given by
\begin{equation}
c_n = -\frac{2}{1-\gamma}\,\left(\frac{\sqrt{1-\gamma}}{2}\right)^n (2n-1) + ...
\end{equation}
for $n \geq 1$, and $c_0 = 1$.  The size of the $n^{th}$ instability band can therefore be estimated as
\beq
\Delta \omega_n \sim \frac{\epsilon}{\pi}|c_n| \sim \left(\frac{\sqrt{1-\gamma}}{2}\right)^n |(2n-1)|\sqrt{\frac{K_{eff}}{l(l+2)}}
\eeq
for $n \geq 1$, and $\Delta \omega_{n=0}\sim (1-\gamma)$.

Allowing $K_{eff}$ to vary in a narrow band $I = \left[K^0_{eff}, K^0_{eff} + \delta K_{eff}\right]$ around some small value $K^0_{eff}$, for a given mode $l = l_{*}$ the set $\delta_{l_{*}} K_{eff}$ of values for $K_{eff}$ such that $\omega_{l_{*}}^2(K_{eff}) = \frac{l_{*}(l_{*}+2)}{K_{eff}}$ lies within some nearby instability band can be estimated as
\beq
\frac{\delta_{l_{*}} K_{eff}}{\delta K_{eff}} \approx \left(\frac{\sqrt{1-\gamma}}{2}\right)^{\sqrt{l_{*}(l_{*}+2)/K^0_{eff}}}\,\left| 2\sqrt{\frac{l_{*}(l_{*}+2)}{K^0_{eff}}}-1\right|\,\sqrt{\frac{K^0_{eff}}{l_*(l_*+2)}}
\eeq
and so the total fraction of the set of $K_{eff}$ that hits an instability band for some value of $l$ as $l$ ranges from $1$ to $\infty$ is controlled by
\beq
\sum_{l = 1}^{\infty}\left(\frac{\sqrt{1-\gamma}}{2}\right)^{\sqrt{l(l+2)/K^0_{eff}}}\, \left|2-\sqrt{\frac{K^0_{eff}}{l(l+2)}}\,\right|\,
\eeq
which is exponentially small as long as $(1-\gamma) \ll 1$.  Once again, measure theoretically speaking, given a value of $\gamma$ in the quivering limit, most values of $K_{eff}$ will do the trick, and this time we do not need to worry about having $K_{eff} \ll 1$.  Actually, we do not even need the parameter $K_{eff}$ in this limit:\ treating $(1-\gamma)$ as the free parameter to be varied in a narrow range around a small value, the argument above carries through almost unmodified to show that for $(1-\gamma) \ll 1$ most values of $(1-\gamma)$ will avoid the entire infinity of resonances.

\subsection{Incorporating backreaction}

As we discussed in \S \ref{subsec:stability}, density perturbations from the domain walls and/or cosmic strings are not precisely described by the equation of motion for a free massless scalar. The main differences are that these fluctuations may have $c_s \neq 1$, and also metric perturbations become important. Let us now briefly comment on the stability for this case.

Incorporating gravitational backreaction on the scalar metric perturbations and assuming a speed of sound $c^2_s > 0$ for the adiabatic mode one obtains a Hill equation with forcing term
\beq
f(\hat{\eta}) = \frac{(l+3)(l-1)c^2_s - 1}{K_{eff}}-\frac{7+9c^4_s}{4}\left(\frac{a'}{a}\right)^2 - \frac{3c^2_s - 1}{2}\frac{a''}{a}\,.
\eeq
The exact value of $c^2_s$ depends on the shear resistance and is model-dependent -- but as we discussed in the previous section, in order to avoid introducing unstable modes when $c_s l \lesssim 1$ we wish to keep it an $O(1)$ fraction of the speed of light.  Then the arguments from before can be easily generalized to prove the existence of choices for the parameters $K_{eff}$, $\gamma$ such that we have stability in the large bounce and quivering limits.

We can also generalize the discussion to give a more realistic treatment of the network dynamics, including effects such as shear modes, as occur in the solid dark matter model of \cite{Bucher:1998mh}. It is also straightforward to generalize the discussion to include isocurvature perturbations when multiple components are present.  However, we do not expect these points to qualitatively alter our conclusions.

\subsection{Nonlinearities}\label{subsec:nonlinear}

We have shown that we can have oscillating cosmologies that are stable at the linear level. It is possible that this stability is violated by non-linear couplings between the oscillating scale factor and the perturbative modes discussed above.  Much like the linear case, instabilities in the non-linear regime occur when a combination of the mode frequencies coupled together by a non-linear interaction fall into an instability band of the scale factor. When this occurs, the oscillations of the scale factor will resonantly excite these modes, leading to an instability.

It is beyond our means to prove the eternal stability of our solutions in the presence of such non-linear instabilities.  But, for the following reasons, we believe that our solutions might be stable for a long time and may in fact be truly eternal at the linear and at the nonlinear level.\footnote{Nonperturbative decays are discussed below in \S \ref{sec:nonpert}.} A general way to study non-linear instabilities is to use KAM theory -- see e.g.~\cite{introkam} for a review.  The applicability of KAM theory though is limited by the requirement that the frequency of the excitable modes depend non-trivially on their amplitude. For a free theory, this condition is obviously not satisfied.  But, in an interacting theory, there will in general be non-linear interactions between these excitable modes (for example, a quartic interaction for a probe scalar).  At leading order, these interactions will not couple to the oscillating scale factor and are thus not a source of new instabilities. Their presence however will lead to a non-trivial dependence between the frequency of the mode and its amplitude, permitting the application of KAM theory.

The KAM conditions boil down to ensuring that linear combinations of the excitable modes do not come sufficiently close to a resonant multiple of the oscillating scale factor. When these conditions are satisfied, KAM theory guarantees the eternal stability of the solution, even in the non-linear regime. Further, as shown in~\cite{Nekhoroshev}, when the KAM conditions fail, the non-linear solution will remain close to the (stable) linear solution for an exponentially long time, with a lifetime that depends upon the size of the failure of the KAM conditions.  Unfortunately, we do not have the expertise to mathematically perform the KAM analysis on our cosmological solutions. But this result suggests that even if our cosmology was not eternally stable, it should still survive for an exponentially long time.

