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

The holy grail of contemporary theoretical physics is
the synthesis of quantum mechanics and general relativity.
String theory provides an arena to explore this synthesis
within a perturbatively self--consistent framework. The
important advantage of string theory is that its internal
consistency conditions mandate the appearance of the gauge
and matter structures that exist in the observed world,
hence facilitating the development of a phenomenological
approach. It seems likely that there exist numerous
mathematical structures that can be used to
developed consistent theories of quantum gravity,
and that mathematical consistency alone will not
be sufficient for the development of a physically relevant
theory, {\it i.e.} one that is relevant for experimental
observations. It is therefore sensible to construct string models
that aim to reproduce the physical characteristics of the
observed physical world. Such toy models can in turn be
used to explore the mathematical structures underlying string
theory and their possible relation to observable phenomena.
Of particular interest is the relation of string vacua to their
effective field theory limits. In this respect it should be noted
that contemporary understanding of string theory is confined to
its static limits. Dynamical questions in nature, such as the
cosmological evolution in the early universe, are explored
at present primarily in the effective field theory limit.

Given the vast space of potential string vacua, it makes further
sense to study classes of models that exhibit certain common structures
and that produce examples of quasi--realistic phenomenological models.
While it is folly to expect at present any of the models to be fully realistic,
it is feasible that some characteristics of a given class of models may
be imprinted in the observable data. The task of string phenomenology is to
identify such imprints in different classes of string compactifications,
and how the experimental data may discern between them.
One class of string compactification that have been explored in
this spirit are the $\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifolds.

The $\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifolds have been studied in the
free fermionic formulation of the heterotic string \cite{fff}
since the late eighties. They produce a large space of
quasi--realistic three generation models with different
unbroken $SO(10)$ and $E_6$ subgroups
\cite{fsu5, fny, alr, slm, lrs, acfkr, su62, frs,
  slmclass, lrsclass, lrsfertile}. Quasi--realistic
$\mathbb{Z}_2\times \mathbb{Z}_2$ orbifold models have also been constructed by using the
free bosonic formulation \cite{stefanetal}. The two formulations
provide complementary tools to explore the same physical spaces,
and detailed dictionaries have been developed to translate models
from one formulation to the other \cite{z2xz2}. The $\mathbb{Z}_2\times \mathbb{Z}_2$
orbifolds exhibit a rich symmetry structure, which has been the interest
of numerous mathematical studies \cite{mathematical}.
The rich symmetry structure of the $\mathbb{Z}_2$ orbifold may also be relevant
for the Standard Model flavour problem \cite{fh, nillesflavours}.
Indeed, it was demonstrated that quasi--realistic fermion mass and
mixing spectrum may be obtained from the fermionic $\mathbb{Z}_2\times \mathbb{Z}_2$
orbifolds \cite{fh}. The $\mathbb{Z}_2\times \mathbb{Z}_2$ orbifold compactifications
have also been studied in other string limits \cite{as}.

While the majority of studies of phenomenological string models
have been of $N=1$ supersymmetric vacua, non--supersymmetric
compactifications have been of increased interest over the past
decade. The starting point for most of these non--supersymmetric
constructions is the $SO(16)\times SO(16)$ heterotic string in ten
dimensions \cite{dh, itoyama, nonsusy, aafs, ADM, FR}. However,
it is well known that string theory also admits solutions that are
tachyonic in ten dimensions \cite{dh, kltclas, gv}. It was recently
argued that these ten dimensional vacua may also serve as starting points for
the constructions of viable phenomenological models, provided that the
tachyonic modes are projected from the physical spectrum in the
phenomenological four dimensional models \cite{spwsp, stable, so10tclass}.
Indeed, a tachyon free three generation Standard--like Model
was constructed in this class \cite{stable},
whereas in ref. \cite{so10tclass} a broad classification of models
with unbroken $SO(10)$ gauge group was presented
including the analysis of their vacuum energy, and
numerous models that satisfy the condition $n_b=n_f$, {\it i.e.} models
with equal numbers of massless bosons and fermions.

In this paper, we extend the analysis of the aforementioned classification
of this class of vacua. Our interest here is in the existence of models
that do not contain any fermions at all. Such models are known in the
literature as type 0 models and have been of interest in other string theory
limits \cite{type0string}. Such models are of particular interest to explore
the boundaries of the space of $\mathbb{Z}_2\times \mathbb{Z}_2$ orbifold compactifications.
It is plausible that progress on some of the phenomenological
issues in string theory, in particular in relation to the
cosmological evolution and vacuum selection, will be obtained
by improved understanding of these vacua. Moreover, it is likely
that further insight
can be achieved by exploring some of the features of these vacua in
connection with the phenomenological string vacua.
In this paper we therefore pursue this line of investigation. We present
several type 0 models in this class. We further adapt the systematic
classification method that was developed using the free fermionic rules
\cite{gkr, fknr, fkr, acfkr, frs, slmclass, lrsclass, lrsfertile, so10tclass}
to this class of models. This requires careful analysis of tachyonic
states that proliferate
in these configurations. While we do not find any model which
is completely free of tachyonic states, we present a model with a minimal
set of tachyonic states.
Similar to the
analysis in ref. \cite{FR}, we may explore the possibility that these
tachyonic states become massive when the moduli is moved away from
the free fermionic point.
Another issue that we analyse in detail is the calculation of the vacuum
energy and the finiteness properties of the string one--loop amplitude.
Naturally, these are divergent due to the existence of tachyonic states.
However, once the tachyonic states are removed by hand the amplitudes are
finite and exhibit a form of misaligned supersymmetry.

Our paper is organised as follows: in Section \ref{model} we present an explicit example of a type 0 model that does not contain any massless fermionic states.
    In Sections \ref{analyticConds}--\ref{equivalence}
we show that the type 0 constraints are in fact very
restrictive and showing that for the chosen set of basis vectors the type 0 model is
in fact unique. Hence, we demonstrate the existence of a huge redundancy in the space of
Generalised GSO phases that span the models. We show that the type 0 model in this
configuration does contain a tachyonic state that transform in the vector representation
of the hidden sector gauge group. A question of interest that we undertake in the following
sections is the existence of a tachyon free type 0 model. We investigate this question by adopting the free fermionic classification methodology to these models that enables the scan
of large space of models and the extraction of models with specific characteristics. We explore
the existence of tachyon free type 0 models in both $\tilde S$--models as well as
$S$--models that utilise the SUSY generating basis vector, Our scan yields a null result,
which suggests that such a tachyon free model may only exist away from the free fermionic
point. In Section \ref{MSSec} we demonstrate the existence of a misalligned supersymmetry
in the type 0 models that guarantee the finiteness of the one--loop amplitude, aside from
the divergence due to the tachyonic states. Section \ref{conclude} concludes our paper.

\section{Type 0 $\mathbb{Z}_2\times \mathbb{Z}_2$ Heterotic String Orbifold}\label{model}

In the free fermionic contruction \cite{fff} models are specified
in terms of boundary condition basis vectors and one--loop Generalised
GSO phases. Details of the formalism are to be found in the literature
and are not repeated here. We adopt the conventional notation used in the
free fermionic constructions
\cite{fsu5, fny, alr, slm, lrs, gkr, acfkr, su62, frs, slmclass, lrsclass,
  lrsfertile}. The first model that we present uses the {${\overline{\rm NAHE}}$}--set
that was introduced in \cite{spwsp, stable}. In this set the basis
vector $S$ that generates spacetime supersymmetry in NAHE--based models
\cite{nahe} is augmented with four periodic right--moving fermions,
which amounts to making the gravitinos massive. This introduces a general
$S\rightarrow {\tilde S}$ map in the space of models that was discussed in
detail in ref. \cite{so10tclass}. The set of basis vectors is given by
\begin{align}\label{basis}
\mathds{1}&=\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, w^{1,\dots,6}\ | \ \overline{y}^{1,\dots,6},\overline{w}^{1,\dots,6},
\overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,8}\},\nonumber\\
\tilde{S}&=\{{\psi^\mu},\chi^{1,\dots,6} \ | \ \overline{\phi}^{3,4,5,6}\},\nonumber\\
{b_1}&=\{\chi^{34},\chi^{56},y^{34},y^{56}\; | \; \overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\\
{b_2}&=\{\chi^{12},\chi^{56},y^{12},w^{56}\; | \; \overline{y}^{12},
\overline{w}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\
{b_3}&=\{\chi^{12},\chi^{34},w^{12},w^{34}\; | \; \overline{w}^{12},
\overline{w}^{34},\overline{\eta}^3,\overline{\psi}^{1,\dots,5}\},\nonumber\\
z_1&=\{\overline{\phi}^{1,\dots,4}\},\nonumber\\
x&=\{\overline{\psi}^{1,\dots,5}, \overline{\eta}^{1,2,3}\}.\nonumber
\nonumber
\end{align}
A model may then be specified through the assignment of modular invariant GGSO phases $\CC{v_i}{v_j}$ between the basis vectors. A type 0 configuration arises for the assignment
{\begin{equation}
\small
\CC{v_i}{v_j}=
\begin{blockarray}{ccccccccc}
&\mathbf{1}& \tilde{S} & b_1 & b_2&b_3&z_1 & x \\
\begin{block}{c(rrrrrrrr)}
\mathbf{1}& 1&  1& -1& -1& -1& -1& -1\ \\
\tilde{S} & 1& -1& -1& -1& -1&  1&  1\ \\
b_1       &-1&  1& -1& -1& -1&  1&  1\ \\
b_2       &-1&  1& -1& -1& -1&  1&  1\ \\
b_3       &-1&  1& -1& -1& -1&  1&  1\ \\
z_1       &-1& -1&  1&  1&  1& -1&  1\ \\
x         &-1&  1& -1& -1& -1&  1&  1\ \\
\end{block}
\end{blockarray}
\label{ggsophases}
\end{equation}}

All models in this basis will have gauge bosons arising from the Neveu--Schwarz (NS) sector that produce the
vector bosons of a $SO(10)\times U(1)^3\times SO(4)^3\times
SU(2)^8$ gauge symmetry. Additional vector bosons may arise from the sectors
in the $z_1$,
$z_3={\bf 1}+{\tilde S}+b_1+b_2+b_3=\{{\bar\phi}^{1,2,7,8}\}$
and
$z_4={\bf 1}+{\tilde S}+b_1+b_2+b_3+z_1=\{{\bar\phi}^{3,4,7,8}\}$, which can affect the observable and the hidden gauge group factors or just the hidden, depending on the right-moving oscillator. A solely observable gauge enhancement may also arise from the sector $x$. In the above model, the hidden $SU(2)^8$ gauge symmetry
is enhanced to $SO(16)$ by vector bosons arising
in $z_1$, $z_3$ and $z_4$.
The four dimensional
gauge group is therefore
$$SO(10)\times U(1)^3\times SO(4)^3\times SO(16).$$
The NS--sector and the three enhancement sectors above produce in total
sixteen tachyonic states that transform in the 16 representation
of the hidden $SO(16)$ gauge symmetry. The vectorial fermionic sectors in the model are
\begin{eqnarray}
\label{VectFerms}
& &{\tilde S};\nonumber\\
& &{\tilde S} +z_1;\\
& &{\tilde S}+z_4\sim S+z_2,\nonumber
\end{eqnarray}
while the massless fermionic spinorial sectors are
\begin{eqnarray}
\label{SpinFerms}
& &{\tilde S}+b_{1,2,3}+x;\nonumber\\
& &{\tilde S}+b_{1,2,3}+x+z_1;\\
& &{\tilde S}+b_{1,2,3}+x+z_4 ;\nonumber\\
& &{\tilde S}+z_3\sim S+z_1+z_2;\nonumber
\end{eqnarray}
where we defined $z_2=\{{\bar\phi}^{5, \cdots, 8}\}$
and $S=\{\psi^{1,2}, \chi^{1,\cdots, 6}\}$, neither of which are
basis vector combinations in the additive group.
We note that the $S$--vector coincides with the supersymmetry generator
in supersymmetric free fermionic models, as well as those that are
compactifications of the ten dimensional $SO(16)\times SO(16)$
heterotic string.
These definitions comply with the terminology used in the classification
of the supersymmetric free fermionic heterotic string models.
We emphasise that the absence of the $S$--vector from the additive group is
the crucial feature of the $\tilde S$--models. As discussed in refs.
\cite{spwsp, stable, so10tclass} the absence of the $S$--vector is the characteristic
property of vacua that descend from the tachyonic ten dimensional vacua.
The massless states from all the fermionic sectors are projected out from
the physical spectrum by the choice of GGSO phases in eq. (\ref{ggsophases}).
In addition to the NS--sector, sectors giving rise to spacetime massless
bosonic states are
\begin{eqnarray}
& &x;\nonumber\\
& &z_{1,3,4};\nonumber\\
& &b_{1,2,3};\nonumber\\
& &b_{1,2,3}+x;\label{Bosons}\\
& &b_{1,2,3}+x+z_1;\nonumber\\
& &b_{1,2,3}+x+z_3;\nonumber\\
& &b_{1,2,3}+x+z_1+z_3.\nonumber
\end{eqnarray}
They give rise to scalar spacetime bosons
that transform in representations of the four dimensional gauge symmetry.
They are of no particular interest here and we do not list their detail explicitly.

