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

There is growing evidence that some disordered systems at criticality
share two unusual features: The existence of logarithmic operators
in the spectrum of the theory and the existence of an infinite number of
relevant operators with negative conformal dimensions.
Logarithmic operators seem to be connected to hidden continuous symmetries.
In \cite{ckt} logarithmic operators were found to generate a change
in a coupling constant of the effective WZNW action for $SU(2r)$,
$r\rightarrow 0$, obtained after use of the replica trick. The existence
of an infinite number of conformal operators \cite{mcw} means that there is
an infinite number of relevant perturbations which render the critical
point unstable. Such a set of conformal dimensions is also related to
the phenomenon of multifractality.

Our aim in this paper is to show that both features follow naturally from
a supersymmetric treatment of the disorder. We consider  the
two-dimensional
random bond Ising model at criticality and the random Dirac model in $2+1$
dimensions. These gaussian models
allow a supersymmetric formulation of averages over disorder
\cite{efetov}. One identifies a global superalgebra symmetry of the
effective action and assumes  it is enhanced
to an affine symmetry at a new
critical point \cite{mcw,db}, as happened at the pure
critical point.
One then derives the Sugawara stress-energy tensor
and obtains the set of conformal dimensions associated with the primary fields.
Additional constraints such as locality  allow
to further restrict the operator content of the theory. This provides
a class of, possibly disordered, critical points in two spatial dimensions.
The specific structure of  the superalgebra implies that
such dimensions  can  generically be negative. The presence of logarithmic
operators is also straightforward from the point of view of
this algebra. Unlike ordinary Lie algebras, superalgebras  have
indecomposable (not fully reducible) representations. It is then
possible to show on general grounds  that such representations
imply the existence of logarithmic operators and logarithms in correlation
functions.

This paper is organized as follows. In section 2 we briefly recall the
supersymmetric approach for the random bond Ising model and the random Dirac
model. In section 3 we write down the $osp(2/2)_k$ current algebra and find
the Sugawara stress-tensor. In section 4 we briefly review the representations
of $osp(2/2)$ and obtain the conformal weights of primary fields. In section 5
we derive the quantization of $k$ from locality constraints. In section 6
we show how indecomposable representations lead to logarithmic
operators and consider an atypical representation. In section 7
we obtain some logarithm-dependent four-point
functions. We conclude in section 8.

\section{The supersymmetric method for disordered systems}

The supersymmetric method is applicable to
models which are gaussian at fixed disorder, but it provides a
good starting point for disentangling properties of a large class
of disordered conformal field theories.
The study of the random bond Ising model reveals that an appropriate
algebraic framework for studying gaussian disordered
systems at criticality should be based on affine Lie
superalgebra with zero superdimension. In the case of
the random Ising model this algebra is $osp(2N/2N)$.
The fact that the algebra has an equal number of bosonic and fermionic
generators ensures that the Virasoro central charge
of the Sugawara stress-tensor vanishes. This is needed
by construction for a disordered system. The vanishing of the
central charge does not imply the triviality of the theory
since it is not unitary.
The relevance of affine Lie superalgebras
was independently realized in refs. \cite{mcw,db}.

We now briefly recall the supersymmetric method for the  study
of the random bond Ising model and the random Dirac model.
These models have been  analyzed using the replica method in
refs. \cite{DoDo,Lu1,Shan}. The random bond Ising model
has also been studied in the context of massless scattering
theories \cite{mussardo}.

\vskip 10pt
{\it The random bond Ising model}

In the scaling limit, near criticality, the Ising model is
described by a massive real Majorana fermion with mass
$\overline{m}\sim \tau = \frac{(T_C-T)}{T_C}$ were $T_C$ is the critical temperature.
In  the presence of disorder, {\it i.e.} when the coupling
constants between sites belong to a random set,
the mass becomes a function of space position.
The random Ising model is defined by the action $(z=x+iy)$:
\begin{eqnarray}
S[m(x)]=\int \frac{d^2x}{4\pi}
\left( \psi \partial_\zb \psi + \overline{\psi} \partial_z
\overline{\psi} + i\, m (x) \overline{\psi} \psi \right)
\label{act1}
\end{eqnarray}
where $\psi$ and $\overline{\psi}$ are grassmanian fields.
The mass $m(x)$ is chosen to be a quenched random
variable with a gaussian measure:
\begin{eqnarray}
P[m] = \exp\left({- \inv{4g} \int \frac{d^2x}{2\pi}
(m(x)-{\overline m})^2 }\right) \; . \label{mesur}
\end{eqnarray}

The energy operator $\epsilon(x) = i\overline{\psi}(x)\psi(x)$  has
dimension one. The Harris criterion
tells us that randomness in the bond interaction is
marginal in the 2d Ising model. It turns out that it is not
exactly marginal but only marginally irrelevant.
At criticality the disorder only induces
logarithmic corrections to the pure system.

In order to compute averages of products of
correlation functions one introduces a
number of copies of fermions and of their supersymmetric
partners equal to the number of correlation
functions in the product \cite{efetov,db}.
One then rewrites these averages as fermionic
and bosonic path integrals. For the Ising model with $\overline{m}=0$
one obtains the following effective action  for the disorder
average of the product of two correlation functions:
\begin{eqnarray}
S_{\rm eff} &=& \int \frac{d^2x}{2\pi}\left(
\psi_- \partial_\zb \psi_+ + \overline{\psi}_- \partial_z \overline{\psi}_+
+\eta\partial_\zb \gamma + {\bar \eta }\partial_z{\bar \ga}\right)
+\frac{g}{8} \int\frac{d^2x}{\pi}~\Phi_{\rm pert} \nonumber\\
&=& S_* + \frac{g}{8} \int\frac{d^2x}{\pi}~\Phi_{\rm pert}
\label{Seff1}
\end{eqnarray}
with
\begin{equation}
\Phi_{\rm pert}=\left(\overline{\psi}_- \psi_+ - \psi_- \overline{\psi}_+
+{\bar \eta }\gamma-\eta{\bar \ga} \right)^2 \label{pert1} \;.
\end{equation}
The $\psi$ are complex fermions and $\eta$ and $\gamma$ are complex
bosonic fields.
This action can be viewed as a perturbation of the (non-unitary)
conformal field theory specified by the action $S_*$.
This fixes the normalization of the fields to be:
\begin{eqnarray}
\psi_-(z) \psi_+(w) \sim \inv{z-w} \quad,\quad
\gamma(z)\eta(w) \sim \inv{z-w} \; . \label{norm}
\end{eqnarray}
The central charge of the Virasoro algebra is zero. Note that since
the fermions $\psi_\pm$ have dimension one half the perturbing
field  $\Phi_{\rm pert}$ has dimension two.
It is therefore marginal.