A physical reason to expect the demise of these cosmologies due to non-linearities is continuous entropy production.  This is simply a re-statement of the above mathematical issue -- the oscillations of the scale factor may resonantly excite modes, leading to the generation of entropy. The produced entropy will back react on the metric, potentially leading to runaway behavior.  But, these processes will not occur if these modes are far from the bands of instability of the system, and are thus subject to the results of a KAM-type analysis. In most physical systems, thermodynamic intuition suggests that the system will have these kinds of instabilities. Indeed, we except a disturbance of the system to dissipate through such excitations.  This intuition though is based on the fact that a typical thermodynamic system has a large number of modes that are nearly degenerate with the disturbance, making it easy for a linear combination of these frequencies to lie within the band of instability of the disturbance.

Our oscillating cosmologies are interestingly different from such thermodynamic systems.  Here, we are interested in excitations caused by the oscillating scale factor which is the lowest frequency in the system. The potentially problematic excitable frequencies all correspond to higher harmonics on the sphere, making it easier to evade the degeneracies encountered in thermodynamic systems.  In fact, for sufficiently high frequencies, we expect notions of decoupling to become important -- a low frequency oscillation (such as that of the scale factor) cannot efficiently excite high frequency oscillations. This implies that the most problematic excitations will be low lying spherical harmonics that are roughly degenerate with the scale factor.  While it is unclear if these modes lead to instabilities, we note that they could be projected out by considering more complex geometries such as orbifolds of $S^3$ by a freely acting group.  Orbifolding will not change the equations of motion, but will project out modes from the spectrum (such a possibility was also considered in \cite{usfirst} and in \S \ref{subsec:stability} to project out linear instabilities). In particular, it will preserve the frequency of the oscillating scale factor but can eliminate low lying modes that are roughly degenerate with it. The resulting system is expected to be more stable since the potentially problematic modes are all at high frequencies compared to that of the scale factor.  Another possibility is to introduce additional sources of energy density such as a frustrated network of strings. As we have seen in \S \ref{sec:multifluid}, these sources can parametrically make the frequencies of the fluctuations higher than the frequency of the scale factor. This should also enhance the non-linear stability of the system. It is an interesting open question whether such strategies could produce a cosmology that is eternal to all orders in perturbation theory.

\section{Comments on nonperturbative decays}\label{sec:nonpert}

A simple harmonic universe that is stable at the linear and nonlinear level may still
be vulnerable to nonperturbative decays on exponentially long timescales~\cite{usfirst}. An interesting example of such a process was found by Mithani and Vilenkin~\cite{Mithani:2011en}, who used the effective potential of the scale factor to construct an instanton that mediates tunneling to zero size.  Here we wish to add a few comments on this decay channel and show how it can be avoided using well-controlled corrections that become important at short scales.

The main point is that the endpoint of the instability is singular and both its amplitude and its existence may be sensitive to the physics operating near $a \sim l_{Pl}$. While~\cite{Mithani:2011en} assumed that the effective GR description is valid all the way up to the Planck scale, it is very plausible that other effects (higher derivative terms, quantum corrections, etc.), which become important before reaching $M_{Pl}$, will modify the Mithani-Vilenkin instanton. We will provide a simple example of this, by showing that the Casimir energy from a collection of scalar fields associated to the compact $S^3$ of the SHU will lift that instanton, in a controlled regime where our semiclassical approximation is valid. We will also comment briefly on other possible decay modes.

Let us begin by reviewing the solution of~\cite{Mithani:2011en} and then show that it can be avoided very simply using Casimir energy from scalar fields. In the minisuperspace approximation, the Wheeler-de Witt wavefunction for the SHU obeys (ignoring ordering ambiguities)
\beq
\left(-\frac{d^2}{da^2} + V(a)\right)\psi(a) = 0
\eeq
where
\beq\label{eq:Va2}
V(a) = \left(\frac{3\pi}{2 G_N}\right)^2 a^2 \left(K_{eff} - \frac{8\pi G_N}{3}(c_{dw}\, a + \Lambda a^2)\right)\,.
\eeq
This follows from (\ref{eq:GRH}) and (\ref{eq:Va}).

Setting $K_{eff} = 1$ and tuning for simplicity parameters $\{ c_{dw}, \Lambda \}$ so that the Lorentzian solution is in the static limit $a_{stat} = \frac{1}{\omega}$, the Euclidean Friedmann equations admit the solution
\beq\label{eq:instanton1}
a(\tau) = \frac{1}{\omega}(1 - e^{-\omega \tau})
\eeq
\noindent and the Euclidean action is given by
\beq
|S_{inst}| = \int^{\infty}_0 d\tau a[\dot{a}^2 + (\omega a - 1)^2] = \frac{3}{32 G_N^2 |\Lambda|}.
\eeq
This assumes that (\ref{eq:Va2}) is valid even for very small values of $a$.
Mithani and Vilenkin interpret this as a tunneling amplitude to zero scale factor, and check that the amplitude agrees with the WKB amplitude one calculates from $V(a)$:
\beq
\int_0^{a_{stat}} \sqrt{V(a)} da = \frac{3}{32 G_N^2 |\Lambda|}
\eeq

The solution is singular at the point of vanishing scale factor, and will be sensitive to the physics resolving the singularity as well as to other sources and corrections which may become important before reaching a Planckian size.  In particular a contribution to the energy such that $\delta V(a) \propto a^{n \leq 0}$ would lift the minimum of $V(a)$ at zero scale factor and prevent the Mithani-Vilenkin process from taking place. It is interesting that for the SHU there is a very natural contribution of this type, namely the Casimir energy associated to the compact spatial slices. More specifically, the Casimir contribution from $N$ massive scalar fields has the right parametrics (and the correct sign, as checked e.g. in~\cite{Herdeiro:2005zj}) to accomplish this:
\beq
\delta V(a)\sim \left(\frac{3\pi}{2 G_N}\right)^2 a^2 \left(\frac{8\pi G_N}{3}\frac{N}{a^2}\right)
\eeq
We have included multiple fields so that when the Casimir energy begins to dominate $V(a)$ at around $a \sim \sqrt{8 \pi G_N N}$, we can still be well within the semiclassical regime as long as $N \gg 1$. This contribution to the effective potential lifts the tunneling instability (\ref{eq:instanton1}). Furthermore, the results of \S \ref{sec:stability} imply that the introduction of these additional scalars does not lead to new linearized instabilities.