In Section \ref{StModels} and \ref{SModels} we perform a more general
search for similar type 0 heterotic string models using the
free fermionic classification methodology.
In particular, we search for type 0
models without tachyons. Our search is conducted
using the ${\tilde S}$ based models as well as models that use $S$. First, however, it is interesting to study a bit more closely the basis (\ref{basis}), and what additional constraints type 0 vacua may satisfy.

\subsection{Analytic conditions on type 0 vacua} \label{analyticConds}

Since we are interested in the construction and analysis of type 0 vacua, we focus on the massless
fermionic sectors and will seek to project them out, leaving only bosonic states at the massless level. The Hilbert space in the free fermionic construction is given by the collection of GGSO-projected states $\ket{S_\xi}$
\begin{equation}
    \mathcal{H}=\bigoplus_{\xi\in\Xi}\prod^{k}_{i=1}
\left\{ e^{i\pi v_i\cdot F_{\xi}}\ket{S_\xi}=\delta_{\xi}
\CC{\xi}{v_i}^*\ket{S_\xi}\right\}.
\end{equation}
For the type 0 case, this will only have contributions from sectors $\xi$ in the additive space $\Xi$ with bosonic spin statistic index $\delta_{\xi}=+1$ at the massless level.

For the basis (\ref{basis}), These definitions comply with the terminology used in the classification
of the supersymmetric free fermionic heterotic string models. In particular, we note that the $S$--vector coincides with the supersymmetry generator
in supersymmetric free fermionic models, as well as those that are
compactifications of the ten dimensional $SO(16)\times SO(16)$
heterotic string which we analyse in Section \ref{SModels}.

However, since we want to find choices of GGSO phases for which the fermionic sectors may be projected out to leave type 0 vacua, it is worth exploring explicitly the analytic conditions on their projection. Another important part of the analysis will be to consider the presence of tachyonic sectors in our models which we turn to in Section \ref{tachAnalysis}. Due to our basis being relatively simple, writing down analytic conditions for generating type 0 models seems tractable \textit{a priori} and we will see that, indeed, it is entirely solvable.

In order to project the fermionic massless sectors given in equations (\ref{VectFerms}) and (\ref{SpinFerms}) we can write down analytic conditions from the GGSO projection equation for the existence of type 0 models. For the fermionic vectorial sectors (\ref{VectFerms})
\begin{align}
\{\bar{y},\bar{w}\}\ket{\tilde{S}} \text{ projected }
\iff & \left( 1-\CC{\tilde{S}}{x}\right)\left( 1-\CC{\tilde{S}}{z_3}\right)=0 \label{VF1} \\
\{\bar{\psi}^{1,...,5},\bar{\eta}^{1,2,3}\}\ket{\tilde{S}} \text{ projected }
\iff & \left( 1+\CC{\tilde{S}}{x}\right)\left( 1-\CC{\tilde{S}}{z_3}\right)=0 \label{VF2} \\
\{\bar{\phi}^{1,2,7,8}\}\ket{\tilde{S}} \text{ projected }
\iff & \left( 1-\CC{\tilde{S}}{x}\right)\left( 1+\CC{\tilde{S}}{z_3}\right)=0 \label{VF3} \\
\{\bar{y},\bar{w}\}\ket{\tilde{S}+z_1} \text{ projected } \iff & \left( 1-\CC{\tilde{S}+z_1}{x}\right)\left( 1-\CC{\tilde{S}+z_1}{z_4}\right)=0 \label{VF4} \\
\{\bar{\psi}^{1,...,5},\bar{\eta}^{1,2,3}\}\ket{\tilde{S}+z_1} \text{ projected }
\iff & \left( 1+\CC{\tilde{S}+z_1}{x}\right)\left( 1-\CC{\tilde{S}+z_1}{z_4}\right)=0 \label{VF5} \\
\{\bar{\phi}^{3,4,7,8}\}\ket{\tilde{S}+z_1} \text{ projected }
\iff & \left( 1-\CC{\tilde{S}+z_1}{x}\right)\left( 1+\CC{\tilde{S}+z_1}{z_4}\right)=0 \label{VF6} \\
\{\bar{y},\bar{w}\}\ket{\tilde{S}+z_4} \text{ projected }
\iff & \left( 1-\CC{\tilde{S}+z_4}{x}\right)\left( 1-\CC{\tilde{S}+z_4}{z_1}\right)=0 \label{VF7}
\end{align}
\begin{align}
\{\bar{\psi}^{1,...,5},\bar{\eta}^{1,2,3}\}\ket{\tilde{S}+z_4} \text{ projected } \iff & \left( 1+\CC{\tilde{S}+z_4}{x}\right)\left( 1-\CC{\tilde{S}+z_4}{z_1}\right)=0 \label{VF8} \\
\{\bar{\phi}^{1,2,3,4}\}\ket{\tilde{S}+z_4} \text{ projected }
\iff & \left( 1-\CC{\tilde{S}+z_4}{x}\right)\left( 1+\CC{\tilde{S}+z_4}{z_1}\right)=0 \label{VF9}
\end{align}
Using the ABK rules on equations (\ref{VF1}), (\ref{VF2}) and (\ref{VF3}) we deduce that
 \begin{equation}\label{StConds}
        \CC{\tilde{S}}{x}=1 \ \text{ and }  \ \CC{\tilde{S}}{b_1}\CC{\tilde{S}}{b_2}\CC{\tilde{S}}{b_3}=-1
\end{equation}
using these results and the ABK rules on equations (\ref{VF4}), (\ref{VF5}) and (\ref{VF6}) gives the further results
\begin{equation}\label{z1Conds}
        \CC{z_1}{x}=1 \ \text{ and }  \ \CC{z_1}{b_1}\CC{z_1}{b_2}\CC{z_1}{b_3}=1.
\end{equation}
Finally, the first bracket of equations (\ref{VF7}), (\ref{VF8}) and (\ref{VF9}) implies the result
\begin{equation}\label{xConds}
         \CC{x}{1}\CC{x}{b_1}\CC{x}{b_2}\CC{x}{b_3}=1.
\end{equation}
Meanwhile, for the fermionic spinorial sectors (\ref{SpinFerms}) we have
\begin{align}
&\tilde{S}+b_i+x \text{ projected }\nonumber \\
\iff & \left( 1-\CC{\tilde{S}+b_i+x}{b_j+b_k+x}\right)\left( 1-\CC{\tilde{S}+b_i+x}{z_3}\right)=0 \label{SF1} \\
&\tilde{S}+b_i+z_1+x \text{ projected }\nonumber \\
\iff & \left( 1-\CC{\tilde{S}+b_i+z_1+x}{b_j+b_k+x}\right)\left( 1-\CC{\tilde{S}+b_i+z_1+x}{z_4}\right)=0 \label{SF2}  \\
&\tilde{S}+b_i+z_4+x \text{ projected } \nonumber\\
\iff & \left( 1-\CC{\tilde{S}+b_i+z_4+x}{b_j+b_k+x}\right)\left( 1-\CC{\tilde{S}+b_i+z_4+x}{z_1}\right)=0  \label{SF3} \\
&\tilde{S}+z_3 \text{ projected } \nonumber\\
\iff & \left( 1-\CC{\tilde{S}+z_3}{b_1+b_2+b_3+x}\right)\left( 1-\CC{\tilde{S}+z_3}{x}\right)=0  \label{SF4}
\end{align}
where $i\neq j \neq k=1,2,3$. Using results (\ref{StConds}),(\ref{z1Conds}) and (\ref{xConds}) in equation (\ref{SF3}) implies that
\begin{equation}\label{z1Fixed}
\CC{z_1}{b_1}=\CC{z_1}{b_2}=\CC{z_1}{b_3}=1
\end{equation}
and using results (\ref{StConds}) and (\ref{xConds}) in equation (\ref{SF1}) allows us to deduce the results
\begin{equation}\label{bibjConds}
 \CC{\tilde{S}}{b_1}\CC{b_1}{b_2}\CC{b_1}{b_3}=-1, \ \ \ \CC{\tilde{S}}{b_2}\CC{b_2}{b_1}\CC{b_2}{b_3}=-1 \ \ \ \text{and} \ \ \ \CC{\tilde{S}}{b_3}\CC{b_3}{b_1}\CC{b_3}{b_2}=-1
\end{equation}
since (\ref{StConds}) means there are only two independent equations here, we can use this result to fix  two of the phases: $\CC{b_1}{b_2},\CC{b_1}{b_3}$ and $\CC{b_2}{b_3}$.

Gathering together the results (\ref{StConds}), (\ref{z1Conds}), (\ref{xConds}), (\ref{z1Fixed}) and (\ref{bibjConds}) we find the following necessary and sufficient conditions on the projection of massless fermions within models derived from the basis (\ref{basis})
\boxalign{\begin{align}
\label{Type0Conds}
 \CC{\tilde{S}}{x}&=1, \ \ \ \CC{z_1}{x}=1,  \ \ \  \CC{z_1}{b_1}=\CC{z_1}{b_2}=\CC{z_1}{b_3}=1 \nonumber\\
 \CC{\tilde{S}}{b_1}&=-\CC{\tilde{S}}{b_2}\CC{\tilde{S}}{b_3} \nonumber\\
  \CC{x}{1}&=~~\CC{x}{b_1}\CC{x}{b_2}\CC{x}{b_3} \\
  \CC{b_2}{b_3}&=-\CC{\tilde{S}}{b_2}\CC{b_1}{b_2}\nonumber\\
  \CC{b_3}{b_1}&=-\CC{\tilde{S}}{b_2}\CC{\tilde{S}}{b_3}\CC{b_1}{b_2}\nonumber.
\end{align}}
These conditions mean that 9 of the 21 GGSO phases are fixed in order to obtain type 0 vacua. Hence, the number of possible type 0 models is reduced to $2^{12}=4096$.

Armed with the conditions (\ref{Type0Conds}) we can look now at the bosonic sectors (\ref{Bosons}) and in fact prove that all these sectors must appear in all 4096 possible type 0 models. To prove this we can go through the projection conditions as we did above for the fermionic sectors. In particular, taking the sectors $b_i$, for $i=1,2,3$ we get
\begin{equation}
b_i \text{ survives }
\iff \CC{b_i}{\tilde{S}+b_j+b_k}=1 \ \ \text{ and } \ \ \CC{b_i}{z_1} =1 \label{B1}
\end{equation}
these conditions coincide exactly with conditions (\ref{bibjConds}) and (\ref{z1Fixed}), respectively, from the projection of fermions analysis above. A similar result can easily be found for the other spinorial bosonic sectors: $b_{1,2,3}+x+z_1,b_{1,2,3}+x+z_3$ and $b_{1,2,3}+x+z_4$.