The conformal field theory specified by $S_*$ is
invariant under an affine supersymmetric algebra whose
conserved currents are:
\begin{eqnarray}
G_\pm(z) = \eta(z)\psi_\pm(z) \quad&,&\quad
\widehat G_\pm(z) = \gamma(z)\psi_\pm(z)\; , \nonumber\\
K(z) = \eta^2(z) \quad&,&\quad \widehat K(z)=\gamma^2(z)\; , \nonumber\\
J(z)=:\psi_-(z)\psi_+(z): \quad&,&\quad
H(z)=:\gamma(z)\eta(z):\;\; .\label{curr}
\end{eqnarray}
The dots refer to fermionic and bosonic normal ordering.
There are four fermionic currents, $G_{\pm}$ and $\widehat G_{\pm}$,
which are generators of supersymmetric transformations, and four bosonic ones.
They form a representation of the affine $osp(2/2)$ \cite{snr,marcu}
current algebra at level one.

The perturbing field  can also be written as a bilinear
in the currents,
\begin{equation}
\Phi_{\rm pert}=2\left[ \overline{J} J - \overline{H} H +\frac{1}{2} (\overline{K} \widehat K
+\overline{\widehat K} K) + \overline{G}_- \widehat G_+ -
\overline{\widehat G}_- G_+ + \overline{G}_+ \widehat G_- - \overline{\widehat G}_+ G_-
\right]\,.
\end{equation}
In other words the perturbation (\ref{pert1}) is
a current-current perturbation. Therefore the
action $S_{\rm eff}$ preserves a global $osp(2/2)$ symmetry.

\vskip 10pt
{\it The random Dirac model}

The random  model of Dirac fermions
has been introduced in connection with
the quantum Hall transition \cite{luetal}.
Its action is:
\begin{eqnarray}
S &=&\int \frac{d^2 x}{2\pi}\left(
  \psi_- \partial_\zb \psi_+ + \overline{\psi}_- \partial_z \overline{\psi}_+
\right.\nonumber\\
 &&\left. +i\frac{m(x)}{2} (\overline{\psi}_- \psi_+ -\psi_- \overline{\psi}_+ )
 +i\frac{V (x)}{2} (\overline{\psi}_- \psi_+ +\psi_- \overline{\psi}_+ )\right)\; .
\end{eqnarray}

The random variables $m$ and $V$ have a
gaussian distribution with widths $g_M$ and $g_V$.
We denote by $\Phi_M$ and $\Phi_V$ the
perturbing fields coupled to the constants
$g_M$ and $g_V$ after averaging over the disorder.
In the two-copy sector we can write them in terms of
the $osp(2/2)$ currents:
\begin{eqnarray}
\Phi_V &=& 2 \overline{H} H - 2 \overline{J} J + \overline{K} \widehat K + \overline{\widehat K} K
+ 2 \overline{G}_- \widehat G_+ - 2 \overline{\widehat G}_+ G_-
+ 2\overline{\widehat G}_- G_+  -2 \overline{G}_+ \widehat G_- \; ,\\
\Phi_M &=& 2 \overline{J} J  - 2 \overline{H} H  + \overline{K} \widehat K +
\overline{\widehat K} K  + 2 \overline{G}_- \widehat G_+ - 2 \overline{\widehat G}_+ G_-
- 2\overline{\widehat G}_- G_+  + 2 \overline{G}_+ \widehat G_- \; .
\end{eqnarray}
We see that $\Phi_M=\Phi_{\rm pert}$.

Both $\Phi_M$ and $\Phi_V$ preserve a global $osp(2/2)$ symmetry, whose
generators are
\begin{equation}
\begin{tabular}{rrrr}
$H_0+\overline H_0\,,$  & $\quad J_0+\overline J_0\,,$ &
$\quad K_0+\overline K_0\,,$ &
$\quad {\widehat K}_0+\overline{\widehat K}_0\,,$  \\
$G_{+0}+\overline G_{+0}\,$ &  $\quad {\widehat G}_{+0}+
\overline{\widehat G}_{+0}\,,$ &
$\quad G_{-0}+\overline G_{-0}\,,$ & $ \quad {\widehat G}_{-0}+
\overline{\widehat G}_{-0}\,.$
\end{tabular}
\end{equation}
for the  perturbation $\Phi_M$, and
\begin{equation}
\begin{tabular}{rrrr}
$H_0+\overline H_0\,,$  & $\quad J_0+\overline J_0\,,$ &
$\quad K_0-\overline K_0\,,$ &
$\quad {\widehat K}_0-\overline{\widehat K}_0\,,$  \\
$G_{+0}+\overline G_{+0}\,,$ &  $\quad {\widehat G}_{+0}-
\overline{\widehat G}_{+0}\,,$ &
$\quad G_{-0}-\overline G_{-0}$\,, & $ \quad {\widehat G}_{-0}+
\overline{\widehat G}_{-0}\,.$
\end{tabular}
\end{equation}
for the perturbation $\Phi_V$. Here the index $0$ denotes the zero
modes of the currents $J^a(z)=\sum_n J^a_nz^{-n-1}$.
If both perturbations are present,
the interaction preserves an $u(1/1)$ symmetry generated by
\begin{equation}
H_0+\overline H_0\,, \quad J_0+\overline J_0\,,  \quad G_{+0}+
\overline G_{+0}\,, \quad
{\widehat G}_{-0}+\overline{\widehat G}_{-0}\,.
\end{equation}

The perturbative study of the
Dirac theory with a random potential and a random mass
is very similar to the perturbative study
of the random mass Ising model.
However the crucial difference is that contrary to
the randomness of the mass, the randomness of the potential
is marginally relevant.
At $g_M=0$, the one-loop beta function is given by $\dot g_V=8 g_V^2$.
This means that $g_V$ grows at large distances. The infrared fixed
point has not yet been determined \cite{luetal}.
Should the affine symmetry, present at the UV fixed point of the pure system,
be restored at the IR point, then we
believe this point should belong to the set we find in
section \ref{quantiz}.

\section{The current algebra approach}

The critical free theory described by the action $S_*$ has a current
algebra symmetry, $osp(2/2)_1$, on both its  holomorphic and antiholomorphic
sectors. When randomness is introduced in the pure system the effective action
is no longer critical, and the current algebra symmetry is reduced
to a global $osp(2/2)$ symmetry. However  at a new critical point the
conformal invariance restores the current algebra symmetry.
The value of the level $k$ is not preserved by the renormalization
flow. Therefore its value at the IR fixed point could be different
from its value at the UV point.
It is possible to extract
properties at the new critical point by studying the current algebra
and its associated stress-energy tensor.
In this context we consider $osp(2/2)_k$ at arbitrary  $k$
and obtain the Sugawara tensor.