In the presence of Casimir energy, there still exists a Euclidean solution interpolating between the static universe and $a = 0$.\footnote{We thank A.~Guth and A.~Vilenkin for pointing this out to us.} This solution has finite action but, unlike (\ref{eq:instanton1}), now the instanton has a singular geometry with $a(\tau) \propto \tau^{1/2}$ as the Euclidean time $\tau \to 0$. It is therefore not clear that this solution should be allowed.\footnote{As a simpler example, in gauge theories only very special singularities are allowed, such as certain singular monopoles from 't Hooft lines.} The question of which singularities are physical in GR is an important open problem that requires a theory of quantum gravity. Without such an understanding here, the standard approach of only path-integrating over smooth configurations implies that this instanton does not contribute. This illustrates again our main point that discussing the behavior near $a \to 0$ requires a UV complete theory of gravity, where various new effects are expected to become important at or before Planckian scales. As another example, higher order curvature corrections may become important as well, and it would be interesting to understand which way these can push.

A smiliar set of questions could be asked of the milder singularity appearing in the solution (\ref{eq:instanton1});\ of course, in this case it is also possible to imagine scenarios where this can be resolved and the decay process can be present. For instance,~\cite{Mithani:2011en} proposed that the singularity may be a coordinate singularity from the projection of a higher-dimensional model~\cite{Garriga:1998ri}, in which case the UV completion comes from the physics of the extra dimensions and need not affect the amplitude of the four-dimensional instanton.  The solution can also be regularized by adding a small amount of dust or radiation, such that the endpoint of the tunneling process occurs at small but nonzero scale factor; the instanton is no longer singular, but the amplitude will be nearly unchanged~\cite{Dabrowski:1995jt, Mithani:2011en}.

On the other hand, there may also be other nonperturbative processes that dominate the decay rate. For instance, these may involve black hole nucleation or collapse of the metastable string and/or wall network. While the rate of these processes is model-dependent, we would expect them to progress more rapidly than the gravitational decay modes. It would be interesting to explore the rates and tunabilities of various decay modes in the context of an explicit model, either in field theory or in string theory.

In any case, we believe that the modification of the SHU that we just described, together with the seemingly special thermodynamic properties of the static limit, make it a promising candidate for exploring the theoretical limits on the (meta)stability of an eternal universe, an issue to which we hope to return in the future

\section{Conclusions and future directions}\label{sec:concl}

In this work we have extended the model and results of~\cite{usfirst} to allow for solutions with large hierarchies between the minimum and maximum sizes of the universe, and studied their stability properties. Stable oscillating universes with a large hierarchy were obtained by combining a network of frustrated cosmic strings, matter with $-1 < w < 1/3$, a negative cosmological constant, and positive curvature. We have found that these cosmologies can be made fully stable at the linearized level, and we ruled out the possibility of parametric resonances in a large range of parameter space.  A point worth stressing once again is that while for analytic simplicity we have focused on the simple harmonic universe (SHU), where a frustrated network of domain walls contributes $w = -2/3$, we expect our results can be extended to this more general class of models.

The simple harmonic universe could suffer from non-perturbative instabilities, which would lead to a metastable and exponentially long-lived, though not eternal, universe.  While there are processes that lead to such instabilities \cite{Mithani:2011en}, we have shown that the simplest of these can be cured through suitable modifications of short distance physics that are naturally present in our model.  We have not ruled out the possibility of non-perturbative decay more generally, but it is not yet clear to what extent this question is model-dependent, and whether the limitations are universal or specific.  A broader class of instabilities may arise from continuous entropy production.  While we were unable to show that these instabilities do not exist in our model, we also could not find concrete arguments that establish their existence.  It would be interesting to further study these classes of instabilities.

It would also be instructive to develop further a microscopic realization of the simple harmonic universe, in the context of string theory or even at the level of field theory.  Furthermore, while have not pursued the inclusion of inflation and the hot Big Bang phase into the simple harmonic universe, it would definitely be interesting to do so. Decays of the SHU into inflating universes may also lead to new possibilities in the string landscape, an alternative that we plan to analyze in the future. It would also be worth pursuing more general questions regarding the structure of GR, in particular whether there are quantum counterparts of the singularity theorems, or more general notions about the arrow of time and the definition of entropy in such a setup.

Finally, it is of interest to understand whether an eternal universe may be possible on general grounds -- see \cite{Wilczek} for an interesting discussion of spontaneous breaking of time translation at the classical and quantum level.  Recent work arguing for the existence of a past singularity and a `beginning of time' \cite{Mithani:2011en, Susskind:2012yv, Mithani:2014jva} in the context of the universe has focused on the generation and role of entropy or on the presence of specific decay modes.  It is not clear, however, that these arguments apply to our class of solutions, particularly in the static limit and with the addition of Casimir energy described in \S \ref{sec:nonpert}.  We hope that our study of these models will stimulate further discussion of these issues.

\section*{Acknowledgements}

We especially thank Shamit Kachru for early collaboration on these and related issues, and we further thank Adam Brown, Xi Dong, Sergei Dubovsky, Brian Greene, Alan Guth, Shamit Kachru, Ali Masoumi, Audrey Mithani, Alex Vilenkin, and Erick Weinberg for very helpful discussions.  PWG, BH, and GT thank the KITP for hospitality during the program ``Primordial Cosmology,'' during which a portion of this work was completed.  PWG~acknowledges the support of NSF grant PHY-1316706, the Hellman Faculty Scholars program, and the Terman fellowship.  BH~is supported in part by the DOE Office of Science under grant DE-FG02-92-ER40699.  SR~was supported by ERC grant BMSOXFORD no.\ 228169.  GT~is supported in part by the National Science Foundation under grant no.~PHY-0756174.