For the bosonic vectorial sectors: $b_i+x$, $i=1,2,3$ we have the conditions
\begin{align}
\{\bar{y},\bar{w},\bar{\psi}^{1,...,5},\bar{\eta}^{i},\bar{\phi}^{5,6}\}\ket{b_i+x} \text{ survives }
\implies & \frac{1}{4}\left( 1+\CC{b_i+x}{z_3}\right)\left( 1+\CC{b_i+x}{z_1}\right)=1 \label{B2} \\
\{\bar{\phi}^{1,2}\}\ket{b_i+x} \text{ survives }
\implies & \frac{1}{4}\left( 1-\CC{b_i+x}{z_3}\right)\left( 1-\CC{b_i+x}{z_1}\right)=1 \label{B3} \\
\{\bar{\phi}^{3,4}\}\ket{b_i+x} \text{ survives }
\implies & \frac{1}{4}\left( 1+\CC{b_i+x}{z_3}\right)\left( 1-\CC{b_i+x}{z_1}\right)=1 \label{B4} \\
\{\bar{\phi}^{7,8}\}\ket{b_i+x} \text{ survives }
\implies & \frac{1}{4}\left( 1-\CC{b_i+x}{z_3}\right)\left( 1+\CC{b_i+x}{z_1}\right)=1 \label{B5}
\end{align}
the type 0 conditions (\ref{Type0Conds}) guarantee that $\CC{b_i+x}{z_3}=1$ and $\CC{b_i+x}{z_1}=1$ and thus the first case survives.

Finally let us show that the bosonic sector $x$ survives. In order to make this sector massless there must be a left moving oscillator, which could make it a gauge boson if this oscillator is $\psi^\mu$. However, since for type 0 models $\CC{x}{\tilde{S}}=1$ only the states of the type $\{y^i/w^i\}_{1/2}\ket{x}$ can survive, which they must do due to $\CC{x}{z_1}=\CC{x}{z_3}=1$.

For the $z_1$ massless sector the conditions for type 0 models necessitate that states of the type $\{y/w\}\{\bar{y}/\bar{w}\}\ket{z_1}$ and extra gauge bosons of the type $\{ \psi^\mu\}\{\bar{\phi}^{5,6,7,8}\}\ket{z_1}$ survive. Similarly for the $z_3$ massless sector type 0 models must have states of the type $\{y/w\}\{\bar{y}/\bar{w}\}\ket{z_3}$ and extra gauge bosons of the type $\{ \psi^\mu\}\{\bar{\phi}^{3,4,5,6}\}\ket{z_3}$. Finally, for the $z_4$ massless sector type 0 models must have states of the type $\{y/w\}\{\bar{y}/\bar{w}\}\ket{z_4}$ and extra gauge bosons of the type $\{ \psi^\mu\}\{\bar{\phi}^{1,2,5,6}\}\ket{z_4}$. Therefore, all type 0 models derived from this basis (\ref{basis}) have a hidden sector enhancement of $SU(2)^4\rightarrow SO(16)$.

This analysis tells us that all 4096 possible type 0 models contain all bosonic sectors \ref{Bosons} with the specific set of oscillators given above. In other words, their massless spectra are identical. Doing the counting of all the bosonic states can be shown to give 4264, which is thus the constant term in the $q$-expansion of the partition function in all 4096 cases. Having seen how restrictive the type 0 conditions $\ref{Type0Conds}$ are at the massless level it makes sense to analyse what happens with the tachyonic sectors for our type 0 models.

\subsection{Tachyon analysis for type 0 vacua}\label{tachAnalysis}
The tachyonic sectors for models derived from the basis (\ref{basis}) come from the sectors $\ket{z_1}, \ \ket{z_3}$ and $\ket{z_4}$ as well as the untwisted tachyon of the NS sector. We can immediately see that all vacua in this basis will contain an untwisted tachyon $\{\bar{\phi}^{5,6}\}\ket{NS}$. This can be seen as being related to the absence of $z_2=\{\bar{\phi}^{5,6,7,8}\}$ in the basis which would allow for the projection of this tachyon since we would be equipped with the GGSO projection with the phase $\CC{NS}{z_2}=1$.

In regard to the tachyons from the sectors $\ket{z_1}, \ \ket{z_3}$ and $\ket{z_4}$, we see that the type 0 conditions (\ref{Type0Conds}) necessitate their presence. For example, all phases that could project the $z_1$ tachyon: $\CC{z_1}{b_1},\CC{z_1}{b_2}, \CC{z_1}{b_3}, \CC{z_1}{x}$ are all equal to +1 and thus leave it in the Hilbert space. Therefore, we conclude that all type 0 in this construction contain the tachyons from the sectors $\ket{z_1}, \ \ket{z_3}$ and $\ket{z_4}$, along with the model-independent untwisted tachyon  $\bar{\phi}^{5,6}\ket{NS}$.

\subsection{Equivalence of type 0 models}\label{equivalence}
Having shown that the massless spectrum and tachyonic sectors are identical for all the 4096 choices of GGSO phases consistent with the type 0 conditions (\ref{Type0Conds}), we might wonder whether these models are in fact identical at all massive levels. Calculating the partition function for all 4096 type 0 models proved that they indeed all have the same partition function
\begin{equation}\label{PF}
     Z = 2\bar{q}^{-1} + 16q^{-1/2}\bar{q}^{-1/2} + 4264 + 45056 q^{1/4}\bar{q}^{1/4} + \cdots
\end{equation}
and thus there is only one type 0 model in our construction with degeneracy 4096.
This is a good example of the non-uniqueness
of the free fermionic construction definition of a model since the partition function (\ref{PF}) is invariant under the 12 phases: $\CC{1}{\tilde{S}}$, $\CC{1}{b_1}$, $\CC{1}{b_2}$, $\CC{1}{b_3}$, $\CC{1}{z_1}$, $\CC{\tilde{S}}{b_2}$, $\CC{\tilde{S}}{b_3}$, $\CC{\tilde{S}}{z_1}$, $\CC{b_1}{b_2}$, $\CC{x}{b_1}$, $\CC{x}{b_2}$, $\CC{x}{b_3}$. This result will ultimately be related to the many symmetries underlying models defined by the basis (\ref{basis}).

\section{Classification of Type 0 $\tilde{S}$-Models}\label{StModels}
Having found that type 0, tachyonic models exist for the simple basis (\ref{basis}), we can consider a more general basis
\begin{align}\label{basisStTi}
\mathds{1}&=\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, w^{1,\dots,6}\ | \ \overline{y}^{1,\dots,6},\overline{w}^{1,\dots,6},
\overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,8}\},\nonumber\\
\tilde{S}&=\{{\psi^\mu},\chi^{1,\dots,6} \ | \ \overline{\phi}^{3,4,5,6}\},\nonumber\\
{T_1}&=\{y^{1,2},w^{1,2}\; | \; \overline{y}^{1,2},\overline{w}^{1,2}\},\nonumber\\
{T_2}&=\{y^{3,4},w^{3,4}\; | \; \overline{y}^{3,4},\overline{w}^{3,4}\},\nonumber\\
{T_3}&=\{y^{5,6},w^{5,6}\; | \; \overline{y}^{5,6},\overline{w}^{5,6}\},\nonumber\\
{b_1}&=\{\chi^{34},\chi^{56},y^{34},y^{56}\; | \; \overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\\
{b_2}&=\{\chi^{12},\chi^{56},y^{12},w^{56}\; | \; \overline{y}^{12},
\overline{w}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\
{b_3}&=\{\chi^{12},\chi^{34},w^{12},w^{34}\; | \; \overline{w}^{12},
\overline{w}^{34},\overline{\eta}^3,\overline{\psi}^{1,\dots,5}\},\nonumber\\
z_1&=\{\overline{\phi}^{1,\dots,4}\},\nonumber
\nonumber
\end{align}
where the $T_i$, $i=1,2,3$ allow for internal symmetric shift in the compactified coordinates around the 3 tori. The only other difference to the basis (\ref{basis}) is that $x$ is now a linear combination:
\begin{equation}
    x=b_1+b_2+b_3+T_1+T_2+T_3
\end{equation}
and we have the same combinations $z_3=1+\tilde{S}+b_1+b_2+b_3$ and $z_4=z_3+z_1$. We can further note that the space of independent GGSO phase configuration is now $2^{36}\sim 6 \times 10^{10}$ for this basis.

The addition of the $T_i$'s has some key consequences in relation to finding tachyon-free type 0 vacua. It multiplies the number of massless fermionic sectors and also increases the number of ways to project the (fermionic) sectors. Furthermore, we now have 15 tachyonic sectors: $z_1,z_3,z_4,T_i,z_1+T_i, z_3+T_i$ and $z_4+T_i$, $i=1,2,3$ rather than just the 3 for basis (\ref{basis}). We can notice that the model--independent NS tachyon $\{\bar{\phi}^{5,6}\}\ket{NS}$ remains in this construction so the minimal number of tachyons is to only have the NS tachyon.
\subsection{Fermionic sector analysis}
Using a similar methodology to Section \ref{analyticConds}, we wish to analyse the conditions on the projection of all fermionic sectors from these models. Due to increased size of the space of models from the added complexity of having of $T_{i=1,2,3}$ in the basis, we developed a computer algorithm to scan efficiently over the space of vacua and check for the absence of fermionic massless states.

We note that massless fermionic vectorials in these models arise from the sectors
\begin{eqnarray}
\label{VectFermsTi}
& &{\tilde S};\nonumber\\
& &{\tilde S} +z_1;\\
& &{\tilde S}+z_4\nonumber
\end{eqnarray}
and the massless fermionic spinorial sectors from
\begin{eqnarray}
\label{SpinFermsTi}
& &{\tilde S}+b_i+b_j+T_k+pT_i+qT_j;\nonumber\\
& &{\tilde S}+b_i+b_j+z_1+T_k+pT_i+qT_j;\\
& &{\tilde S}+b_{1,2,3}+x+z_4 ;\nonumber\\
& &{\tilde S}+z_3.\nonumber
\end{eqnarray}
Our computer algorithm can then be further applied to analyse the tachyonic sectors arising in type 0 models. The results of this computerised scan are presented in the following section.

\subsection{Results of classification}
By implementing the projection conditions on the massless fermionic sectors (\ref{VectFermsTi}) and (\ref{SpinFermsTi}) in a computer scan we can collect data for the number of fermionic states remaining in the Hilbert Space of a model and see how many are fermion-free and thus type 0. The distribution of the number of fermionic states for a scan of $10^7$ is displayed in Figure \ref{StNumFerms}. In this sample we find a total of 24508 which are free of fermionic states.

\begin{figure}[!htb]
\centering
\includegraphics[width=0.8\linewidth]{NumFermsGraphSt.eps}
\caption{\label{StNumFerms}\emph{Frequency plot for the number of fermionic states in a model from a sample of $10^7$ randomly generated GGSO configurations.}}
\end{figure}

In order to gather a slightly larger sample of type 0 models in this basis we take a larger sample of $10^8$ models which still does not take much computing time. From this sample, we find 245685 type 0 models which gives a probability $\sim 2.46\times 10^{-3}$ for type 0 vacua in the total space. We now wish to classify these type 0 configurations according to which tachyonic sectors remain in their spectra (along with the model-independent untwisted tachyon), as shown in Table \ref{StTachs}.
\begin{table}[!htb]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
&&&\\[-1.6ex]
$z_k$ Tachyon & $z_k+T_i$ Tachyon &$\{\bar{\lambda}^a\}\ket{T_i}$ Tachyon &Frequency\\[0.8ex]
\hline
0& 2& 2&    42773\\ \hline
1& 2& 1&   33513\\ \hline
1& 2& 2&    19402\\ \hline
1& 0& 2&    17405\\ \hline
1& 0& 1&   17140\\ \hline
1& 1& 2&  12056\\ \hline
0& 3& 1&    11996\\ \hline
3& 0& 1&     7141\\ \hline
0& 1& 3&     6044\\ \hline
3& 0& 2&     5708\\ \hline
1& 2& 3&     5575\\ \hline
1& 2& 0&     5175\\ \hline
1& 1& 1&     5170\\ \hline
1& 4& 2&     5071\\ \hline
0& 4& 2&     5017\\ \hline
0& 0& 2&     4262\\ \hline
0& 2& 3&     4253\\ \hline
1& 4& 1&     4226\\ \hline
3& 0& 0&     3827\\ \hline
0& 3& 2&     3405\\ \hline
0& 1& 1&     3389\\ \hline
1& 4& 0&    3322\\ \hline
1& 1& 3&     2625\\ \hline
1& 3& 3&     2179\\ \hline
0& 3& 3&     1774\\ \hline
0& 4& 3&     1724\\ \hline
3& 3& 2&     1713\\ \hline
1& 3& 2&     1631\\ \hline
3& 0& 3&     1529\\ \hline
3& 3& 3&     1168\\ \hline
1& 5& 2&      913\\ \hline
1& 5& 3&      888\\ \hline
0& 4& 1&      854\\ \hline
1& 0& 3&      840\\ \hline
1& 4& 3&      795\\ \hline
1& 6& 3&      583\\ \hline
0& 0& 3&      308\\ \hline
3& 6& 3&      291\\ \hline
\end{tabular}
\caption{\label{StTachs} \emph{Number of tachyonic sectors for 245685 type 0 $\tilde{S}$-models, where $k=1,3,4$, $i=1,2,3$ and $\bar{\lambda}^a$ is any right-moving oscillator with NS boundary condition.}}
\end{table}

These results clearly show that all type 0 models have both the model-independent untwisted tachyon and some combination of at least 2 twisted tachyonic sectors. One might wonder how general this result is since our sample size of $10^8$ only covers about $1:687$ models in the total space of GGSO phase configurations. Recalling the 4096 degeneracy factor from the analysis of models in the basis (\ref{basis}), we can reasonably suppose that type 0 models are highly constrained and degenerate also in the current construction where the $T_{i=1,2,3}$ are incorporated in the basis.