\subsection{The $osp(2/2)_k$ algebra}

In this section we write down the singular terms of
the Operator Product Expansions \cite{ginsparg} satisfied by the currents
of the affine $osp(2/2)$ algebra at level $k$.
The non-trivial OPE of the currents
are easily obtained from the currents (\ref{curr}). The level $k$
appearing below is equal to 1 for these currents. We find:
\begin{eqnarray}
J(z)J(w) \sim \frac{k}{(z-w)^2}
\quad &;& \quad
H(z)H(w) \sim \frac{-k}{(z-w)^2} \nonumber\\
J(z)G_\pm(w) \sim \frac{\pm1}{z-w} G_\pm(w)
\quad &;& \quad
J(z)\widehat G_\pm(w) \sim \frac{\pm1}{z-w} \widehat G_\pm(w)\nonumber\\
H(z)G_\pm(w) \sim \frac{1}{z-w} G_\pm(w)
\quad &;& \quad
H(z)\widehat G_\pm(w) \sim \frac{-1}{z-w} \widehat G_\pm(w)\label{opecurrent}\\
H(z) K(w) \sim \frac{2}{z-w} K(w)
\quad &;& \quad
H(z)\widehat K(w) \sim \frac{-2}{z-w}\widehat K(w)\nonumber\\
\widehat G_\pm(z) G_\mp(w) &\sim& \frac{k}{(z-w)^2}
+ \frac{1}{z-w}(H(w)\pm J(w)) \nonumber\\
\widehat K(z) K(w) &\sim& \frac{2k}{(z-w)^2}
+\frac{4}{z-w} H(w) \nonumber\\
G_-(z) G_+(w) \sim \frac{1}{z-w} K(w)
\quad &;& \quad
\widehat G_-(z) \widehat G_+(w) \sim \frac{1}{z-w} \widehat K(w) \nonumber\\
K(z) \widehat G_\pm(w) \sim \frac{-2}{z-w} G_\pm(w)
\quad &;& \quad
\widehat K(z) G_\pm(w) \sim \frac{2}{z-w} \widehat G_\pm(w)\nonumber
\end{eqnarray}

Requiring the algebra to be associative constrains the possible
central terms extensions to the ones appearing in the OPE's (\ref{opecurrent}).
We can rewrite these OPE's in a more compact form as
\begin{equation}
J^a(z) J^b(w) \sim k \frac{\kappa^{ab}}{(z-w)^2} +f^{ab}_{\;\;\;\; c}\;
\frac{J^c(w)}{z-w}\; .\label{jj}
\end{equation}
The $f^{ab}_{\;\;\;\; c}$ are the structure
constants of the Lie superalgebra $osp(2/2)$,
and $\kappa^{ab}$ is proportional to its non-degenerate Killing form.

\subsection{The Sugawara stress-energy tensor}

We now construct the Sugawara stress-energy tensor \cite{suga}.
It is bilinear
in the currents defined in the previous section. Because of singularities
which appear at coinciding points one has to consider a regularized
version where normal ordered products of currents appear. This
is equivalent to a point splitting procedure where the singular parts
appearing in the OPE's of the currents are subtracted.
We take the usual definition for the  normal ordered product
of two fields $A(z)$ and $B(w)$:
\begin{equation}
:AB:(w) \equiv \oint_w\frac{dz}{2\pi i}\frac{A(z)B(w)}{z-w}
\label{nord}
\end{equation}
It should be noted that this ordering prescription does not coincide in general
with the Wick prescription used in (\ref{curr}).
The Sugawara stress-energy tensor $T(z)$ is given
by
\begin{equation}
T(z)= \frac{1}{\kappa} \kappa_{ab} :J^a(z) J^b(z): \label{suga}
\end{equation}
where $\kappa_{ab}$ is the inverse of  $\kappa^{ab}$.
The constant $\kappa$ is determined by requiring $J^a(z)$ to be
a primary field of conformal weight one:
\begin{equation}
T(z)J^a(w) \sim \frac{J^a(w)}{(z-w)^2} + \frac{\partial J^a(w)}{z-w}\;.
\label{tjprim}
\end{equation}
When calculating OPE's one has to add a minus sign each time two odd
generators are permuted.
We find
\begin{eqnarray}
T(z) &=&\frac{1}{4-2k}\left( : H(z)H(z)-J(z)J(z)-\frac{1}{2}(K(z)\widehat{K}(z)
+\widehat{K}(z)K(z))\right. \nonumber\\
&+& \left. \widehat{G}_+(z)G_-(z) -G_-(z)\widehat{G}_+(z) +
\widehat{G}_-(z)G_+(z) -G_+(z)\widehat{G}_-(z):\right) \;.
\end{eqnarray}
The Sugawara tensor also satisfies
\begin{equation}
T(z)T(w)\sim \frac{c/2}{(z-w)^4} + 2 \frac{T(w)}{(z-w)^2} +
\frac{\partial T(w)}{z-w} \;.
\end{equation}
For a general Lie superalgebra, it is easy to show that the Virasoro central
charge $c$ is proportional to the
superdimension; this is the difference between the number
of even and odd generators. Therefore $c$ vanishes for
the $osp(2/2)$ algebra, and more generally for any superalgebra
with an equal number of even and odd generators.
More precisely one has:
\begin{equation}
c=2 k\,\frac{\kappa_{ab}\kappa^{ab}}{\kappa}=
2 k \,\frac{{\rm sdim}\, G}{\kappa}=0 \; .
\end{equation}
Because $c$ vanishes
$T(z)$ is a primary field, contrary to the case $c\neq 0$ where $T$ is
just a level-two descendant of the unit operator.

\section{Primary Fields}

We briefly describe the representations of $osp(2/2)$
and find the corresponding conformal weights.

\subsection{Some $osp(2/2)$ representations \label{reps}}

Unlike ordinary Lie algebras, the are two types of representations
for most superalgebras. The {\em typical} representations are irreducible and
are similar to those of ordinary Lie algebras.
The {\em atypical} representations have no counterpart
in the ordinary Lie algebra setting. They can be irreducible or not
fully reducible (read reducible but indecomposable).

The superalgebra $osp(2/2)$ is isomorphic to the superalgebra $spl(2/1)$.
The representation theory of the latter algebra was studied
in \cite{snr,marcu}.
The quadratic Casimir of $osp(2/2)$  is
\begin{equation}
C_2 =\frac{1}{2}\left( H^2 -\frac{1}{2} (K\widehat K+\widehat K K)-J^2+
\widehat G_+G_--G_-\widehat G_++\widehat G_-G_+-G_+\widehat G_-\right)\;.
\end{equation}
The four even generators $K,\widehat K,H,J$ form a
$su(2)\oplus u(1)$ subalgebra.
The correspondence with the notation of \cite{snr}
is: $Q_+=-\widehat K/2$, $Q_-=K/2$, $Q_3=-H/2$, $B=J/2$,
$V_+=\widehat G_+/\sqrt{2}$, $V_-=-G_+/\sqrt{2}$, $W_+=-\widehat G_-/\sqrt{2}$,
$W_-=G_-/\sqrt{2}$.

Let $b$ and $q_3$ be the eigenvalues of $B$ and $Q_3$.
Generically, a representation $(b,q)$,
$b \in \bC  \; ,\; q=0,\frac{1}{2},1,\frac{3}{2},...$,
contains four $su(2)\oplus u(1)$ multiplets:
\begin{eqnarray}
|b,q,q_3\rangle \; &,&\; q_3=-q,-q+1,...,q-1,q \;\; {\rm if}\;\; q\geq 0 \;,
\label{rep1}  \\
|b+\frac{1}{2},q-\frac{1}{2},q_3\rangle \; &,& \;q_3=-q+\frac{1}{2},...,q-\frac{3}{2},q-\frac{1}{2}
\;\; {\rm if}\; \; q\geq \frac{1}{2}\;, \label{rep2} \\
|b-\frac{1}{2},q-\frac{1}{2},q_3\rangle \; &,& \;q_3=-q+\frac{1}{2},...,q-\frac{3}{2},q-\frac{1}{2}
\;\;{\rm if}\;\; q\geq \frac{1}{2}\;,\label{rep3} \\
|b,q-1,q_3\rangle \; &,& \; q_3=-q+1,...,q-2,q-1 \;\;{\rm if}\;\; q\geq 1\; .
\label{rep4}
\end{eqnarray}
The action of the four even generators on these multiplets is the one implied
by the notation. The four odd generators mix the different multiplets.
The vector $|v\rangle=|b,q,q\rangle$ is a highest weight
vector, {\it i.e.} it satisfies
\begin{equation}
\widehat K |v\rangle=\widehat G_+ |v\rangle=\widehat G_- |v\rangle=0\;.
\end{equation}
The quadratic Casimir of this representation is $C_2=2(q^2-b^2)$.