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redpajama_set_name: RedPajamaArXiv


The truth is even the fluffy words on the ABI website about the new Beaumont don't really come close to capturing the feeling you get when you step inside.
Although it is 14′ wide, there is so much space inside – thanks to the great design and layout – one still can't quite work out how they have made the home look so spacious and yet so cosy and intimate at the same time.
Top quality finishes, from the furnishings to the little touches, there is something to impress at every turn. A fantastic living area sets the tone for the rest of the holiday home. The guest bedroom, with twin beds is stylish, there's a family shower room, a large master bedroom which has a walk-in dressing area leading to the en-suite bathroom.


text:
NY high court rejects prison phone rate refund lawsuit
The New York State Court of Appeals dismissed a lawsuit last week brought by prisoners' families seeking refunds for the exorbitant phone rates it cost them to speak to their loved ones behind bars. The class action suit, Walton v. NYSDOCS, was filed in 2004 by the Center for Constitutional Rights on behalf of prisoners' families and friends, who for years had paid inflated rates for collect calls due to the state's monopoly contract with MCI/Verizon.
Lawyers for the families said that the excessively high rates amounted to an unfair and "unlawful tax" that resulted in millions of dollars being paid over a number of years. The prison telephone service contract stipulated that MCI pay the state 57.5 percent of the fees received from prisoners' collect calls. Then MCI would be allowed to charge well over the market rate: a $3 connection surcharge and between 16 and 36 cents per minute. The rate of a collect call from a federal prison is 7 cents per minute. Most of New York's calls are from prisons upstate to loved ones in New York City.
Mostly due to the long struggle waged by the New York Campaign for Telephone Justice, upon arrival to office in 2007, Gov. Eliot Spitzer halted the arrangement. Rates then fell to about half of what they had been. In June 2007, the state Legislature made Gov. Spitzer's decision into a law, the Family Connections bill.
Lawyers for the plaintiffs said that families were paying up to $300 or $400 per month to speak to their loved ones who were incarcerated. Since 1996, they said, the state had collected about $200 million in profits from kickbacks outlined in the phone service arrangement with MCI. The lawyers maintain that families are owed compensation for this unnecessary burden.
The high court's 5-1 decision affirmed a lower court's ruling that the families had failed to assert a legitimate claim under the the state Constitution. The court said the the policy was bad, but not unconstitutional.
Filed under Activism, Criminal Justice
Tagged as Center for Constitutional Rights, Eliot Spitzer, Family Connections bill, Kieran K. Meadows, MCI, New York Campaign for Telephone Justice, New York State Court of Appeals, New York State Department of Correctional Services, prison, Verizon, Walton v. NYSDOCS

redpajama_set_name: RedPajamaCommonCrawl


Q. 1 What can I do to stop chewing the inside of my lower lip and cheek? I am compulsive about it, and just can't stop. Have you seen this in other patients? What do you recommend to break the habit? Could it cause any kind of long-term damage?
Many people do bite or chew on the insides of the lower lip or cheek, perhaps out of boredom or nerves. This habit is often initially prompted by a misdirection of the teeth that causes the person to mistakenly bite into the lower lip while chewing. This obviously causes pain and swelling, which results in blood filling the area and forming a lump or bump, known medically as an oral hematoma. Although this is simply the result of the trauma, it can extend into your chewing cycle and the wound can easily be opened again. I find that when I bite my lower lip by mistake I often repeat that process two to three times before the wound is healed. As you stated, it can become a nervous habit, especially when there is a palpable bump, but it is important to try to stop yourself because you could potentially develop scar tissue that remains there permanently, or becomes some type of growth, either benign or cancerous. Long-term chronic trauma can lead to abnormal cellular growth.
Applying petroleum jelly (Vaseline) to your inner lip over the excess tissue so that it reminds you not to play with it.
Having your dentist remove the excess tissue and place some sutures in the area so you won't have extra tissue to bite on. The sutures will also remind you not to keep hitting them.
Wearing a lower mouth appliance, like a night guard, to keep your lip away from the biting surface of your teeth.
It is also important to have the area examined, and possibly surgically remove the tissue and have it biopsied, to make sure there are no abnormal cells in the area already.
Q. 2 My 82-year-old mother has a palate condition that has developed over the years. I have found very little information about it to help her. She has a lump in the center of the upper roof of her mouth that has grown larger over time. It is oval in shape, and has caused her discomfort in eating certain foods, specifically crusted foods like toast, because it gets irritated. Have you ever heard of such a growth, if it is indeed a growth? Might it be surgically removed? Is there anything I can get to assist in numbing the feeling in that area to make her mouth feel better? Oral gels that desensitize the area don't seem to help.
This very common condition is called torus palatinus, also known as oral tori. It causes bony growths to develop on the direct palate of the arch of the upper jaw or on the tongue side of the lower jaw on one or both sides. These growths can be any size or shape, though often they are rounded like pebbles. Only a very thin layer of tissue covers them and this can easily be irritated, scratched, cut, or burned, depending on the type of food that is eaten.
While this condition is harmless and not associated with any other diseases, it does get in the way if a denture or partial denture is needed. It may even be difficult to take X-rays in the area depending on the size of the growth. It is possible to remove them, though most people do not unless it's necessary for a denture or partial denture.
Having said all this, it is still critical that your mother visit an excellent dentist who can evaluate and diagnose her condition in person. Without seeing her, I cannot tell if a growth of any kind — whether benign or not — is in fact what she's dealing with. More than likely, it is one of these tori and they can either be left alone or removed, based on her needs.
Q. 3 I had two root canals and temporary crowns put on, and then took a plane to Florida. Since returning I have felt a lot of pressure in my ears, and sometimes I have the same sort of feeling in the area of my tooth repair. Could this have anything to do with the dentistry that was done?
It is very common for someone to experience pain in their teeth or jaw if they fly shortly after root-canal therapy. The difference in air pressure can cause gases to build up inside the teeth and cause pain. I recommend waiting at least five to seven days to make sure there is no postoperative pain before getting on a plane. If the root canal is free of bacterial infection and any pain has been eliminated or minimized, then it is usually fine to fly.
The trouble and pressure in your ears, however, may not be related to your teeth; this issue may actually be caused by an inner-ear problem. While it is rare for an earache to cause tooth pain, there are always exceptions. It is more likely, however, that tooth pain in the lower jaw might feel like there is ear involvement.
My advice is to report back to the dentist who performed the root-canal treatment and put in the temporary crowns. The success of the procedure and final restorations should then be evaluated. If they check out fine and there is no dental issue, then I would go see an internist or an ear, nose, and throat (ENT) physician who can analyze whether your problems might be related to your inner ear or even to something else.
Q. 4 Does anything happen to your gums or teeth if you regularly chew ice?
Chewing ice actually has the potential to be fairly detrimental to the health of your teeth. Because saliva is warm and will usually melt the ice in your mouth, nine times out of ten nothing happens. Chewing anything as cold and hard as ice can potentially shock a nerve in a tooth, however, and you may end up requiring root canal therapy. Teeth are not flat, and when we chew and crunch on ice or anything else, the inclined planes on the tops of the teeth facilitate the chewing motion. If a piece of ice is between your teeth and you bite down at an unusual angle, you could potentially chip off part of the enamel on a tooth or crack off a filling. You might also crack, or split, the tooth itself. For this reason, I certainly recommend that you avoid chewing ice, especially on a regular basis.