To see this we took $10^4$ type 0 models out from the 245685 total sample and calculated their partition functions and found a total of 109 distinct ones. In Figure \ref{Deg} a comparison between the degeneracy of these $10^4$ type 0 models and those of a random sample of $10^4$ models is shown and the number of different type 0 models are seen to converge fast to just over 100, This shows, just as in the earlier case, that the subspace of type 0 vacua is highly symmetric. This result strongly justifies the generality of our results from the $10^8$ sample for the tachyonic analysis and makes it highly likely that our 245685 type 0 models from the $10^8$ sample captures all such unique models. In Section \ref{MSSec}, we will further discuss the structure of these type 0 models from the point of view of the partition function and one-loop vacuum energy.

\begin{figure}[t]
\centering
\hspace{1cm}
\includegraphics[width=0.85\linewidth]{DegeneracyPlot.eps}
\caption{\emph{The degeneracy of models for a random sample of models versus type 0 models for a sample of $10^4$ models each. We see that the space of type 0 models is indeed highly degenerate. }}
\label{Deg}
\end{figure}

\section{Classification of Type 0 $S$--Models}\label{SModels}
Having explored a space of $\tilde{S}$-models in the previous section we now wish to do the same analysis for models deriving from the ten-dimensional $SO(16)\times SO(16)$ non-supersymmetric heterotic string, which we will refer to as S-models since their basis contains the SUSY--generating vector $S$. The precise basis we use is:
\begin{align}\label{basisS}
\mathds{1}&=\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, w^{1,\dots,6}\ | \ \overline{y}^{1,\dots,6},\overline{w}^{1,\dots,6},
\overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,8}\},\nonumber\\
S&=\{{\psi^\mu},\chi^{1,\dots,6} \},\nonumber\\
{T_1}&=\{y^{1,2},w^{1,2}\; | \; \overline{y}^{1,2},\overline{w}^{1,2}\},\nonumber\\
{T_2}&=\{y^{3,4},w^{3,4}\; | \; \overline{y}^{3,4},\overline{w}^{3,4}\},\nonumber\\
{T_3}&=\{y^{5,6},w^{5,6}\; | \; \overline{y}^{5,6},\overline{w}^{5,6}\},\nonumber\\
{b_1}&=\{\chi^{34},\chi^{56},y^{34},y^{56}\; | \; \overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\\
{b_2}&=\{\chi^{12},\chi^{56},y^{12},y^{56}\; | \; \overline{y}^{12},
\overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\
z_1&=\{\overline{\phi}^{1,\dots,4}\},\nonumber\\
z_2&=\{\overline{\phi}^{5,\dots,8}\},\nonumber
\nonumber
\end{align}
which is in fact identical to that used in ref. \cite{FR} and the same as that used in the supersymmetric classifications of \cite{gkr, fknr, fkr, acfkr, frs, slmclass, lrsclass, lrsfertile, so10tclass} up to the swap $T_{1,2,3}\rightarrow e_{1,...,6}$.
As noted in these works, we will make regular use of the important linear
combination $x$ \cite{xmap}, which appears as the combination
\begin{equation}
    x=1+S+\sum_{i=1}^3T_i+z_1+z_2
\end{equation}
in this basis and we further have the combination $b_3=b_1+b_2+x$ to give the generator of the third orbifold plane.

The untwisted gauge bosons in this construction generate a gauge group of $U(1)^6\times SO(10)\times U(1)^3 \times SO(8)^2$ and the full space of models is again given by the combinations of modular invariant GGSO phase configurations $2^{36}\sim 6\times 10^{10}$.

An important difference from the $\tilde{S}$-construction is that we do not have a model-independent tachyon as the NS tachyon is automatically projected. This leaves the door open for possible tachyon-free type 0 models.

Now we turn to the massless fermionic vectorial sectors which for our $S$-models arise from
\begin{eqnarray}
\label{SVectFerms}
& &S\nonumber\\
& &S+z_1 ;\\
& &S+z_2\nonumber\\
& & S+b_i+x+pT_j+qT_k\nonumber
\end{eqnarray}
and the massless fermionic spinorial sectors from
\begin{eqnarray}
\label{SSpinFerms}
& &S+x\nonumber\\
& &S+z_1+z_2\nonumber\\
& &S+b_i+pT_j+qT_k;\\
& &S+b_i+z_1+x+pT_j+qT_k ;\nonumber\\
& &S+b_i+z_2+x+pT_j+qT_k;\nonumber
\end{eqnarray}
where $i\neq j \neq k\in \{1,2,3\}$ and $p,q\in \{0,1\}$. We note that there are more fermionic sectors in this construction than in the $\tilde{S}$ case. In particular, the familiar $\mathbf{16}/\overline{\mathbf{16}}$ of $SO(10)$ from the sectors $S+b_i+pT_j+qT_k$ and vectorial $\mathbf{10}$ of $SO(10)$ from the sectors $(S+)b_i+x+pT_j+qT_k$ arise in this construction. However, for the $\tilde{S}$-models they were deliberately chosen to be absent and there are in fact no spinorial fermionic sectors with non-trivial representation under the $SO(10)$ observable gauge group factor.

As in the case of the $\tilde{S}$-models we use a computer algorithm to scan for type 0 configurations where these fermionic sectors are projected out. The results are presented in the following section.

\subsection{Results of classification}
As in the case of the $\tilde{S}$-models we generate a distribution for the number of fermionic states across a sample of $10^7$ randomly generated GGSO phase configurations. This is shown in Figure \ref{NumFermsS}. Comparing this to Figure \ref{StNumFerms} for the $\tilde{S}$ case we see a more dense distribution with more possible values for the fermionic states.
\begin{center}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.8\linewidth]{NumFermsGraphS.eps}\caption{\label{NumFermsS}\emph{Frequency plot for the number of fermionic states for $S$-models from a random sample of $10^7$ GGSO configurations.}}
\end{figure}
\end{center}

As in the $\tilde{S}$ case we generate a larger sample of type 0 models by taking a scan of $10^8$ GGSO configurations. From this scan we find 54590 type 0 models which is probability $\sim 5.46\times 10^{-4}$ which makes them approximately 5 times rarer than in the $\tilde{S}$ case. The likely reason for this and main difference in general between the $S$ case and $\tilde{S}$ case is the already mentioned fact that in the $S$-models we have more fermionic massless sectors to project.

Despite being rarer we already noted that we do not have any model-independent tachyons for these $S$-models which leaves the possibility of tachyon-free type 0 vacua open. The data for the numbers of tachyons is shown in Table \ref{STachs} and we see again that no tachyon-free models exist and that the minimal number of tachyonic sectors is 2, which always includes at least 1 vectorial tachyon of the type $\{\bar{\lambda}^a\}\ket{T_i}$.

\begin{table}[!htb]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
&&&\\[-1.6ex]
$z_k$ Tachyon & $z_k+T_i$ Tachyon &$\{\bar{\lambda}^a\}\ket{T_i}$ Tachyon &Frequency\\[0.8ex]
\hline
1& 1& 2&    11605 \\ \hline
1& 0& 2 &   10471\\ \hline
1& 1& 1 &   4431\\ \hline
1& 0& 3 &  4388\\ \hline
2& 0& 2 &  4066\\ \hline
2& 0& 1 &   3749\\ \hline
1& 0& 1 &    3363\\ \hline
1& 1& 3 &    2384\\ \hline
1& 2& 2 &   1870\\ \hline
2& 0& 3 &    1509\\ \hline
1& 2& 3 &    1318\\ \hline
1& 2& 1 &    1080\\ \hline
2& 2& 1 &     871\\ \hline
2& 2& 2 &     538\\ \hline
0& 1& 3 &     488\\ \hline
0& 1& 2 &     454\\ \hline
0& 2& 1 &     299\\ \hline
1& 3& 1 &     291\\ \hline
1& 3& 2 &     290\\ \hline
0& 0& 2 &    236\\ \hline
1& 3& 3 &     189\\ \hline
0& 4& 3 &     151\\ \hline
0& 2& 3 &     151\\ \hline
0& 2& 2 &     135\\ \hline
0& 0& 3 &     135\\ \hline
2& 4& 1 &    128\\ \hline
\end{tabular}
\caption{\label{STachs} \emph{Number of tachyonic sectors for 54860 type 0 $S$-models, where $k=1,2$ and $i=1,2,3$.}}
\end{table}

As in the case of the $\tilde{S}$-models, we observe a degeneracy in the space of type 0 models and from the sample of 54590 type 0 $S$-models, we found just 89 independent partition functions and the same convergence pattern as shown in Figure \ref{Deg}, therefore we can confidently claim that the lack of tachyon-free type 0 models is a generic result in this class of vacua. It is however worth remembering that our models are defined at a free fermionic point in moduli space and so it may be that such tachyonic instabilities may disappear when a model is translated into a bosonic language and defined away from the fermionic point. The process for doing this in the same basis as we are employing here for the $S$-models is detailed in ref. \cite{FR}.

\section{Misaligned Supersymmetry in Type 0 Models}\label{MSSec}

The partition function of string models encapsulates all the
information one knows about its structure, symmetries and spectrum.
Thus to fully understand these type 0 models it is essential to get a handle on their partition function. The analysis of the partition function is particularly instrumental in non-supersymmetric constructions, since it gives a complementary tool to count the total number of massless states, and its integration over the fundamental domain corresponds to the cosmological constant.

The partition function for free fermionic theories is given by the integral
\begin{equation}
    Z = \int_\mathcal{F}\frac{d^2\tau}{\tau_2^2}\, Z_B  \sum_{Sp.Str.} \CC{\alpha}{\beta} \prod_{f} Z \sqbinom{\alpha(f)}{\beta(f)},
    \label{ZInt}
\end{equation}
where $d^2\tau/\tau_2^2$ is the modular invariant measure and $Z_B$ is the bosonic contribution arising from the worldsheet bosons.  The sum over spin structures represents the contributions from the worldsheet fermions. The integral is over the fundamental domain of the modular group
$$ \mathcal{F} = \{\tau\in\mathbb{C}\,|\,|\tau|^2>1 \;\land\;|\tau_1|<1/2\}. $$
This ensures that only physically inequivalent geometries are counted. The above expression (\ref{ZInt}) specifically represents the one-loop vacuum energy of our theory and so we may refer to it as the cosmological constant.

The practical way to perform this integral is as presented in \cite{D,so10tclass} using the expansion of the $\eta$ and $\theta$ functions in terms of the modular parameter, or more conveniently in terms of $q\equiv e^{2\pi i \tau}$ and $\bar{q}\equiv e^{-2\pi i \bar{\tau}}$. This leads to a series expansion of the one-loop partition function
\begin{equation}
    Z = \int_\mathcal{F} \frac{d^2\tau}{\tau_2^3} \sum_{n.m} a_{mn} \, q^m \bar{q}^n .
    \label{QPF}
\end{equation}
Writing the partition function as above ensures that the $a_{mn}$ physically represent the difference between bosonic and fermionic degrees of freedom at each mass level, i.e. $a_{mn} = N_b - N_f$.  Further details on the $q$--expansions and the general calculation of these integrals can be found in Section 6 of \cite{so10tclass}.  As expected, on-shell tachyonic states, {\it i.e.} states with $m=n<0$, have an infinite contribution. On the other hand off--shell tachyonic states may contribute a finite value to the partition function. It is also important to note that modular invariance only allows states with $m-n\in \mathbb{Z}$.