If $b\neq \pm q$ the representation is denoted by $[b,q]$ and is typical;
the quadratic and cubic Casimirs do not vanish. All the vectors in
the representation can be obtained from the highest weight
vector $|v\rangle$ by applying on it polynomials in the generators.
The representation $[b,q]$ has dimension $8q$.
The  representation $[0,1/2]$ is four-dimensional
and contains one spin $1/2$ and two spin $0$ multiplets.

{\it Atypical representations}

When $b=\pm q$ several kinds of atypical representations arise.
Both Casimirs vanish, and yet these representations are not the
trivial one-dimensional representation.
One kind has dimension $4q+1$ and is denoted by $[q]_{\pm}$.
To obtain $[q]_+$ (resp. $[q]_-$)
one drops the two multiplets
(\ref{rep3}) and (\ref{rep4}) (resp. (\ref{rep2}) and (\ref{rep4})).
These representations are irreducible.

{\it Atypical indecomposable representations}

Generally, they are semidirect sums of atypical irreducible
representations. They can contain two, three or four terms
and they  arise in tensor products of irreducible
representations.
An interesting example, containing four irreducible representations,
arises in the tensor product of two
representations $[0,1/2]$. The result
is the direct sum of $[0,1]$ and
an eight-dimensional representation which is the semidirect sum
of $[1/2]_-$, $[1/2]_+$ and two $[0]$ representations.
We call it $[0,-1/2,1/2,0]$ (see fig. 1 b).
The vector $s$ is
invariant; it is annihilated by all the generators. The quadratic Casimir
$C_2$ vanishes on all states except $t$: $C_2\ t=4 s$.
We have
\begin{eqnarray*}
s&=&\frac{1}{2} \left( |\frac{1}{2},0\rangle\otimes |-\frac{1}{2},0\rangle-|0,\frac{1}{2}\rangle\otimes
|0,-\frac{1}{2}\rangle + |0,-\frac{1}{2}\rangle\otimes|0,\frac{1}{2}\rangle + |-\frac{1}{2},0\rangle\otimes
|\frac{1}{2},0\rangle \right) \\
t&=&\frac{1}{2} \left( |\frac{1}{2},0\rangle\otimes |-\frac{1}{2},0\rangle+|0,\frac{1}{2}\rangle\otimes
|0,-\frac{1}{2}\rangle - |0,-\frac{1}{2}\rangle\otimes|0,\frac{1}{2}\rangle + |-\frac{1}{2},0\rangle\otimes
|\frac{1}{2},0\rangle \right)
\end{eqnarray*}

\subsection{Tensor product of $osp(2/2)$ representations}

The tensor product of two irreducible representations of a
superalgebra is not
necessarily completely reducible.
Ref. \cite{snr} gives a sufficient condition for a tensor
product of two $osp(2/2)$ representations to be completely reducible,
and the irreducible components. The result is the following
\begin{eqnarray}
\label{tpr}
&[b,q]\otimes [b',q']=[b+b',q+q']\oplus [b+b',q+q'-1]\oplus\ldots
\oplus [b+b',|q-q'|]& \nonumber \\
&\oplus [b+b',q+q'-1] \ldots \oplus [b+b',|q-q'|+1]&\nonumber \\
&\oplus [b+b'+1/2,q+q'-1/2] \ldots \oplus [b+b'+1/2,|q-q'|+1/2]&\nonumber \\
&\oplus [b+b'-1/2,q+q'-1/2] \ldots \oplus [b+b'-1/2,|q-q'|+1/2]& \\
& {\rm if} \;\; \pm b>q\geq \frac{1}{2}, \quad \pm b'>q' \geq \frac{1}{2} \; .&\nonumber
\end{eqnarray}
For the other values of $b$, $b'$, this expression still gives the
correct content of $su(2)\oplus u(1)$ charges of the tensor product.
This expression also permits to obtain some information on the
reducibility: if two components have different Casimirs,
then they belong to different (maybe indecomposable) representations.
Vanishing Casimirs are, generally, signs of some pathologies.

\vskip 0.5cm
\epsfxsize 16.0 truecm
\epsfbox{tensorp.eps}
\vskip 0.5cm
{\small {\bf Fig.1:} a) The tensor product of the representations
$[b,1/2]$ and $[b',1/2]$. If $b>1/2$, $b'>1/2$, the result is the direct
sum of $[b+b',1]$ (white dots), $[b+b'+1/2,1/2]$ (black dots) and
$[b+b'-1/2,1/2]$ (grey dots). The arrows represent the action of the
generators.
 b) The representation $[0,-1/2,1/2,0]$.
The vector $t$ (black dot) is cyclic and $s$ (white dot) is invariant.
The action of the fermionic generators $\widehat G_+$,
 $\widehat G_-$, $G_+$ and  $G_-$
is represented by arrows.}
\vskip 0.4cm

\subsection{$osp(2N/2N)$ representations}

When calculating the disorder average of product of correlation
functions for the Ising and Dirac models, we can introduce an
arbitrary number of copies of
fermions and bosons. One is led is to consider $osp(2N/2N)$ as symmetry.
The theory is consistent if the results
of the calculations do not depend on $N$.
One step in this direction is to show that the conformal dimensions of the
primary fields are independent of $N$.

The main difference between the superalgebras $osp(2/2)$ and $osp(2N/2N)$
with $N>1$ is that the even subalgebra of the first one is not simple.
This is why representations of the former are indexed by a
continuous parameter, while the representation of the
latter are characterized only by discrete variables.
According to the classification of the superalgebras \cite{kac}, $osp(2/2)$
is the first of the series $C(N+1)$, while $osp(2N/2N)$ is a superalgebra
of the type $D(N,N)$.

The finite dimensional typical representation of the superalgebras were
classified by Kac \cite{kac}.
For $D(N,N)$, the typical representations are characterized by
their highest weights, $\Lambda=\sum_{i=1}^Na_i\delta_i+\sum_{i=1}^Na_{N+i}
\epsilon_i\,;$ $\delta_i$ and $\epsilon_i\,,$ $i=1,N$ form
an orthogonal basis with $(\epsilon_i,\epsilon_j)=-\delta_{ij}\,,$
$(\delta_i,\delta_j)=\delta_{ij}\,,$ $(\epsilon_i,\delta_j)=0$.
The `numerical marks' $a_i$ satisfy the following conditions:

i) $a_i \in {\Bbb Z}_+$, $i\neq N$,

ii) $j=a_N-a_{N+1}-...-a_{2N-2}-\frac{1}{2}(a_{2N-1}+a_{2N}) \in {\Bbb Z}_+$,

iii) $a_{N+j+1}=...=a_{2N}=0$, if $j\leq N-2$; $a_{2N-1}=a_{2N}$,
if $j=N-1$.