MAUREEN THERESA BIRD worked in company NETWORK SUPPORT INTERNATIONAL LIMITED.
MAUREEN THERESA BIRD was from 1991.04.30 to 2000.11.28 at NETWORK SUPPORT INTERNATIONAL LIMITED employed as Secretary.
MAUREEN THERESA BIRD Score for MAUREEN THERESA BIRD is 3 stars.


Thousands of people gathered in central London to watch the Lord Mayor's Show.
The 682nd Lord Mayor, Nick Anstee, travelled in a 252-year-old gold coach on a procession through the Square Mile up to the Royal Courts of Justice.
He stopped first at St Paul's Cathedral before pledging allegiance to the Crown at the Royal Courts.
He said he wants to "recover the trust" in the City which has been "damaged". Plans for fireworks on the River Thames have been cancelled due to bad weather.
Mr Anstee, a chartered accountant who has lived and worked in the Square Mile for 32 years, said: "The City of London is a brand in its own right.
"Many have observed that over the past 18 months the brand has actually been damaged.
"There is an element of mistrust of the financial services industry and to help recover that trust is a very important part of what I am seeking to do over the next 12 months."
An RAF flypast opened the show which included various street performances and floats from schoolchildren and livery companies in the City.
A total of 6,100 people participated in the show, including 2,000 servicemen and women.
A fireworks display on the River Thames was cancelled by the Port of London Authority due to bad weather.
This was the first time in more than 20 years the display had been called off.
A spokeswoman said: "There are high winds in London already and they are forecast to get worse in terms of wind speed."
In 1215 King John granted a charter allowing the City to elect its own mayor and since then the Lord Mayor has made his journey every year.
The annually-elected Lord Mayor is the head of the City of London Corporation and acts as global ambassador for the whole UK-based financial services industry.


text:
package org.wisdom.template.thymeleaf.impl;

import org.apache.commons.io.FilenameUtils;
import org.slf4j.LoggerFactory;
import org.thymeleaf.TemplateProcessingParameters;
import org.thymeleaf.resourceresolver.IResourceResolver;
import org.wisdom.api.templates.Template;
import org.wisdom.template.thymeleaf.ThymeleafTemplateCollector;

import java.io.IOException;
import java.io.InputStream;
import java.net.MalformedURLException;
import java.net.URL;

/**
 * The resource resolver for Thymeleaf.
 * The engine is relying on this class to load the template file.
 */
public class WisdomURLResourceResolver implements IResourceResolver {
    public static final String THYMELEAF_FILE_RESOLVER = "THYMELEAF_FILE_RESOLVER";
    private final ThymeleafTemplateCollector engine;

    public WisdomURLResourceResolver(ThymeleafTemplateCollector collector) {
        this.engine = collector;
    }

    @Override
    public String getName() {
        return THYMELEAF_FILE_RESOLVER;
    }

    @Override
    public InputStream getResourceAsStream(TemplateProcessingParameters templateProcessingParameters,
                                           String resourceName) {

        // Check whether we have a 'parent' template
        // If so and if the given template name is 'simple' (not absolute), we compute the full path.
        final Template mayBeParentTemplate = (Template) templateProcessingParameters.getContext().getVariables()
                .get("__TEMPLATE__");
        if (mayBeParentTemplate != null && !isAbsoluteUrl(resourceName)) {
            // Compute the new name, we retrieve the 'full' path of the parent. It's not the complete URL just the path.
            String path = mayBeParentTemplate.name();
            if (path.contains("/")) {
                // If the path contains a /, remove the part after the / and append the given name
                resourceName = path.substring(0, path.lastIndexOf('/')) + "/" + resourceName;
            }
            // Else Nothing do do, we already have the path (root).

            // We normalize to manage the .. case
            resourceName = FilenameUtils.normalize(resourceName);
        }
        ThymeLeafTemplateImplementation template = engine.getTemplateByResourceName(resourceName);

        if (template == null) {
            LoggerFactory.getLogger(this.getClass()).error("Cannot resolve the template {}, " +
                            "neither {} nor {}.thl.html exist in the template directory or is available in bundles.",
                    resourceName, resourceName, resourceName);
        } else {
            try {
                return template.getURL().openStream();
            } catch (IOException e) {
                LoggerFactory.getLogger(this.getClass()).error("Cannot resolve the template {} ({}): cannot open the " +
                        "file.", resourceName, template.getURL().toExternalForm(), e);
            }
        }
        return null;
    }

    private boolean isAbsoluteUrl(String resourceName) {
        // We identify url using the : character. It's not perfect but should cover most cases.
        return resourceName.contains(":");
    }
}

redpajama_set_name: RedPajamaGithub


The ATSC 3.0 Capacity calculator helps broadcaster select their desired modulation parameters by displaying the full channel capacity of any given configuration along with RF performance information. The calculator also helps guide broadcasters by indicating which configurations choices are permitted by the standard.
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OK! magazine is the home of celebrity news. Each weekly issue of OK! Magazine is packed with the biggest stories, the best photographs and the hottest stars from the world of showbiz. Whether it's movies, music, TV, celebrity weddings, babies, fashion, break ups or make-ups, we bring you the truth about stars with exclusive access to every aspect of celebrity. If you want to get up close and personal to the stars you're in the right place. OK! Magazine publishes 51 issues weekly per year.
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text:
Q: Check whether class members are null from Object list Is there way to check whether the particular class members are null ?
I have a class with N number of properties (members). I got to create many Objects for the class and may assign the values for those properties.. I am using List to store all the objects.
Now, I want to check whether the object in particular index from the list has any of its property as null or not set ? Can this be accomplished without using loop or checking each property manually ?
Thanks in advance..