In theories with spacetime supersymmetry, it is ensured that the bosonic and fermionic degrees of freedom are exactly matched at each mass level. That is, we necessarily have that $a_{mn} = 0$ for all $m$ and $n$, which in turn causes the vanishing of the cosmological constant as one expects. For our non-supersymmetric models, this level-by-level cancellation is not ensured and so such theories in general produce a non-zero value for $\Lambda$. It is, however, not obviously clear that they should produce finite partition functions, or what form the state degeneracies should take. It has, however, been shown \cite{MSUSY2,MSUSY, carlo}, that the partition functions of non-supersymmetric closed string theoroies possess a special feature called misaligned supersymmetry.

As one expects, the degeneracy of states grows rapidly going up the infinite tower of massive states. This growth, in theory, could counteract the suppression received from the decreasing contributions from the integrals of (\ref{QPF}) and cause divergences. The mechanism of misaligned supersymmetry, however, causes the states in the massive tower to oscillate between an excess of bosons and an excess of fermions. This behaviour is referred to as boson-fermion oscillation. Instead of cancelling level-by-level as in the supersymmetric case, the cancellation is misaligned causing the oscillation meaning a large positive contribution is followed by an even larger negative contribution and so on.

In the case of type 0 models presented above, due to the presence of on-shell tachyonic states, the partition function (\ref{QPF}) diverges. However, this divergence is contained purely in the tachyonic mode, i.e. the degeneracy of states $a_{mn}$ for $m,n>0$ still behaves in a similar fashion to any other non-tachyonic heterotic theory. This is also emphasised by the presence of misaligned supersymmetry in the massive spectra of our models. Such misalignment of the physical spectrum in heterotic theories is well documented in the literature \cite{MSUSY2,MSUSY,ADM,so10tclass}. It however remains to see whether this mechanism is replicated for type 0 heterotic theories described in previous sections. We find that indeed both sets of type 0 models generated by (\ref{basis}) and (\ref{basisStTi}) exhibit such misalignment of their on-shell massive tower of states. The misalignment pattern appears to have no correlation to whether a specific model has massless fermions or not. The origins of it are, however, not clear in the above cases.

It was proved in \cite{MSUSY2} that non-SUSY heterotic strings without physical tachyons should always produce such misalignment of their massive spectra. In our case, due to the presence of on-shell tachyons the emergence of misaligned supersymmetry may seem somewhat unexpected and of unknown origin. The analysis presented in the proof of misaligned supersymmetry in \cite{MSUSY2, carlo} relies on modular invariance together with the stated lack of physical tachyons. Since our theory is of course still modular invariant, we can speculate that the emergence of the misalignment should be a consequence of this, however, rigorous analysis is still lacking under these conditions. For the time being, we only present this as an observations for the theories under consideration in this paper, but further investigation may lead to a deeper understanding of the relationship between on-shell tachyons and misaligned SUSY.

As an example, for the $\tilde{S}$-models of Section \ref{StModels} we observe the two general patterns shown in Figure \ref{BFOsc}, or in general a combination of these two.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{Type0BF.eps}
\caption{\emph{The boson-fermion oscillation of misaligned supersymmetry for the on-shell states of two $\tilde{S}$ models to 8$^{th}$ order in the q-expansion. The overall sign of $\pm\log(\mid a_{mn}\mid)$ is chosen according to the sign of $a_{mn}$.}}
\label{BFOsc}
\end{figure}
This is true whether or not the choice of GGSO coefficients project all massless fermions. The only observable difference we find for type 0 models from the partition function point of view is the larger value of  $a_{00}$. This is of course fully expected due to the lack of fermions at the massless level. The misalignment pattern is mostly similar for the $S$--models of Section \ref{SModels}.  It is important to note that, unlike for tachyon--free non--supersymmetric theories, this oscillatory behaviour does not result in a finite value for the cosmological constant due to the presence of the physical tachyons described in Sections \ref{StModels} and \ref{SModels}.

We would like to emphasise that the misaligned supersymmetry found in our
string models is independent of the existence of the tachyon in the spectrum.
In fact, our results demonstrate this fact unequivocally. It is well
known that the absence of a tachyon is sufficient for the existence
of misaligned supersymmetry \cite{MSUSY2,MSUSY, carlo}. However,
our results demonstrate that it is in fact not necessary. It may of course
be the case that the misaligned supersymmetry in vacua that contain tachyons
is only exhibited for special values of the moduli parameters and that moving
away from those points will spoil this property. However, as exhibited
by our results, at least at the free fermionic point, the oscillatory
pattern between fermions and bosons of the massive string spectrum,
persist in vacua that contain tachyons as well.

\section{Conclusions}\label{conclude}
The $\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifold produced a rich space of phenomenological
three generation models and serves as a testing ground to explore how the Standard Model
of particle physics may be incorporated in a theory of quantum gravity. The $\mathbb{Z}_2\times \mathbb{Z}_2$
orbifold also exhibits a rich symmetry structure which has been of interest from a purely
mathematical point of view. It is of course of much further interest to explore the role
of these mathematical structures in the phenomenological properties of the $\mathbb{Z}_2\times \mathbb{Z}_2$
orbifold models. In this paper we explored the existence of type 0 models
in this class of string
compactifications. Type 0 string vacua are those that do not contain any massless fermionic
states and have been of interest in other string theory limits, {\it e.g.} the issue of tree
level stability have been studied in the framework of type II orientifolds, whereas
other authors have advocated that there is holographic duality of the
type 0B string theory and four dimensional non--supersymmetric Yang--Mills
theory \cite{glmr}. In this paper we demonstrated the existence of type 0
$\mathbb{Z}_2\times \mathbb{Z}_2$ heterotic string orbifolds. We showed that there exist a large degree
of redundancy in the space of GGSO projection coefficients when the type 0 restrictions
are implemented. We explored the existence of such configurations in several constructions.
The one presented in Section \ref{model} correspond to essentially a unique configuration
out of a priori $2^{21}$ discrete GGSO choices. We demonstrated this uniqueness analytically
in Section \ref{analyticConds} as well as by the corresponding analysis of the
partition function in Section \ref{equivalence}.
In Sections \ref{StModels} and \ref{SModels} we performed a wider classification in
$\tilde S$--models and $S$--models, where the first class correspond to compactifications
of a tachyonic ten dimensional heterotic string vacuum, whereas the second correspond to
compactifications of the ten dimensional non--tachyonic $SO(16)\times SO(16)$. We showed
that the type 0 models in both cases contain physical tachyons at the free fermionic point
in the moduli space. These vacua are therefore necessarily unstable.
we demonstrated the existence of a misalligned supersymmetry
in the type 0 models that guarantee the finiteness of the one--loop amplitude, aside from
the divergence due to the tachyonic states.
Given their rather
restrictive structure, the type 0 models may be instrumental in exploring the string dynamics in
cosmological scenarios in the spirit of the analysis performed in ref. \cite{FR}. Of further
interest will be to explore vacua with a minimal number of bosonic states. Whereas the
string models always contain massless bosons from the untwisted sector, we may look for
models in which the bosonic states from twisted sectors are projected out. Another
possibility is to look for models in which the fermions from the observable sectors are
balanced by scalars charged under the hidden sector.

\section*{Acknowledgments}

We would like to that Carlo Angelantonj for fruitful discussions.
The work of VGM is supported in part by EPSRC grant EP/R513271/1.
The work of BP is supported in part by STFC grant ST/N504130/1.

\bigskip

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Enjoying a central location in Brighton, 100 metres from Victoria Gardens, the seafront Cecil House Hotel features a terrace and views of the sea. All rooms here come with a TV and feature either en-suite or shared bathroom facilities. There is a 24-hour front desk at the property. In the same building, guests will find Smokeys Brighton, an American Barbecue Restaurant. Hotel guests benefit from a 10% discount on food, excluding special offers. The beach is just steps from the Cecil House Hotel, while Churchill Square Shopping Centre is 300 metres away. The nearest airport is London Gatwick Airport, 46 km from Cecil House Hotel.


Celtic striker Moussa Dembele will miss the Champions League qualifier against Rosenborg with a hamstring problem.
The 22-year-old picked up the injury in last week's second leg of the opening 6-0 aggregate victory over Alashkert.
The Frenchman is expected to be out for up to two weeks more.
"It was just a minor strain," said manager Brendan Rodgers. "Sometimes the hamstring can be six to eight weeks but this one, from the time it happened, we were told two to three weeks maximum."
Dembele scored twice in Celtic's 3-0 victory in Glasgow before limping off in the second half.
Celtic host Norwegian side Rosenborg on Wednesday (19:45 BST kick-off) before travelling to Trondheim for the return on 1 August.
"He'll certainly miss both legs of this round," added Rodgers. "It's unfortunate because he's looked very good over pre-season, but he'll be back fit and strong in the next few weeks."
Fellow striker Leigh Griffiths is a doubt for the first leg at Celtic Park, as is winger Jonny Hayes, both of whom have thigh injuries.
Should Celtic progress, they will face AEK Athens of Greece in the third qualifying round, while the loser will drop into the Europa League qualifiers and take on Cork City.


text:
Q: stop asynctask being called twice I have found my AsyncTask being called twice. To stop this, in my MainActivity,I have defined
 public Boolean isloading = false;

and asynctask is called as:
   if (!isloading) {
      new GetFlickr(MainActivity.this, imageViewWeakReference, MainActivity.this)
          .execute(Double.toString(Lat), Double.toString(Long));
    }

When asynctask was inside the MainActivity class (i.e., in same file), there was no problem setting isloading = true in PreExecute, as:
  @Override
    protected void onPreExecute() {
      super.onPreExecute();
      pBar = new ProgressBar(getApplicationContext());
      pBar = findViewById(R.id.progressBar);
      pBar.setProgress(0);
      pBar.setVisibility(View.VISIBLE);
      isloading = true;
    }

But, I have moved asynctask to a seperate file, and can't set the value to true there.
Any help please?

A: The trivial thing is just to set isLoading xhecking the value, in the calling code.
if (!isloading) {
      isLoading = true
      new GetFlickr(MainActivity.this, imageViewWeakReference, MainActivity.this)
          .execute(Double.toString(Lat), Double.toString(Long));
}

This works so long as only a single thread can call the coad that starts the async thread.  If multiple threads can, then you need to move to a lock like a synchronized block.

redpajama_set_name: RedPajamaStackExchange


text:
Russia , also officially the Russian Federation , is a country in Eurasia. At , Russia is the largest country in the world by surface area, covering more than one-eighth of the Earth's inhabited land area, and the ninth most populous, with over 140 million people at the end of March 2016. The European western part of the country is much more populated and urbanised than the eastern, about 77% of the population live in European Russia. Wikipedia*
Russia , also officially the Russian Federation , is a country in Eurasia. At , Russia is the largest country in the world by surface area, covering more than one-eighth of the Earth's inhabited land area, and the ninth most populous, with over 140 million people at the end of March 2016. The European western part of the country is much more populated and urbanised than the eastern, about 77% of the population live in European Russia..
Sort By Last Name (A-Z)Number of Programs
Sergei Adomovich
Member, Parliament, Russia
Nikolai Alexeevich
Yevgeniy Ambartsumov
Member, Presidium, Supreme Soviet, Russia
Anatoly Antonov
Ambassador, United States, Russia
Larisa Arap
Activist, Russia
Alexei Arbatov
Member, Duma, Russia
Olga Beklemischeva
Member, Duma, Yabloko Party, Russia
Sergey Belaev
Leader, Our Home is Russia Party, Russia
Sergey Belyaev
Leader, Duma, Nash Dom Rossia Party, Russia
Valentin Berezhkov
Interpreter, Diplomatic Service, Russia
See all Russian Federation people

redpajama_set_name: RedPajamaCommonCrawl


As the world of transport changes, how can governments respond to ensure a safe and timely transition?
We track and report on the tides and trends driving the world's infrastructure markets.
How smart is our infrastructure?
Bernard Salt joins industry experts to discuss the priorities for Australian infrastructure over the next 5 years.
This edition covers questions facing governments, infrastructure owners, investors, developers and operators as we move into an era of change and uncertainty.
Paul Foxlee on how we can use infrastructure technology to improve our communities.
Globally, the infrastructure sector is grappling with consolidation, surplus capacity, access to capital, pressures on profitability, complex information technology requirements and ever-increasing government regulation. The demands of infrastructure projects have implications for all layers of government, as well as for financiers, investors and various public and private-sector enterprises.
KPMG's infrastructure professionals are involved in many exciting projects across the world, in various sectors and stages of the infrastructure lifecycle. Our team of professionals advise on designing, executing and managing major infrastructure projects and transactions. We have participated in many of Australia's largest and most complex government asset sales, privatisations and public-private partnerships (PPPs).
Our capabilities cover all key sectors, including energy and natural resources, water, telecommunications and transport. We have worked on major social infrastructure projects such as schools, hospitals and public housing.
To understand the effects of zero emission and autonomous vehicles, new models are needed.
Paul is the National Sector Leader for our Transport & Infrastructure group.
Government agencies are finding smarter ways to structure mega deals.
Maximise the value of your organisation's assets with our integrated approach.
A road pricing trial will build support for agreeing on and implementing the right scheme.
Evaluating 25 countries on their readiness for autonomous vehicles.