The first two conditions express the fact that $\Lambda$ is a
dominant weight for the even algebra $Sp(2N)\otimes O(2N)$.

The value of the quadratic Casimir for the representation with
highest weight $\Lambda$ is given, up to a normalization,
by $(\Lambda, \Lambda+2\rho)$, with $\rho=\rho_0-\rho_1$ and $\rho_{0(1)}$
is half the sum of the even (odd) positive roots.
Considering the positive roots associated to the distinguished
Dynkin diagram (the Dynkin diagram with a single fermionic root),
we obtain
$$\rho=\sum_{i=1}^N (N-i)(\epsilon_i-\delta_{N-i+1})\;.$$
We normalize the value of the quadratic Casimir of the adjoint
representation $(2,0,...,0)$ to be $1$; the Casimir of the representation
$(a_1,...,a_{2N})$ will be
$$C_2(\Lambda)={1\over 4}\sum_{i=1}^N
\left[(a_i-i+1)^2-(a_{N+i}+N-i)^2\right]\;.$$
For the fundamental representation $(1,0,...,0)$ we have $C_2=1/4$,
independent of $N$.

In order to obtain compatibility between the results obtained in the
$osp(2/2)$ and $osp(2N/2N)$ frameworks, the continous parameter $b$
of the $osp(2/2)$ representations has to be constrained to the discrete
values compatible with the value of the discrete parameters of the
$osp(2N/2N)$ representations.

\subsection{Conformal dimensions of the primary fields}

The primary fields $\phi^\alpha(w)$ are highest weight vectors
of affine $osp(2/2)_k$ and of the Virasoro algebra.
In terms of operator products one has:
\begin{eqnarray}
J^a(z) \phi^\beta(w) &\sim& (T^a)^{\gamma\beta}\;\frac{\phi^\gamma(w)}{z-w}\;,
\label{jprim}\\
T(z) \phi^\alpha(w) &\sim& \Delta\frac{\phi^\alpha(w)}{(z-w)^2}
+\frac{\partial \phi^\alpha(w)}{(z-w)}\;.\label{tprim}
\end{eqnarray}
Using (\ref{jj}) and (\ref{jprim}) one then shows that the matrices
$T^a$ form a representation of the
$osp(2/2)$ algebra. But this applies to any algebra.
Using the Sugawara form (\ref{suga}) of the stress-energy tensor
and (\ref{jj}) one obtains the conformal weights corresponding to
a particular representation. For the representations
$[b,q]$ and $[q]_{\pm}$ the Casimir $C_2$ is diagonal and
we obtain:
\begin{equation}
\Delta_{(b,q)}=\frac{1}{\kappa} \kappa_{ab}T^a T^b=\frac{2(q^2-b^2)}{2-k}\;.
\end{equation}

We are also able to calculate the conformal dimensions of the primary
fields for all $N$. Due to the fact that the dual Coxeter number of
$osp(2N/2N)$ equals 1 for all $N$, these conformal dimensions do
not change if we change the number of copies to $N'>N$; for example
$$\Delta_{(1,0,...,0)}={1\over 4-2k}$$
is the conformal dimension of the fields in the fundamental representation.

The conformal dimensions are potentially negative;
this reflects the non-unitarity of the theory we considered.

\section{Local Operators and Quantization of $k$ \label{quantiz}}

As seen in section two the interactions generated by
the random mass and random scalar potential conserve diagonal
$osp(2/2)$, $J_x^a=J_0^a\pm\bar J_0^a$. We expect to find
in the spectrum of the theory, at the new fixed point for the
potential interaction, particles
which carry the same charges $H_x$ and $J_x$ as the original ones (bosons and
fermions).  The fields corresponding to these particles are
no more  holomorphic or antiholomorphic, but they can be made
up of the new primary fields.
Suppose that they are of the form
\begin{equation}
\label{champs}
\phi(z,\bar z)=\phi_{[b,q]}(z)\phi_{[b',q']}(\bar z)
\end{equation}
where $\phi_{[b,q]}(z)$ ($\phi_{[b',q']}(\bar z)$) are primary
fields belonging to the representation
$[b,q]$, respectively $[b',q']$ of the two chiral algebras $J^a(z)$
and $\bar J^a(\zb)$.

The charges of the bosons and fermions with respect to the currents
$J^a_x$ are:
\begin{equation}
\label{fb}
\begin{tabular}{rrr}
$\  $&$J_x $ &$ H_x$ \\
$\psi_+(z) \qquad $ & $1/2$ &$0$ \\
$\psi_-(z) \qquad$  & $-1/2$ &$0 $ \\
$\gamma(z) \qquad$  & $0$ &$1/2 $ \\
$\eta(z) \qquad$  & $0$ &$-1/2 $
\end{tabular}
\end{equation}
We have to identify the representation $[0,1/2]$
in the tensor product of the
representations $[b,q]$ and $[b',q']$.
Using equation (\ref{tpr}) we obtain the following possibilities
\begin{equation}
\label{spinb}
 a) \quad q=q', \quad b+b'\pm 1/2=0\;,\;\;\;\;
b) \quad |q-q'|=1/2, \quad b+b'=0\;.
\end{equation}

As in  \cite{mcw}, in order to obtain a sensible theory,
we impose the constraint of locality
on the fields $\phi(z,\bar z)$.
The OPE of two such composite fields can be written as
\begin{equation}
\label{ope}
\CO_a(z,\zb) \CO_b(0,0)=\sum_c C_{ab}^c\, z^{\Delta_c-\Delta_a-\Delta_b}
\ \zb^{\bar\Delta_c-\bar\Delta_a-\bar\Delta_b} \CO_c(0,0)\;.
\end{equation}
The operators $\CO_a$ and $\CO_b$ are said to be mutually local
if the OPE is single-valued, which means that the spin difference
is an integer
\begin{equation}
s_c-s_a-s_b=\Delta_c-\Delta_a-\Delta_b-\bar \Delta_c+\bar \Delta_a+ \bar \Delta_b
\in {\Bbb Z}\;.
\end{equation}
Let us take $\CO_a(z,\zb)=\CO_b(z,\zb )
=\phi_{[b,q]}(z)\phi_{[b',q']}(\zb)$. We evaluate the left and right
chirality OPE's separately. The
operators appearing in the OPE for the left chirality are the
operators associated to the representations $\alpha$
appearing in the tensor product
$[b,q]\otimes[b,q]$ (see eq. (\ref{tpr}))
\begin{eqnarray*}
\phi_{[b,q]}(z)\phi_{[b,q]}(0)
& \sim & \sum_{\alpha} z^{\Delta_{\alpha}-2\Delta_{[b,q]}}\; \phi_{\alpha}(0)
\end{eqnarray*}
and similarly for the right chirality, with $[b,q]$ replaced by
$[b',q']$.
We ignored the possible existence of logarithms in this
OPE (when two conformal dimensions are degenerate).
We also supposed that all the primary fields allowed by the
$osp(2/2)$ selection rules (and their
descendants) appear in
the OPE. For a Kac-Moody algebra
this is not always the case; the operator content of the
theory depends on the value of the level $k$.
In fact the calculation shows that the constraint of locality
depends very weakly on how many of the operators $\phi_{\alpha}$
really appear in the OPE.