A: Assume you have a list of items of type A, listA. Below is a LINQ expression that will find all items in the list that have any property set to null:
var output = listA.Where(a => typeof (A).GetProperties()
                                        .Any(p => p.GetValue(a) == null))
            .Select((a, index) => index)
            .ToList();

A: Here is how you can check if an item at an index has any null properties:
var item = myItems[indexToCheck];
bool hasNullProperty = typeof(MyClass)
    .GetProperties()
    .Any(p => p.GetValue(item) == null);

A: listOfObjects[i].HaveAnyOfPropertiesNullOrNotSet()
I see no loop here.

A: It could use some work, but here is general function you can use.  I'll be honest, I'm not sure if this violates your restriction that you don't use a loop / check each member individually (seeing as how it does both).
public static bool AreAnyPropertiesNull(object obj)
{
    foreach (var prop in obj.GetType().GetProperties())
    {
        object propertyValue = prop.GetValue(obj, null);
        if (propertyValue == null)
            return true;
    }
    return false;
}

redpajama_set_name: RedPajamaStackExchange


Do you always wanted a little guidance so that you can study on your own? So now you don't have to worry because Studentwhiz has presented University of Phoenix MHS 600 Touro Week 3 Entire Class and MHS 600 Touro Week 3 Individual Assignments so that not only you get guidance but also the key features of how to get prepared for exams through MHS 600 Touro Week 3 Quiz and UOP MHS 600 Touro Week 3 Complete. Here we also provide all the latest updates regarding MHS 600 Touro Week 3 Discussion Questions and MHS 600 Touro Week 3 Learning Team Assignments.


Kevin How attended Circuit Stream's 10-week online VR course to continue his education while balancing full-time work and parenting three children. With no background in programming or computer science, Kevin worried he wouldn't be able to keep up, but Circuit Stream's online courses and 1:1 mentorship took Kevin from someone with no programming experience to a VR developer.
Kevin was interested in computer programming, and taught himself what he could through YouTube tutorials. Without ongoing support from an instructor, however, he found his motivation and interest would taper off before completing any projects.
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Join Kevin and others like him in taking the next step in developing your programming knowledge in the field of VR/AR. Download our course syllabus today.


We will be moving to Mrs. Hopper this week for three weeks of Social Studies. We also start Art this week. Last week we started our new math classes, so your child may have a different teacher for that subject. In writing we are working on descriptive writing. The main idea with that is writing using your senses and lots of details to describe an object or event. In reading, we are starting our regular stations. You should see a spelling list and a stations list every week. If you do not see them, be sure to ask your child about it.
This week we will start our reindeer project in Science. We will be doing NWEA and SRI testing to place kids in math groups and should start switching for math next week. Until then, we are working on basic problem solving, place value, and multiplication. In Language Arts, we will be working on figurative speech, basic grammar, and descriptive writing. My class has two more weeks of Science with me and then they will go to Mrs. Hopper for Social Studies. We will start technology with Mrs. Green this week and kids will get to check out a book. During the reading block, we are working on learning how the different stations work; like silent reading, word work, response to literature and SRA cards. It will be a fun and busy week!
Eek! Here we are the night before the first day of school! I am so excited and a little bit nervous. I know that we will have a wonderful year. For the first week of school, we will be learning routines, schedules, how our personalities interact and many other beginning of the year things. We will be jumping right into our subjects as well. We will learn how stations work in this room and we will be writing starting on the first day. We will be learning about how to use interactive notebooks in science and then moving on to ecosystems and habitats. This year my students will have three weeks of science with me, then three weeks of social studies with Mrs. Hopper, and then another three weeks of social studies with Mrs. Fajt. We will do the same rotation every quarter. We won't start switching for math for another week or two, but students will be moving to other classes for math and will be in one of six classes in the two fifth grade pods. Get lots of sleep tonight and eat a good breakfast in the morning! Cannot wait to see you! Also wear tennis shoes because we have PE starting tomorrow!


Brooklyn Self Storage is located one mile off of Route 6, just east of the Brooklyn/Danielson line, and about two miles from the Rte 6 exit off I-395. We acquired this facility a few years ago and completed a significant upgrade with repairs and improvements to our building, roofing, siding, and more. Access has been expanded to 7 days per week, 365 days per year. This property now also includes an automated, secure entry gate as well as video surveillance cameras. Give our office a call between 8 a.m. and 6 p.m. Monday through Saturday. Move ins can be done by appointment as well as completley online.
Head west onto US 6.


Discussion in 'Suggestions' started by jezter6, Oct 10, 2017.
I'm only about an hour away, and would love the chance for those of us who are semi-local to swing by and pick up parts. It would save me a bunch of $ in shipping, and it would be totally cool if we could see the magic behind the warehouse.
While we do not currently have a pick-up option available, feel free to choose the cheapest shipping method at checkout and take advantage of the fact that any orders shipped from our warehouse to your location will only spend 1 Business Day in transit.
We hope this information helps and we appreciate your support.