Also, a change in leadership at Ridge Vineyards, Beaujolais is devastated by hail, and craft beer's dominance begins to wane.
In the midst of this year's Wimbledon tournament, 29-year-old Novak Djokovic has purchased vineyard land in his native Serbia. Located near the village of Šumadija, the heart of the country's wine region, the land was listed as a vineyard over 50 years ago but is currently abandoned woodland that will require a period of regeneration. Vladimir Stojić of the Šumadija Winemakers Association, says grapevines will likely be planted in the spring. Hopefully this new project will provide a happy distraction for Djokovic, after suffering a stunning loss to Sam Querrey and exiting the tournament this weekend.
Is That Bacon Vodka? You're Under Arrest!
Once the home of founders Katie and Gene Trefethen, "Villa Trefethen" is now hosting visitors. The 1922 Napa Valley home is a classic example of California's Arts and Crafts movement. Repairs and restoration are still taking place on the historic winery building, which was damaged in a 2014 earthquake.
Hartwell Estate Vineyards is mourning the loss of winery founder and proprietor, Bob Hartwell. Established in 1986, Hartwell Estate Vineyards garnered critical acclaim and a loyal following by producing exceptional small production Bordeaux varietal wines from its Stags Leap District estate. Mr. Hartwell passed peacefully in his sleep on June 22.
Local officials have called for parts of Beaujolais to be declared a disaster zone after fierce hail storms severely damaged vineyards in the area. Just one month after a previous hail storm, the region was hit again on the evening of June 24. Mélina Condy, from regional wine body Inter-Beaujolais, reported that 3,000 hectares or 20 percent of the region's vineyards were damaged. The commune of Fleurie is believed to have suffered the most. "70 to 80 percent of the vineyards are totally destroyed by the hail," said Frédéric Miguet, mayor of Fleurie.
Tough Times Ahead For Craft Beer?
Sales of craft brews have long been a bright, but may be on the wane. Craft beer sales are up only 6 percent this year, a far cry from 19 percent of just two years ago. "The consensus is that craft beer has slowed down significantly this year," says Trevor Stirling of Bernstein Research. One potential explanation? The fragmentation of the craft beer market.
Listen to what makes a restaurant one of the best when it comes to wine.


text:
Films, 2015 films, Disney Channel Original Movies,
Bad Hair Day (film)
Eric Canuel
Leigh-Allyn Baker
Joel Rice
Denise Carlson
Dawn Wolfrom
Michael Prupas
Eric M. Gardner
Steven H. Wilson
Billy & Matt Eddy
Disney Channel Original Movie
Bad Hair Day is a Disney Channel Original Movie. It is directed by Erik Canuel, produced by Muse Entertainment and written by Eric M. Gardner, Steven H. Wilson, and Billy & Matt Eddy. The film was executive produced by Leigh-Allyn Baker, Joel Rice, Denise Carlson, Dawn Wolfrom, and Michael Prupas. The film was filmed in Montreal, Canada. It premiered on February 13, 2015.
4 Broadcast
Set in Boston, Massachusetts, Monica Reeves (Laura Marano), a high school tech whiz, is determined to become prom queen. She prepares herself the day before prom so she can save time when it comes, and thinks nothing will go wrong. But on the big day, she suddenly wakes up having a bad hair day, and her destroyed prom dress, broken shoes, and everything that can go wrong does go badly wrong (and of course the hair). Liz Morgan (Leigh-Allyn Baker) seeks the necklace that Monica somehow ends up possessing, after buying it at an antique shop, wanting to recover it and get her job back, disguised as an FBI agent asking Monica to answer her questions, but Monica herself knows she is not who she says she is after analyzing her behavior and checking the FBI website. Prom day goes wrong as the pair is pursued by Pierce Peters (Christian Campbell) on a wild ride around the city. At the end of the movie, Monica and Liz are lured into a trap by Pierce but they outsmart him and he is arrested. Monica and Liz go to the prom in a full police procession and the movie ends with a happy ending, the two both learning valuable lessons.
Laura Marano as Monica Reeves, a high school tech whiz who prefers to choose things based on other people's opinions.
Leigh-Allyn Baker as Liz Morgan, a "down-on-her-luck cop" who is tracking down a diamond necklace Monica purchased at a pawn shop.
Christian Campbell as Pierce Peters, a "dogged jewel thief" who is trying to steal Monica's diamond necklace.
Kiana Madeira as Sierra, Monica's best friend.
Christian Paul as Ed, Liz's former partner.
Alain Goulem as Monica's dad who works as a nurse.
Jake Manley as Kyle Timmons, Monica's former boyfriend.
Zoe De Grande Maison as Ashley Mendlebach, Monica's competition and arch enemy.
Filming commenced in the beginning of July 2014 and ended in August.
In an interview with Variety, Baker said, "I wanted to executive produce because I wanted to see what it was like to build and fulfill a vision from the ground up, and since this was Disney Channel's first movie with an adult lead, I wanted that role to be protected and I wanted the movie to have my creative stamp on it." She also liked that she got to work on the Disney movie because it taps into her family unit fanbase. Marano also mentioned that it was her first DCOM. Filming took place in Montreal. Marano noted that in contrast to Austin & Ally which had four cameras and shot about ten scenes a day, the film shot with fewer cameras and fewer scenes, one of which took the entire day.
Marano said that she likes that her character embodies two normally contrasting high school cliches - a prom queen nominee who likes beauty and fashion and is popular, and a computer whiz who has a nerdy tech obsession and codes. She also likes that she relates with her character when deciding whether to go to the state college or MIT.
The film was first released on February 6, 2015 on WATCH Disney Channel, and had its premiere broadcast in the US on February 13. It is scheduled to premiere in Canada on the Family Channel on March 20, 2015. It was also one of the first Disney Channel Original Movies to exclusively premiere on CGI basic cable and computer-generated satellite television via the WATCH Disney Channel App.
Bad Hair Day delivered 3.97 million viewers for its television premiere and has had 435,000 views on the WATCH Disney Channel app since it was released on February 6. Amy Amatangelo of The Hollywood Reporter found the duo of Baker and Marano "played well off each other", and that Marano's character was "refreshingly age-appropriate". She found the movie to be goofy but not slapsticky, and that it was "like a junior version of Thelma and Louise albeit with a much, much happier ending."
This is the final Disney Channel movie to have opening credits until Invisible Sister.
This is also the second DCOM to take place over the course of one whole day, after Phineas and Ferb the Movie: Across the 2nd Dimension.
Disney Channel - Original Movies coming in 2015
Trailer - Bad Hair Day
Tiger Town • Gone Are the Dayes • Love Leads the Way • Black Arrow • Lots of Luck • The Undergrads • The Blue Yonder • The Parent Trap II • Spot Marks the X • Down the Long Hills • Not Quite Human • The Christmas Visitor • Ollie Hopnoodle's Haven of Bliss • A Friendship in Vienna • Not Quite Human II • Spooner
Lantern Hill • Chips, the War Dog • Mother Goose Rock 'n' Rhyme • The Little Kidnappers • Back to Hannibal: The Return of Tom Sawyer and Huckleberry Finn • Bejewelled • Perfect Harmony • Mark Twain and Me • Still Not Quite Human • The Ernest Green Story • Hocus Pocus • Heidi • The Old Curiosity Shop • Escape to Witch Mountain • The Four Diamonds • Freaky Friday • Wish Upon a Star • Susie Q • Northern Lights • Under Wraps • Tower of Terror • You Lucky Dog • Brink! • Halloweentown • Zenon: Girl of the 21st Century • Can of Worms • The Thirteenth Year • Smart House • Johnny Tsunami • Genius • P.U.N.K.S. • Don't Look Under the Bed • Horse Sense
Up, Up and Away • The Color of Friendship • Alley Cats Strike • Rip Girls • Miracle in Lane 2 • Stepsister from Planet Weird • Ready to Run • Quints • The Other Me • Mom's Got a Date with a Vampire • Phantom of the Megaplex • The Ultimate Christmas Present • Zenon: The Zequel • Motocrossed • The Luck of the Irish • Hounded • Jett Jackson: The Movie • The Poof Point • Halloweentown II: Kalabar's Revenge • 'Twas the Night • Double Teamed • Cadet Kelly • Tru Confessions • Get a Clue • Gotta Kick it Up! • A Ring of Endless Light • The Scream Team • You Wish! • Right on Track • The Even Stevens Movie • Eddie's Million Dollar Cook-Off • The Cheetah Girls • Full - Court Miracle • Pixel Perfect • Going to the Mat • Zenon: Z3 • Stuck in the Suburbs • Tiger Cruise • Halloweentown High • Now You See It... • Buffalo Dreams • Kim Possible: So the Drama • Go Figure • Life is Ruff • The Proud Family Movie • Twitches • High School Musical • Cow Belles • Wendy Wu: Homecoming Warrior • Read It and Weep • The Cheetah Girls 2 • Return to Halloweentown • Jump In! • Johnny Kapahala: Back on Board • High School Musical 2 • Twitches Two • Minutemen • Camp Rock • The Cheetah Girls: One World • Dadnapped • Hatching Pete • Princess Protection Program • Wizards of Waverly Place: The Movie
StarStruck • Den Brother • Camp Rock 2: The Final Jam • Avalon High • The Suite Life Movie • Lemonade Mouth • Phineas and Ferb the Movie: Across the 2nd Dimension • Geek Charming • Good Luck Charlie, It's Christmas! • Frenemies • Radio Rebel • Let It Shine • Girl vs. Monster • Teen Beach Movie • Cloud 9 • Zapped • How to Build a Better Boy • Bad Hair Day • Teen Beach 2 • Descendants • Invisible Sister • Adventures in Babysitting • The Swap • Tangled: Before Ever After • Descendants 2 • Zombies • Freaky Friday • Kim Possible • Descendants 3
Zombies 2 • Upside-Down Magic •
Spin • Under Wraps • Christmas Again
Unofficial DCOMs
Kim Possible: A Sitch in Time • Leroy & Stitch • Harriet the Spy: Blog Wars • 16 Wishes • Marvel Rising: Secret Warriors
Retrieved from "https://disney.fandom.com/wiki/Bad_Hair_Day_(film)?oldid=4260427"
Disney Channel Original Movies


Slot car racing is a wonderful hobby, in part because there's always something new to learn. At Slot Car Corner Canada, we try to help our customers as much as we can and provide them with some quality reference material that they can refer to when a question arises.
Over the years, we have created many tools, charts, tutorials and articles about our products and slot racing in general.
This Reference Center is where you'll find all those resources. The categories on the right (or the tiles below) will bring you to specific pages, where you'll find more information. Have fun browsing through them.


text:
#ifndef _MAGICKCORE_IMAGE_H
#define _MAGICKCORE_IMAGE_H

#include "magick/color.h"