The condition $s_c-s_a-s_b \in {\Bbb Z}$ is fulfilled for all
the possible operators $\CO_c(0,0)$ in (\ref{ope}) if
\begin{eqnarray}
\label{quant}
&2l(4q-1)\in {\Bbb Z}\;, \quad 2l(4q-2)\in {\Bbb Z}\;, \quad 4l(q-b)\in {\Bbb Z}\;,\\
&2l(4q'-1)\in {\Bbb Z}\;, \quad 2l(4q'-2)\in {\Bbb Z}\;, \quad 4l(q'-b')\in {\Bbb Z}\;, \\
&{\rm and} \quad\;\; 4l(q^2-b^2-q'^2+b'^2) \in {\Bbb Z}\;,
\end{eqnarray}
where $l=1/(2-k)$.
With the two possibilities in (\ref{spinb}) we obtain the following
restriction on $k$:
\begin{equation}
k=2-\frac{1}{l}\;, \qquad l\in {\Bbb Z}\;. \label{restri}
\end{equation}

\section{Logarithmic operators \label{loga}}

We argue that the appearance of  logarithmic operators is a
general feature of theories with an underlying superalgebra symmetry,
as long as indecomposable representations are concerned.
We give as an example the occurrence of
logarithmic correlations for $osp(2/2)$.

The first logarithmic correlation functions
in conformal field theory were obtained by Rozansky and Saleur in \cite{saleur}
in the framework of the $GL(1,1)$ WZW model. Later
Gurarie \cite{gurarie} pointed out that the logarithms are related
to non-diagonalizable Virasoro generators. See also \cite{ckt} for disordered
models,  \cite{kausch},  and
\cite{flohr} in relation to the fractional quantum Hall effect.
Suppose we are given two operators
$A_1(z)$, $A_2(z)$ with:
\begin{eqnarray}
T(z)A_1(0) &\sim & \frac{\Delta}{z^2}A_1(0)
+\frac{\partial A_1(0)}{z}\;,\label{ta1}\\
T(z)A_2(0) &\sim & \frac{\Delta}{z^2}A_2(0)+
\frac{\partial A_2(0)}{z}+\frac{a}{z^2}A_1(0)\;.\label{ta2}
\end{eqnarray}
Then they transform under the dilatations $z'=\alpha\, z$ as
\begin{equation}
A'_1(z')=\alpha^{-\Delta}A_1(z)\;,\;\;
A'_2(z')=\alpha^{-\Delta}A_2(z)+a\log\alpha \; A_1(z)\;.
\end{equation}
And more generally one has:
\begin{eqnarray}
&\lbrack L_n,A_1(z)\rbrack =(z^{n+1}\partial_z+\Delta(n+1)z^n)A_1(z) \;, \\
&\lbrack L_n,A_2(z)\rbrack =(z^{n+1}\partial_z
+\Delta(n+1)z^n)A_2(z)+ a(n+1)z^n A_1(z)\;.
\end{eqnarray}
This gives rise to differential equations for the
correlation functions of the operators $A_1(z)$ and $A_2(z)$, through the
conformal Ward identities
for $n=-1,0,1$. Let us determine the two-point functions of these
operators. The $n=0$ differential equations for these
correlation functions are
\begin{eqnarray*}
&(z\partial_z+2\Delta)\vev{A_1(z)A_1(0)}=0\;, \;\;
(z\partial_z+2\Delta)\vev{A_2(z)A_1(0)}+a\vev{A_1(z)A_1(0)}=0\;,\\
&(z\partial_z+2\Delta)\vev{A_2(z)A_2(0)}+a\vev{A_1(z)A_2(0)}+
a\vev{A_2(z)A_1(0)}=0\;,
\end{eqnarray*}
while for $n=1$ we have
\begin{eqnarray*}
&(z\partial_z+2\Delta)\vev{A_2(z)A_1(0)}+2 a\vev{A_1(z)A_1(0)}=0\;,\\
&(z\partial_z+2\Delta)\vev{A_2(z)A_2(0)}+ 2 a \vev{A_1(z)A_2(0)}=0\;.
\end{eqnarray*}
This implies
\begin{eqnarray*}
&\vev{A_1(z)A_1(0)}=0\,, \;\vev{A_2(z)A_1(0)}=\vev{A_1(z)A_2(0)}=
C z^{-2\Delta}\,,\\
&\vev{A_2(z)A_2(0)}= (-2a C\log z+C')z^{-2\Delta}\,,
\end{eqnarray*}
with $C$, $C'$ arbitrary constants. This system of correlation functions
appeared in \cite{gurarie,ckt}.
Thus if the operators $A_1$, $A_2$ satisfy the OPE (\ref{ta1}--\ref{ta2})
they can have logarithmic correlators, and $A_2(z)$ is the logarithmic operator
related to $A_1(z)$.

In the following we give an example of such operators
appearing in the indecomposable representations of the $osp(2/2)$
superalgebra.
Consider fields $\phi_i(z)$ in the atypical indecomposable
representation $[0,-1/2,1/2,0]$ (fig. 1b).
The  structure of the generators $T^a$ is
\begin{equation}
T^a= \begin{array}{c}
     \\  \lambda \\ \lambda' \\ s \\ t
     \end{array}
\begin{array}{c} \begin{array}{cccc}\lambda & \lambda' &s & t \end{array} \\
      \left(\begin{array}{cccc}
            t^a & 0 &0 &*\\
            0&t'^a& 0& * \\
            ** & * & 0& 0 \\
            0 & 0 & 0& 0
       \end{array} \right)
     \end{array}
\end{equation}
where $\lambda=[1/2]_+$, $\lambda'=[1/2]_-$ and the stars stand
for possible non-zero matrix elements.
The action of the currents on the fields
takes the form
\begin{equation}
J^a(z) \phi_1(0) = T^a_1 \frac{\phi_1(0)}{z}\;,\quad
J^a(z) \phi_2(0) = T^a_1 \frac{\phi_2(0)}{z} + T^a_2 \frac{\phi_1(0)}{z}\;,
\end{equation}
where the index 1 refers to the invariant subspace. This form is typical
of indecomposable representations of any algebra. We find
the most singular term arising
from the $J-\phi$ contractions in  $\vev{J^a(z)J^b(w)\phi_2(0)\prod{\cal O}}$.
Using the definition
\begin{equation}
T(z)=\lim_{z\rightarrow w}\frac{1}{\kappa} \left( \kappa_{ab} J^a(z)J^b(w)
-k \frac{\kappa_{ab}\kappa^{ab}}{(z-w)^2}\right)
\end{equation}
we conclude that
\begin{eqnarray}
T(z)\phi_1(0) &=& \frac{\Delta}{z^2} \phi_1(0)
+ \frac{1}{z}\partial\phi_1(0)\;,\\
T(z)\phi_2(0) &=& \frac{\Delta}{z^2} \phi_2(0)
+ \frac{1}{\kappa}\frac{\kappa_{ab}(T^a_1 T^b_2+T^a_2 T^b_1)}{z^2} \phi_1(0)
+\frac{1}{z}\partial \phi_2(0) \;,
\end{eqnarray}
where $\Delta=\frac{1}{\kappa}\kappa_{ab} T^a_1 T^b_1$.
This non-diagonal action of the Virasoro algebra is the direct
consequence of the existence of an indecomposable representation
at the affine level.  For the representation at hand one gets
$C_2\, t \sim s$, $C_2\, s=0$, and for the fields
$\phi_s$ and $\phi_t$ with zero $osp(2/2)$ charges:
\begin{equation}
T(z)\phi_s(0) \sim \frac{\partial \phi_s(0)}{z}\;,\;\;\;\;
T(z)\phi_t(0) \sim  a\frac{\phi_s(0)}{z^2}
+\frac{\partial \phi_t(0)}{z}\;.
\end{equation}
According to the previous discussion their correlation functions are
$$\vev{\phi_s(z)\phi_s(0)}=0\;,\;\;
\vev{\phi_t(z)\phi_s(0)}= C\;,\;\; \vev{\phi_t(z)\phi_t(0)}=
-2a C \log z + C'\;.$$
We can also solve the differential equations for
$\vev{\phi_s}$ and $\vev{\phi_t}$ to find $\vev{\phi_s}=0$, $\vev{\phi_t}=c$.