text:
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The Race Beat
[Reading] ➶ The Race Beat ➽ Gene Roberts – Sigilo.us
22 March 2019 Gene Roberts 10
Gene Roberts
This Is The Story Of How America Awakened To Its Race Problem, Of How A Nation That Longed For Unity After World War II Came Instead To See, Hear, And Learn About The Shocking Indignities And Injustices Of Racial Segregation In The South And The Brutality Used To Enforce It It Is The Story Of How The Nation S Press, After Decades Of Ignoring The Problem, Came To Recognize The Importance Of The Civil Rights Struggle And Turn It Into The Most Significant Domestic News Event Of The Twentieth Century.Drawing On Private Correspondence, Notes From Secret Meetings, Unpublished Articles, And Interviews, Veteran Journalists Gene Roberts And Hank Klibanoff Go Behind The Headlines And Datelines To Show How A Dedicated Cadre Of Newsmen First Black Reporters, Then Liberal Southern Editors, Then Reporters And Photographers From The National Press And The Broadcast Media Revealed To A Nation Its Most Shameful Shortcomings And Propelled Its Citizens To Act We Watch The Black Press Move Bravely Into The Front Row Of The Confrontation, Only To Be Attacked And Kept Away From The Action Following The Supreme Court S 1954 Decision Striking Down School Segregation And The South S Mobilization Against It, We See A Growing Number Of White Reporters Venture South To Cover The Emmett Till Murder Trial, The Montgomery Bus Boycott, And The Integration Of The University Of Alabama We Witness Some Southern Editors Joining The Call For Massive Resistance And Working With Segregationist Organizations To Thwart Compliance But We Also See A Handful Of Other Southern Editors Write Forcefully And Daringly For Obedience To Federal Mandates, Signaling To The Nation That Moderate Forces Were Prepared To Push The Region Into The Mainstream.The Pace Quickens In Little Rock, Where Reporters Test The Boundaries Of Journalistic Integrity, Then Gain Momentum As They Cover Shuttered Schools In Virginia, Sit Ins In North Carolina, Mob Led Riots In Mississippi, Freedom Ride Buses Being Set Afire, Fire Hoses And Dogs In Birmingham, And Long, Tense Marches Through The Rural South For Many Journalists, The Conditions They Found, The Fear They Felt, And The Violence They Saw Were Transforming Their Growing Disgust Matched The Mounting Countrywide Outrage As The New York Times, Newsweek, NBC News, And Other Major News Organizations, Many Of Them Headed By Southerners, Turned A Regional Story Into A National Drama.Meticulously Researched And Vividly Rendered, The Race Beat Is An Unprecedented Account Of One Of The Most Volatile Periods In Our Nation S History, As Told By Those Who Covered It.
Is a well-known author, some of his books are a fascination for readers like in the The Race Beat book, this is one of the most wanted Gene Roberts author readers around the world.
10 thoughts on "The Race Beat "
Jan Rice Jan Rice	says:
The mainstream press and the civil rights eraWhen people imagine the past, they do so with benefit of hindsight, according to which the past had a good side and a bad side, and some people out of habit, obstinacy, selfishness, or sheer badness, took the bad side Others were on the side of good by virtue of their suffering or their care for the sufferers Actually, though, the past was full of noise, confusion, mixed m...
Dawn Dawn	says:
The final quote of the book sums it up well If it hadn t been for the media the print media and television the civil rights movement would have been like a bird without wings, a choir without a song The book is amazing many viewpoints are explored, as well as key events, the roles of the president, Supreme Court, the states, governors, law enforcement, marshals, FBI, preachers, editors, reporters, photographers, students, Martin Luther King, the Ku Klux Klan it s all here The nonvio The final quote of the book sums it up well If it hadn t been for the media the print media and television the civil righ...
Mikey B. Mikey B.	says:
This is another significant work on the Civil Rights era Its overriding theme is how the media newsprint and T.V played an essential role in presenting to the general public the sordid racist state that existed in the Southern U.S and by exposing this helped to bring progress to ameliorate the conditions Without the media it is doubtful that the racial climate in the Southern U.S would have improved it certainly would have takentime John Lewis said that without the presence of t This is another significant work on the Civil Rights era Its overriding them...
Annie Annie	says:
This is a must read book It s entertaining, insightful, and still relevant in 2014 I started reading this book before any of the Ferguson protests started, then took a break The book is not a light read I found myself going back to the book in the midst of the protests and found many parallels ...
Patrick Sprunger Patrick Sprunger	says:
The subtitle of The Race Beat reads The press, the civil rights struggle, and the awakening of a nation. The genius of Jim Crow and key to its longevity was its ability to operate undetected Many of the defenders of the fortress segregation of the South lent support without ever knowing the true shape of the institution Most Southerners were like people with their noses pushed up to the edge of an iceberg From their vantage point, they had no way of knowing the hulk s true size and sh...
Alice Alice	says:
The Race Beat tells the story of the press, its coverage of the civil rights movement, and its importance in effecting change by bringing to the nation s attention the wrongs of segregation The argument was compelling, exciting, not too I m banging you over the head, and it made an important point about the importance of the press The writing was easy to read, elegant, but nothing extraordinary It was the content that hooked me The stories within the larger story were fascinating.Just a The Race Beat tells the story of the press, its coverage of the civil rights movement, and its importance in effecting change by bringing to the nation s attention the wrongs of segregation The argument was compelling, exciting, not too I m banging you over the head, and it made an important point about the importance of the press The writing was easy to read, elegant, but nothing extraordinary It was the content that hooked me The stories within the larger story were fascinating.Just a few highlights the Emmett Till trial the attempted enrollments of Autherine Lucy and J Meredith coverage of the Little Rock Nine the role of the press in protecting riders in the Freedom Rides the Alabama libel suit against Salisbury and the New York Times which threatened freedom of the press reporter Chancellor fending off a bunch of thugs with the threat that the whole nation would kn...
David David	says:
If you have to read this for a class or otherwise study this book, don t try to beat the system by listening to the audiobook instead, because bewilderingly long lists of people and newspapers go whizzing by fairly frequently They re hard to keep track of But for personal edification while driving, cooking, or exercising, this audiobook is first rate However, vivid descriptions of beatings and other assorted violent mayhem are wince inducing, which might draw odd looks from the person on the If you have to read this for a class or otherwise study this book, don t try to beat the system by listening to the audiobook instead, because bewilderingly long lists of people and newspapers go whizzing by fairly frequently They re hard to keep track of But for personal edification while driving, cooking, or exercising, this audiobook is first rate However, vivid descriptions of beatings and other assorted violent mayh...
Craig Werner Craig Werner	says:
It s not often I read something about the African American freedom movement that adds something important to my sense of the big picture, but The Race Beat did Roberts writes with a keen sense of the story that illuminates the larger tensions that emerge when you focus on the way that the movement was presented to its various publics some in the places the events w...
Martha Martha	says:
Staggeringly good I know, they don t give out Pulitzers for nothing Tells many now familiar stories of the civil rights movement, but from a totally new perspective, weaving them together with stories of the men who covered them their backgrounds, their personalities, their struggles with the movement and what it meant to them and to the nation as a whole And, mixed in with all that is consistently pointed media analysis, from illumination of t...
Katie Wood Katie Wood	says:
What I liked so much about this book is not only its unique perspective, but also that it tells the story of the civil right movement as a page turner Even though we know how history turned out, at eac...
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text:
What does all American mean in college football?
It mean to be the best player at your position
Q: What does all American mean in college football?
Which president was an all American football player in college?
Gerald Ford was an All-American center for the University of Michigan.
When did All American Football happen?
All American Football happened in 1989.
When was All American Football created?
All American Football was created in 1989.
Which sport has the most athletic academic all-American players?
Football because they have the largest roster of all college teams.
Do all college football players get to play in the NFL?
No. Not all college football players go to the NFL, not even all division 3 college football players go into the NFL
When was College Football All-America Team created?
College Football All-America Team was created in 1889.
What does AA mean during Colin Powell's years in college?
What us president was an outstanding college football player?
When was All American Football League created?
All American Football League was created in 2007.
Who is the American opera singer who was an outstanding college football player?
Bass Morris Robinson was a two time All American offensive lineman at The Citadel.
What football college team can you walk on to?
What Percent of college football players that get drafted?
Approximately 7% of all college football players get drafted into the NFL.
What liked more rugby or football?
If by football you mean soccer, then football would be more liked. If by Football you mean American Football, then Football would still be more liked. It's all a matter of personal opinion. Some people may like football more (regardless if it is proper football (soccer) or American football (as in the NFL)), while others, still, may enjoy rugby more.
What college has produced the most All American football players?
University of Southern California - 156 as of 2008 university of oklahoma-142
Who is the all time leading rusher in college football in 2008 season?
The all time leading rusher in college football is now nate kmic from the college of mount union
How friendly is university of Georgia football are with Georgia military college football?
What is the all-time LSU vs SEC record in football?
Are there American football fans all over the world?
== American Football Fans Worldwide = Yes.
Who were All Americans in college football in the 1800s?
Click on the 'College Football Awards History' link on this page to see the All Americans from 1889 through 2006 plus the winners of other prestigious awards in college football.
What college has the most losses all-time in college football?
Who were the college football national champions?
You can find all champions of the National College Football team on the website of Wikipedia. College football is the highest level of collegiate play in the United States.
Longest college football winning streak?
universality of Wisconsin-Whitewater has the longest winning streak in all of college football
All time wins in college football?
What is the total college football player in all?
about 680
Most college football wins all time?
The University of Michigan has the most college football wins of any NCAA team.
How many ugs are you 50 mgs?
What is chaismata and when do they form?
Are the heads on a 98 and 99 v8 5.7L the same?
What are the three major sectors in which water use is divided?
How do you write 6.23 in division?
How do you get 10000000000 kincash?
What are two caracteristics about bormine?
Does Berkekey accept transcripts with a W on it?