#if defined(__cplusplus) || defined(c_plusplus)
extern "C" {
#endif

#define OpaqueOpacity  ((Quantum) 0UL)
#define TransparentOpacity  (QuantumRange)

typedef enum
{
  UndefinedAlphaChannel,
  ActivateAlphaChannel,
  BackgroundAlphaChannel,
  CopyAlphaChannel,
  DeactivateAlphaChannel,
  ExtractAlphaChannel,
  OpaqueAlphaChannel,
  ResetAlphaChannel,  /* deprecated */
  SetAlphaChannel,
  ShapeAlphaChannel,
  TransparentAlphaChannel,
  FlattenAlphaChannel,
  RemoveAlphaChannel,
  AssociateAlphaChannel,
  DisassociateAlphaChannel
} AlphaChannelType;

typedef enum
{
  UndefinedType,
  BilevelType,
  GrayscaleType,
  GrayscaleMatteType,
  PaletteType,
  PaletteMatteType,
  TrueColorType,
  TrueColorMatteType,
  ColorSeparationType,
  ColorSeparationMatteType,
  OptimizeType,
  PaletteBilevelMatteType
} ImageType;

typedef enum
{
  UndefinedInterlace,
  NoInterlace,
  LineInterlace,
  PlaneInterlace,
  PartitionInterlace,
  GIFInterlace,
  JPEGInterlace,
  PNGInterlace
} InterlaceType;

typedef enum
{
  UndefinedOrientation,
  TopLeftOrientation,
  TopRightOrientation,
  BottomRightOrientation,
  BottomLeftOrientation,
  LeftTopOrientation,
  RightTopOrientation,
  RightBottomOrientation,
  LeftBottomOrientation
} OrientationType;

typedef enum
{
  UndefinedResolution,
  PixelsPerInchResolution,
  PixelsPerCentimeterResolution
} ResolutionType;

typedef struct _PrimaryInfo
{
  double
    x,
    y,
    z;
} PrimaryInfo;

typedef struct _SegmentInfo
{
  double
    x1,
    y1,
    x2,
    y2;
} SegmentInfo;

typedef enum
{
  UndefinedTransmitType,
  FileTransmitType,
  BlobTransmitType,
  StreamTransmitType,
  ImageTransmitType
} TransmitType;

typedef struct _ChromaticityInfo
{
  PrimaryInfo
    red_primary,
    green_primary,
    blue_primary,
    white_point;
} ChromaticityInfo;

#include "magick/blob.h"
#include "magick/colorspace.h"
#include "magick/cache-view.h"
#include "magick/color.h"
#include "magick/composite.h"
#include "magick/compress.h"
#include "magick/effect.h"
#include "magick/geometry.h"
#include "magick/layer.h"
#include "magick/locale_.h"
#include "magick/monitor.h"
#include "magick/pixel.h"
#include "magick/profile.h"
#include "magick/quantum.h"
#include "magick/resample.h"
#include "magick/resize.h"
#include "magick/semaphore.h"
#include "magick/stream.h"
#include "magick/timer.h"

struct _Image
{
  ClassType
    storage_class;

  ColorspaceType
    colorspace;      /* colorspace of image data */

  CompressionType
    compression;     /* compression of image when read/write */

  size_t
    quality;         /* compression quality setting, meaning varies */

  OrientationType
    orientation;     /* photo orientation of image */

  MagickBooleanType
    taint,           /* has image been modified since reading */
    matte;           /* is transparency channel defined and active */

  size_t
    columns,         /* physical size of image */
    rows,
    depth,           /* depth of image on read/write */
    colors;          /* size of color table on read */

  PixelPacket
    *colormap,
    background_color, /* current background color attribute */
    border_color,     /* current bordercolor attribute */
    matte_color;      /* current mattecolor attribute */

  double
    gamma;

  ChromaticityInfo
    chromaticity;

  RenderingIntent
    rendering_intent;

  void
    *profiles;

  ResolutionType
    units;          /* resolution/density  ppi or ppc */

  char
    *montage,
    *directory,
    *geometry;

  ssize_t
    offset;

  double
    x_resolution,   /* image resolution/density */
    y_resolution;

  RectangleInfo
    page,           /* virtual canvas size and offset of image */
    extract_info,
    tile_info;      /* deprecated */

  double
    bias,
    blur,           /* deprecated */
    fuzz;           /* current color fuzz attribute */

  FilterTypes
    filter;         /* resize/distort filter to apply */

  InterlaceType
    interlace;

  EndianType
    endian;         /* raw data integer ordering on read/write */

  GravityType
    gravity;        /* Gravity attribute for positioning in image */

  CompositeOperator
    compose;        /* alpha composition method for layered images */

  DisposeType
    dispose;        /* GIF animation disposal method */

  struct _Image
    *clip_mask;

  size_t
    scene,          /* index of image in multi-image file */
    delay;          /* Animation delay time */

  ssize_t
    ticks_per_second;  /* units for delay time, default 100 for GIF */

  size_t
    iterations,
    total_colors;

  ssize_t
    start_loop;

  ErrorInfo
    error;

  TimerInfo
    timer;

  MagickProgressMonitor
    progress_monitor;

  void
    *client_data,
    *cache,
    *attributes;      /* deprecated */

  Ascii85Info
    *ascii85;

  BlobInfo
    *blob;

  char
    filename[MaxTextExtent],         /* images input filename */
    magick_filename[MaxTextExtent],  /* ditto with coders, and read_mods */
    magick[MaxTextExtent];           /* Coder used to decode image */

  size_t
    magick_columns,
    magick_rows;

  ExceptionInfo
    exception;        /* Error handling report */

  MagickBooleanType
    debug;            /* debug output attribute */

  volatile ssize_t
    reference_count;

  SemaphoreInfo
    *semaphore;

  ProfileInfo
    color_profile,
    iptc_profile,
    *generic_profile;

  size_t
    generic_profiles;  /* this & ProfileInfo is deprecated */

  size_t
    signature;

  struct _Image
    *previous,         /* Image list links */
    *list,             /* Undo/Redo image processing list (for display) */
    *next;             /* Image list links */

  InterpolatePixelMethod
    interpolate;       /* Interpolation of color for between pixel lookups */

  MagickBooleanType
    black_point_compensation;

  PixelPacket
    transparent_color; /* color for 'transparent' color index in GIF */

  struct _Image
    *mask;

  RectangleInfo
    tile_offset;

  void
    *properties,       /* per image properities */
    *artifacts;        /* per image sequence image artifacts */

  ImageType
    type;

  MagickBooleanType
    dither;            /* dithering method during color reduction */

  MagickSizeType
    extent;

  MagickBooleanType
    ping;

  size_t
    channels;

  time_t
    timestamp;

  PixelIntensityMethod
    intensity;      /* method to generate an intensity value from a pixel */

  size_t
    duration;       /* Total animation duration sum(delay*iterations) */
};

struct _ImageInfo
{
  CompressionType
    compression;

  OrientationType
    orientation;

  MagickBooleanType
    temporary,
    adjoin,
    affirm,
    antialias;

  char
    *size,
    *extract,
    *page,
    *scenes;

  size_t
    scene,
    number_scenes,
    depth;

  InterlaceType
    interlace;

  EndianType
    endian;

  ResolutionType
    units;

  size_t
    quality;

  char
    *sampling_factor,
    *server_name,
    *font,
    *texture,
    *density;

  double
    pointsize,
    fuzz;

  PixelPacket
    background_color,
    border_color,
    matte_color;

  MagickBooleanType
    dither,
    monochrome;

  size_t
    colors;

  ColorspaceType
    colorspace;

  ImageType
    type;

  PreviewType
    preview_type;

  ssize_t
    group;

  MagickBooleanType
    ping,
    verbose;

  char
    *view,
    *authenticate;

  ChannelType
    channel;

  Image
    *attributes;  /* deprecated */

  void
    *options;

  MagickProgressMonitor
    progress_monitor;

  void
    *client_data,
    *cache;

  StreamHandler
    stream;

  FILE
    *file;

  void
    *blob;

  size_t
    length;

  char
    magick[MaxTextExtent],
    unique[MaxTextExtent],
    zero[MaxTextExtent],
    filename[MaxTextExtent];

  MagickBooleanType
    debug;

  char
    *tile;  /* deprecated */

  size_t
    subimage,  /* deprecated */
    subrange;  /* deprecated */

  PixelPacket
    pen;  /* deprecated */

  size_t
    signature;

  VirtualPixelMethod
    virtual_pixel_method;

  PixelPacket
    transparent_color;

  void
    *profile;

  MagickBooleanType
    synchronize;
};

extern MagickExport ExceptionType
  CatchImageException(Image *);

extern MagickExport FILE
  *GetImageInfoFile(const ImageInfo *);

extern MagickExport Image
  *AcquireImage(const ImageInfo *),
  *AppendImages(const Image *,const MagickBooleanType,ExceptionInfo *),
  *CloneImage(const Image *,const size_t,const size_t,const MagickBooleanType,
    ExceptionInfo *),
  *DestroyImage(Image *),
  *GetImageClipMask(const Image *,ExceptionInfo *),
  *GetImageMask(const Image *,ExceptionInfo *),
  *NewMagickImage(const ImageInfo *,const size_t,const size_t,
    const MagickPixelPacket *),
  *ReferenceImage(Image *),
  *SmushImages(const Image *,const MagickBooleanType,const ssize_t,
    ExceptionInfo *);

extern MagickExport ImageInfo
  *AcquireImageInfo(void),
  *CloneImageInfo(const ImageInfo *),
  *DestroyImageInfo(ImageInfo *);

extern MagickExport MagickBooleanType
  ClipImage(Image *),
  ClipImagePath(Image *,const char *,const MagickBooleanType),
  IsTaintImage(const Image *),
  IsMagickConflict(const char *),
  IsHighDynamicRangeImage(const Image *,ExceptionInfo *),
  IsImageObject(const Image *),
  ListMagickInfo(FILE *,ExceptionInfo *),
  ModifyImage(Image **,ExceptionInfo *),
  ResetImagePage(Image *,const char *),
  SetImageBackgroundColor(Image *),
  SetImageClipMask(Image *,const Image *),
  SetImageColor(Image *,const MagickPixelPacket *),
  SetImageExtent(Image *,const size_t,const size_t),
  SetImageInfo(ImageInfo *,const unsigned int,ExceptionInfo *),
  SetImageMask(Image *,const Image *),
  SetImageOpacity(Image *,const Quantum),
  SetImageChannels(Image *,const size_t),
  SetImageStorageClass(Image *,const ClassType),
  StripImage(Image *),
  SyncImage(Image *),
  SyncImageSettings(const ImageInfo *,Image *),
  SyncImagesSettings(ImageInfo *,Image *);

extern MagickExport size_t
  InterpretImageFilename(const ImageInfo *,Image *,const char *,int,char *);

extern MagickExport ssize_t
  GetImageReferenceCount(Image *);

extern MagickExport size_t
  GetImageChannels(Image *);

extern MagickExport VirtualPixelMethod
  GetImageVirtualPixelMethod(const Image *),
  SetImageVirtualPixelMethod(const Image *,const VirtualPixelMethod);

extern MagickExport void
  AcquireNextImage(const ImageInfo *,Image *),
  DestroyImagePixels(Image *),
  DisassociateImageStream(Image *),
  GetImageException(Image *,ExceptionInfo *),
  GetImageInfo(ImageInfo *),
  SetImageInfoBlob(ImageInfo *,const void *,const size_t),
  SetImageInfoFile(ImageInfo *,FILE *);