The indecomposable representation we considered
appear in the tensor product of two fundamental representations.
Thus logarithmic terms are expected to appear in the four
point functions
(if the logarithmic operators are not identically
zero, as seems to be the case for the free bosons/fermions $k=1$).

\section{Four-point functions}

We now calculate  a set of four-point correlation functions for
primary fields in the $[0,\frac{1}{2}]$ representation of $osp(2/2)$.
As explained in section \ref{loga} we expect to find a logarithmic
dependence.

\subsection{Differential equations}

Consider the basis
\begin{equation}
|1\rangle =|0,\mbox{$\frac{1}{2}$}\rangle \; ,\; |2\rangle =|-\mbox{$\frac{1}{2}$},0\rangle \; ,\;
|3\rangle =|\mbox{$\frac{1}{2}$},0\rangle \; ,\; |4\rangle =|0,-\mbox{$\frac{1}{2}$}\rangle \; ,
\label{basis}
\end{equation}
where $|1\rangle$ and $|4\rangle$ are even (bosonic) and $|2\rangle$ and
$|3\rangle$ are odd (fermionic). These vectors are labeled
by the eigenvalues of $B$ and $Q_3$. The odd generators are given by:
\begin{eqnarray}
\label{vrep1}
V_+=\sqrt{2}\left( \begin{array}{cccc}
0 & \epsilon & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & \alpha \\
0 & 0 & 0 & 0    \end{array} \right) &,&
W_-=\sqrt{2}\left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
-\beta & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & \gamma & 0  \end{array} \right)\;, \\
\label{vrep2}
W_+=\sqrt{2}\left( \begin{array}{cccc}
0 & 0 & \gamma & 0 \\
0 & 0 & 0 & \beta \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0  \end{array} \right) &,&
V_-=\sqrt{2}\left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
-\alpha & 0 & 0 & 0 \\
0 & \epsilon & 0 & 0  \end{array} \right) \;.
\end{eqnarray}
The four parameters appearing in (\ref{vrep1}--\ref{vrep2}) are constrained
by
\begin{equation}
4 \alpha \gamma =1 \;\; ,\;\;\; 4 \beta \epsilon = 1 \; .
\label{const}
\end{equation}
Thus there are two free parameters which correspond to arbitrary relative
normalizations of the  $su(2)$ doublet $(|1\rangle ,|4\rangle )$,
and the two singlets $|2\rangle$ and $|3\rangle$.

The vacuum invariance
under the zero modes of the affine algebra, or equivalently invariance under
global gauge transformations, imply the following linear equations
\begin{equation}
\sum_{i=1}^4 T^a_i \langle
\phi_1(z_1) \phi_2(z_2)\phi_3(z_3)\phi_4(z_4)\rangle =0 \;.\label{glo}
\end{equation}
It is understood that minus signs arise each time we permute two fermionic
objects, and omit the antiholomorphic dependence and indices of the fields.
Similar equations hold for the `right' (antiholomorphic) chiral generators.
These equations are solved in terms of invariant tensors for the
representations at hand. We have  found the explicit expressions for three
linearly independent tensors which we give in the appendix.

Requiring the
fields of a specific module to be both affine and Virasoro primary
gives a set of equations relating the fields. Such equations give partial
differential equations, the Knizhnik-Zamolodchikov equations \cite{kz},
for the correlation functions.
One gets
\begin{equation}
\left[\frac{1}{2}\frac{\partial}{\partial z_1} - \frac{\kappa_{ab}}{\kappa}
\sum_{j=2}^4 \frac{1}{z_1-z_j}T_1^a T_j^b \right] \langle \phi_1(z_1)
\prod_{j=2}^4 \phi_j (z_j)\rangle =0 \; .
\end{equation}
After a conformal transformation this equation  reduces to a matrix
differential equation for the correlators
\begin{equation}
F^{i_1 i_2 i_3 i_4}(z)=\lim_{w\rightarrow \infty} w^{2\Delta_4}
\langle \phi^{i_1}(z) \phi^{i_2}(0)\phi^{i_3}(1)\phi^{i_4}(w)\rangle\;.
\end{equation}
One gets
\begin{equation}
\frac{1}{2}\partial F(z)= \frac{1}{\kappa} \left(\frac{1}{z} {\cal P} +\frac{1}{z-1}
{\cal Q}\right) F(z)\;,
\end{equation}
where
\begin{equation}
z=\frac{z_{12}z_{34}}{z_{23}z_{41}}\;,\;\;
{\cal P}=\kappa_{ab} T^a_1\otimes T^b_2\;,\;\;
{\cal Q}=\kappa_{ab} T^a_1\otimes T^b_3\;.
\label{kz}
\end{equation}
Similar equations hold for the antiholomorphic sector.
The complete correlator is given by
\begin{equation}
{\cal F}(z_1,z_2,z_3,z_4)= z_{14}^{-2\Delta_1} z_{23}^{-\Delta_1-\Delta_2-\Delta_3+\Delta_4}
z_{24}^{\Delta_1-\Delta_2+\Delta_3-\Delta_4} z_{34}^{\Delta_1+\Delta_2-\Delta_3-\Delta_4}
F(z) \;.
\end{equation}
For the foregoing representations $\Delta_i=1/(2 x)$ and
the $z_{ij}$ prefactor reduces to $z_{14}^{-x^{-1}} z_{23}^{-x^{-1}}$,
where
$$x=2-k\, .$$