Regular attendance is vital to ensure that all children can achieve their full potential. Missing out on lessons leave children vulnerable to falling behind and with gaps in their learning. At St Andrew's we follow the guidance on taking family holidays in term time, which was updated by the Government in 2018.


text:
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text:
8Bitone for iPhone
iOS Central
By John Brandon, Macworld	 Jul 6, 2009 7:18 am PDT
A few iPhone apps are just a bit… unworldly. Leaf Trombone ( ) is one example—it's basically Guitar Hero with leaves. Shazam ( ), which records audio clips and reports back on the song and artist name, is another. They seem like magic on an LCD screen.
Entering this short list of phenoms is 8Bitone Micro Composer, a $9 audio sequencer that proves how software on the iPhone can mimic a physical synthesizer. In fact, 8Bitone almost works better than an analog synthesizer, such as the Roland SH-201, because you can quickly load and save audio clips, create unique sounds. And let's face it: the iPhone is a lot more portable.
8-Bit Is Enough: Relive that old-school sound with 8Bitone, an audio sequencer that recreates the synthesized sounds of yesteryear.
The app, created by Yudo (the same Japanese company behind Vocoder SV-5 and Rectools02), is remarkable because it functions exactly like an old-school 8-bit computer—think: life before the IBM PC and original Mac. 8Bitone lets you generate sounds that imitate—in a wonderfully accurate way—the Nintendo console systems of yesteryear. The app has three main modes: a piano roll screen for composing a track, a sequencer for assembling the tracks, and a synthesizer mode for creating audio samples—basically, the short tones you add to the tracks.
8Bitone only supports four tracks, but that's plenty for just about any micro-music composer. Once you create a song, you can save it and load it for later playback. The app is self-contained as far as I can figure out—you can't export the song to iTunes, for example. You could easily connect the iPhone headphone output to your computer and record clips that way.
If 8Bitone just let you punch in audio patterns and arrange them into a song, it would still be worthwhile. The tool goes much farther: there's a full synthesizer that lets you adjust tones in infinite ways. (The app does require some understanding of how a synthesizer works.)
There are controls for oscillator (the sound signal itself) and envelope—e.g., parameters of the signal, such as the sustain (how long the sample plays), decay (how long it takes before the sample reaches a full sound level), and release (how long the sample plays after you release the key). You can also control the arpeggio, which is a musical term that describes how notes are played in rapid succession rather than all at once. So, with arpeggio, you can mimic the "game over" sound of most Nintendo games—a series of quick audio blips.
So, why would anyone use 8Bitone? For one thing, it's just fun to create a song and learn how all of the control works. Composing a full song takes a couple of hours at least, and if you're serious about it, the final result is satisfying in its own right. I could also see indie bands like The Flaming Lips or Animal Collective using this tool to create unusual audio patterns.
The tool is not entirely perfect. Some of the buttons look grayed out, even though they are actually available. The app has a high learning curve, especially if you are not a musician. Fortunately, Yudo offers quite a few tutorials (there is no voice over, so they are visual how-tos) to help you learn the basics.
This is one of those unusual apps that's hard to explain, and I'll admit that it's not for everyone. But what makes 8Bitone so interesting is how it packs a real old-school synthesizer onto the iPhone, and how the controls and recording options are so intense—you can get lost in audio creation, which makes a plane ride or long road trip go a lot faster. For anyone with a passing interest in learning about or creating music, 8Bitone is worth a serious look.
8Bitone Micro Composer is compatible with any iPhone or iPod touch running the iPhone 2.2.1 software update.
[John Brandon is a 20-year veteran Mac user who used to run an all-Mac graphics department.]