#if defined(__cplusplus) || defined(c_plusplus)
}
#endif

#endif

redpajama_set_name: RedPajamaGithub


All PARCEL O SUMMERLIN VILLAG homes currently listed for sale in Las Vegas as of 04/23/2019 are shown below. You can change the search criteria at any time by pressing the 'Change Search' button below.
"Summerlin Stunner FULLY furnished, a home renovated from top to bottom to an elegant contemporary finish throughout with no expense spared! This Executive Estate home will not disappoint, from the amazing Master retreat downstairs w/ dual over sized Master closets and Resort style Spa bathroom and private entrance to the Spa and Pool. The open plan downstairs features entertaining at its BEST, wine cellar full bar, 2 fireplaces and 90 inch TV's!"
"Gorgeous Single Story in Palisades with Casita that features a full bathroom/WIC and 12 foot ceilings*GE Monogram appliances*2 wine refrigerators*Under cabinet Led lighting in kitchen*Wet bar*Crown Molding throughout home*Courtyard access from office*Grid-Loc flooring in garage*Putting Green*Master features 2 WIC*Amazing courtyard entrance with 2 fountains and an outdoor fireplace for those cool evenings*Travertine flooring in covered patio"


text:
Double Dose of Dario, Kathy Acker's Sex Tape, and More
October 23, 2014 By Nicole Disser
Here's a buncha films we're psyched to see this week and beyond. We won't go completely spooky on you until next week, so enjoy our handful of non-Halloween picks in the bunch while you still can. But don't worry, we've still got at least one slasher on deck for ya.
Oh hey this one's kind of a big deal– it's the Edward Snowden documentary. Laura Poitras is the filmmaker behind CitizenFour, and as the first person Snowden contacted with his big secret, she's developed a special way of seeing the NSA whistleblower. The New Yorker put out a fantastic profile of Poitras that we highly suggest checking out in conjunction with the film, because apparently Poitras, though a huge part of the Snowden story, prefers to remain in the background. The IFC Center will welcome the filmmaker for a few introductions and Q&A sessions at a handful of screenings, so get your tickets early if you'd like to see Poitras in person. Friday, October 24th through Thursday, November 20th at IFC Center, various showtimes; tickets, $14
4th Annual Spectacle Shriek Show
Because one horror film is never enough, Spectacle is bequeathing one lucky audience with almost 12-14 straight hours of scary movies spanning from the 1960s to 2014. The Williamsburg theater has done this in the past, but they promise this year will be the "craziest yet." Check out the full schedule over here. Our first pick for the marathon is Kolobos (screening at 5:30)– a 1999 Dario Argento-inspired slasher film which follows a reality show as it quickly devolves into a murderous nightmare. Another intriguing film on the list is The Strange Color of Your Body's Tears, another film inspired by Italian horror classics, this Belgian movie is a highly stylized creepy erotic thriller. Saturday, October 25th noon till past midnight at Spectacle Theater in Williamsburg; tickets, $5 for individual films and $25 for the entire day, purchase or at the door
This Pre-Rosemary's Baby classic from Roman Polanski is the close-up story of a woman unraveling at the seams. See what happens when she totally loses it and munch on brunch at the same time. Saturday, October 25th and Sunday, October 26th at Nitehawk Cinema in Williamsburg, noon; tickets, $11
(Photo: Light Industry)
In 1974 downtown writer Kathy Acker and artist Alan Sondheim hooked up, so to speak, but instead of just leaving it at that, the pair decided to lock themselves away for 48 hours and document the experience. The result is Blue Tape — a video that's not just about the attraction between the two artists, but the aesthetic debate they engage in. Chris Kraus wrote in Feed: "When Sondheim faces Acker, Blue Tape becomes a shoot-out between high minimalism and what would become punk's new narrative." Tuesday, October 28th at Light Industry in Greenpoint, 7:30 pm; tickets, $7 at the door
I don't know about yous, but this is myyyyy personal favorite horror movie of all time. Dario Argento, the duke of Italian horror film, brings you psychedelic colors, blood splatter, psychosis, black magic, and disembowelments– what more could you want? Oh wait, then there's the killer GOBLIN soundtrack. Be there or be [ ]. Friday October 25th and Saturday October 26th at Nitehawk Cinema, 12:05 am; tickets, $11
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text:
Education Week's blogs > The School Law Blog See our Law & Courts coverage
Expert insights and analysis on legal issues in education, including school-related cases in the U.S. Supreme Court and in lower courts.
« Over a Dissent, Justices Decline Appeal on Public School Graduation at Church | Main | Supreme Court Backs Speech Protection for Sworn Public-Employee Testimony »
U.S. Supreme Court to Weigh Threats on Social Media
By Mark Walsh on June 16, 2014 1:12 PM
In a case with potential implications for public schools, the U.S. Supreme Court on Monday agreed to take up a case involving the free speech rights of those who make threats or deliver violent rants on social media such as Facebook.
The court agreed to hear the appeal of a Pennsylvania man who was convicted under a federal law of making illegal threats based on numerous threats and rants he put on his Facebook page, mostly directed against his estranged wife. The case is Elonis v. United States (No. 13-983).
Among the postings by Anthony Douglas Elonis, who was a 27-year-old amusement park employee at the time, was this one published in November 2010:
That's it, I've had about enough
I'm checking out and making a name for myself
Enough elementary schools in a ten mile radius
to initiate the most heinous school shooting ever imagined
And hell hath no fury like a crazy man in a kindergarten class
The only question is . . . which one?
Elonis testified in court that the posting was a reference to the song "I'm Back" by the rap artist Eminem, in which the artist had criticized his ex-wife and fantasized about participating in the 1999 shootings at Columbine High School in Colorado.
Elonis was convicted for making unlawful threats against schoolchildren. Based on other violent Facebook postings aimed at his estranged wife that didn't mention schools, Elonis was convicted of three other counts of transmitting in interstate commerce threats to injure another person. He was sentenced to almost four years in prison.
After a federal appeals court upheld the conviction, lawyers for Elonis appealed to the Supreme Court, arguing that there was a split among federal courts about whether so-called true threats should be weighed under a subjective or objective standard.
In Elonis' case, the U.S. Court of Appeals for the 3rd Circuit, in Philadelphia, held last year that an utterance is an unprotected "true threat" whenever "a reasonable person would foresee that the statement would be interpreted by those to whom the maker communicates the statement as a serious expression of an intention to inflict bodily harm."
Many other courts have followed that standard, but others have held that speech may be deemed unprotected by the First Amendment as a "true threat" only upon proof that the speaker subjectively intended the speech as a threat.
In a friend-of-the-court brief filed on Elonis' side, the Thomas Jefferson Center for the Protection of Free Expression, based in Charlottesville, Va., urged the justices to grant review of the case to analyze its true-threat jurisprudence in light of "the emergence of social networking and other modern modes of communication that ... underlie the vast majority of contemporary threat cases."
"By accepting the petition in this matter, this court can also address for the first time whether the
nature of the medium through which speech is conveyed affects a true-threat analysis," the group's brief says. "This is an important question because individuals increasingly face prosecution for alleged threats conveyed on new media, including Facebook, YouTube, and Twitter."
School administrators are having to face some of those questions, not always as full federal criminal cases, but as disciplinary matters, and the Supreme Court's decision in the case could affect student cases as well.
Justices Decline to Take Up Challenge to District's Pro-Transgender Policy
Justices Decline to Review Case Involving Strip Search of 4-Year-Old at School
Justices Decline Appeals on Private School Zoning, Public-Sector Speech
Justices Decline Challenge to Exclusive Public-Employee Union Representation
Judge Blocks Texas Law Prohibiting Israel Boycotts, Including for K-12 Contractors
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Scooby is a very friendly cat, he is of medium chunky build . There are no distinctive markings but if he is put under a bright light you can see faint ginger tortoise shell markings in his fur.
Scooby loves his home comforts. Sorry I'm not sure what else to say about Scooby as he is like most black cats and it is very hard..
All she does is meow at the moment so she knows something is wrong.
I have 4 kids who are also missing him deeply and it has been a miserable christmas with out him. Please look out for Scooby so we can have him home where he belongs.


text:
Handbook of the History of Religions in China I
From the Beginnings until the Period of the Five Dynasties and Ten Kingdoms
Zhongjian Mou, Jian Zhang
Paperback 69,90 €
Hier klicken für Informationen zu Lieferbarkeit, Versandkosten und -zeiten
eBook 39,99 €
Verfügbare Formate:
Rezensionsexemplar bestellen ODER
This book is part of an initiative in cooperation with renowned Chinese publishers to make fundamental, formative, and influential Chinese thinkers available to a western readership, providing absorbing insights into Chinese reflections of late, and offering a chance to grasp today's China. In their influential book Handbook of the History of Religions in China, Zhongjian Mou and Jian Zhang present a panorama of the religions existing in China through time. In their fascinating History, they delineate the emergence and development of Daoism, Buddhism, Confucianism, Islam, and Christianity and explore the roles they played in Chinese society and the interrelations between them. In China, also due to the encompassing Confucian idea of "living together harmoniously while maintaining differences," religions—including newly arrived ones—came closer together than anywhere else in the world and reached a unique level of peaceful societal coexistence. Despite many frictions and conflicts, communication and reconciliation were indisputably predominant in China throughout history. Buddhism was peacefully introduced into China and, later on, a harmonious, symbiotic syncretism of Confucianism, Buddhism, and Daoism developed—an exemplary process of how a diverse set of different religions can complement each other and contribute to a better life.
Autor/-in
Zhongjian Mou, an accomplished philosophical historian and religious scholar of China, has pioneered Chinese contemporary religious studies. He was born in Yantai city of Shandong province in May, 1939. From 1957 to 1965, he studied with Feng Youlan, Ren Jiyu, Zhu Bokun at the Philosophy Department of Peking University. From April 1966 to November 1987, he worked at the World Religion Research Institute of the Chinese Academy of Social Sciences (the former Philosophy and Social Sciences Department of the Chinese Academy of Sciences). Starting from 1987, he has been serving as professor and doctoral tutor of the Philosophy and Religion Department of Minzu University of China, once serving as the director of the Research Center of the "985 Project" on Contemporary Major National Religious Issues. He also serves as consultant of the International Confucian Association, consultant of the Chinese Religious Society, academic consultant of the China Confucius Foundation, national condition expert of the Chinese National Library; he is professor of the Chinese National School of Administration, professor of the Training Center of the State Bureau of Religious Affairs, member of the expert committee of the World Confucian Conference, and dean of Nishan Shengyuan Academy.
Jian Zhang, a graduate from the School of Philosophy of Renmin University of China (1982), is now Professor of the School of Continuing Education of Renmin University of China. He has been engaged in the study of the history of Chinese religion and philosophy for a long time. At present, he is also a researcher of the Buddhist and religious teaching research base of the Ministry of Education, a visiting professor of Minzu University of China, member of International Confucian Association, and Honorary Chairman of the Chinese Society of Shixue.
Zhongjian Mou, Jian Zhang, Chi Zhen
Herausgeber/-in
Ole Döring
Gesellschaft/Politik
Soviet and Post-Soviet Politics and Society
Ukrainian Voices
Balkan Politics and Society
Impulse. Debatten zu Politik, Gesellschaft, Kultur
Changing Europe
Göttinger Junge Forschung
Jahrbücher des Göttinger Instituts für Demokratieforschung
Explorations of the Far Right
CROP International Poverty Studies
Chinesische Perspektiven
Chinesische Perspektiven: Politik
Chinesische Perspektiven: Philosophie
Chinesische Perspektiven: Ökonomie
Chinesische Perspektiven: Geschichte
Analyzing Political Violence
Beyond the Social Sciences
Critical Studies on Latin America
Von Antidiskriminierung zu Diversity und Inklusion
Schemapädagogik kompakt
Systemische Impulse für die Soziale Arbeit
Integrale Pädagogik
European Studies in the Caucasus
Dialogisches Lernen
Body-Feeling und Body-Bildung
Ökonomische Forschungsbeiträge zur Umweltpolitik
Ecological Energy Policy
Interdisziplinäre Schriftenreihe des Centrums für interkulturelle und europäische Studien
Außenhandelspolitik und -praxis
Städtebau - Architektur - Gesellschaft
Kultur - Bildung - Gesellschaft
Qualität und Qualitätssicherung in der Sozialen Arbeit
Literatur/Sprache
Natur/Technik


At the Uitmarkt Amsterdam and The Hague UIT Festival, Dancing on the Edge will raffle tickets for Badke, the opening performance of the upcoming festival. Everyone who visits us at our stands can participate! In Amsterdam you can find us on Saturday 31 August from 12:00 – 18:00, and Sunday 1 September from 13:00 – 18:00 at the Museumplein in tent 116. In The Hague you can find us on Sunday 8 September from 13:00 – 18:00 at the information market.


The chestnut gall wasp, Dryocosmus kuriphilus one of Canada's newest invasive species was found to be more invasive than previously known. The wasp reproduces without the presence of a male. A study was done by University of Extremadura, in Spain. In a very short time one or very few female chestnut gall wasps give rise to genetically homogeneous populations which are threatening the chestnut sector in Europe. This study was published in ScienceDaily 21 May 2018.
The wasps are producing genetically identical daughters. To date, no males have been found in Italy where is was introduced in 2006. This shows the insect is both highly vulnerable and highly reproductive.
At this point, the only defense recommended by the authors is maintaining all stock for at least one year to ensure the tiny wasp eggs are not transported. Fleshy balls form on the tips of branches of infected stock. As the invasive is new to Canada, nothing is listed for use to control it in this country.