The correlators have the $osp(2/2)\times osp(2/2)$ invariant decomposition
\begin{equation}
F^{i_1i_2i_3i_4,j_1j_2j_3j_4}(z,\zb)=
\sum_{A,B=1}^{3} I_A^{i_1i_2i_3i_4} \bar{I}_B^{j_1j_2j_3j_4} F_{AB}(z,\zb).
\end{equation}
The tensors are given in the appendix and the nine scalar functions satisfy
the differential equations:
\begin{equation}
x\frac{\partial F}{\partial z} =
\left[\frac{1}{z} P +\frac{1}{z-1} Q\right] F\;,
\end{equation}
where $F$ now denotes the vector $(F_{1B},F_{2B},F_{3B})$
for all $B$.
There are similar equations for the antiholomorphic dependence, for the vector
$(F_{A1},F_{A2},F_{A3})$.
We give the matrices $P$ and $Q$ in the appendix.
Let $f_A(z)=F_{AB}(z)$ for any given $B$. It is straightforward
to reduce this first-order matrix differential equation
to the following set of equations:\footnote{It is amusing to note that
the third-order differential equations for $f_1(z)$ and $f_2(z)$
reveal two {\em apparent} singularities at $z=2$ and $z=\frac{1}{2}$.}
\begin{eqnarray}
&x^3 z^3 (1-z)^3 \partial^3 f_3(z) + x^2 (1+2 x) z^2 (1-z)^2(1-2 z)\partial^2
f_3(z) +x z(1-z)[-1-x& \nonumber\\
& +2 x z-2 x(2+x)z(1-z)]\partial f_3(z)
+(-x -1 +2 z+2 x z(1-z))f_3(z) = 0&
\end{eqnarray}
\begin{eqnarray}
f_2(z) &=& -\frac{1}{4\epsilon\gamma x z(1-z)}(x^2 D^2 f_3(z) +2x(1-z) D f_3(z)
+(1-2 z)f_3(z) )\\
f_1(z)&=&\frac{1}{4\epsilon\gamma} (x Df_3(z) -f_3(z) ) +(z-2) f_2(z)
\end{eqnarray}
where $D=z(1-z)\partial $.
As expected the equation
for $f_3$ is fuchsian with three regular singularities at $0$, $1$
and $\infty$.\footnote{See \cite{ince} or any book on differential equations.}
The corresponding indices are $(-x^{-1},-x^{-1},1+x^{-1})$,
$(-x^{-1},1-x^{-1},x^{-1})$ and $(0,0,1+ 2 x^{-1})$. The degeneracy of some
indices, {\it i.e.} when the difference of two indices belongs to ${\Bbb Z}$,
signals the eventual presence of logarithmic solutions. Thus one is naturally
lead to distinguish between different values of $x$'s.

We were able to express the functions $f_i$ in terms of generalized
hypergeometric functions.
We obtained:
\begin{eqnarray}
&F^{1423}(z)=f_3(z)= \mbox{$\frac{1}{2}$} (1-x\,D)F^-(z)\;,& \nonumber\\
&x^3 z^3(1-z)^3\partial^3 F^- + x^2(3x+1)z^2(1-z)^2(1-2z)
\partial^2 F^-&\nonumber\\
&+ z(1-z)(x^2(x+1)(1-6z(1-z))-x)\partial F^- -(1-2z)F^- =0\;.&\label{de1}
\end{eqnarray}

Equation (\ref{de1}) is fuchsian with three regular singular points at
0, 1 and $\infty$. The indices at 0 and 1 are $(-\frac{1}{x},-\frac{1}{x},
\frac{1}{x})$. The  solutions of equation (\ref{de1}) can be written as
generalized hypergeometric functions:
\begin{eqnarray}
F^-_0(z) &=& (z(1-z))^{-\frac{1}{x}}\; {}_3F_2\left(\frac{1}{2},-\frac{1}{x},
1-\frac{1}{x};1,-\frac{2}{x};4z(1-z)\right)\\
F^-_1(z) &=& F^-_0(z) \log[4z(1-z)] + (z(1-z))^{-\frac{1}{x}}\sum_{n\geq 0}
c_n (4z(1-z))^n\\
F^-_2(z) &=& (z(1-z))^{\frac{1}{x}} \; {}_3F_2\left(\frac{1}{x},1+\frac{1}{x},
\frac{1}{2}+\frac{2}{x};1+\frac{2}{x},1+\frac{2}{x};4z(1-z)\right)
\end{eqnarray}
where
\begin{eqnarray*}
& & {}_3F_2(a_1,a_2,a_3;b_1,b_2;z) \equiv \sum_{n\geq 0}\frac{(a_1)_n (a_2)_n
(a_3)_n}{(b_1)_n (b_2)_n n!} z^n\;,\;\; (a)_n = a(a+1)\cdots (a+n-1)\;, \\
&c_n& =\frac{d}{ds}
\left(\frac{\Gamma(n+s+\mbox{$\frac{1}{2}$})\Gamma(n+s-\frac{1}{x})\Gamma(n+s+1-\frac{1}{x})
\Gamma(s+1)^2\Gamma(s+1-\frac{2}{x})}
{\Gamma(n+s+1)^2\Gamma(n+s+1-\frac{2}{x})\Gamma(s+\mbox{$\frac{1}{2}$})\Gamma(s-\frac{1}{x})
\Gamma(s+1-\frac{1}{x})}\right)_{s=0}\;.
\end{eqnarray*}
It must be noted that although the solutions look symmetric under
the exchange $z\leftrightarrow 1-z$ it is {\em not}
possible to have three linearly independent solutions with such a symmetry.
When the series defining the hypergeometric functions do not truncate there
are singularities at $u= 4z(1-z) =1$ of the type $\log(1+\sqrt{1-u})$.
And since the reciprocal image
of the interior of the $u$-unit disc consists of two disjoint lobes, in the
$z$ plane,  centered
around 0 and 1, the change $z \rightarrow 1-z$ changes the determination
of the hypergeometric function around $z=1$  from what it is around
$z=0$. This feature can be seen explicitly
for the cases $x=1$ and $x=1/2$ below.

Before analyzing some specific solutions let us note that after the
change of function $F^-(z)=(z(1-z))^{-\frac{1}{x}} G(z)$
one obtains
\begin{eqnarray}
&z^2 (z-1)^2 \partial^3 G + z(1-z)(K_1 z+K_2 (z-1))\partial^2 G
+(L_1 z^2+L_2 (z-1)^2&\nonumber\\
&+L_3 z(z-1))\partial G + (M_1 z + M_2 (z-1))G=0\;,&
\end{eqnarray}
where
\begin{equation}
K_1=K_2=3-\frac{2}{x}\;,\;\; L_1=L_2=1-\frac{2}{x}\;,\;\;
L_3=4(1-\frac{1}{x})^2\;,\;\; M_1=M_2=\frac{2}{x}(\frac{1}{x}-1)\;.
\end{equation}
Equations of this type appeared for correlations functions containing
a level three null vector in minimal models (see ref.~\cite{dotfat}).
There are integral representations for the solutions
\begin{equation}
G(z)=\int_{C_1}dt_1 \int_{C_2}dt_2 t_1^{\frac{1}{x}-1}(t_1-1)^{\frac{1}{x}-1}
(t_1-z)^{\frac{1}{x}} t_2^{\frac{1}{x}-1}(t_2-1)^{\frac{1}{x}-1}
(t_2-z)^{\frac{1}{x}} (t_1-t_2)^{-\frac{4}{x}} \;.
\end{equation}
These integrals can however diverge, depending on the contours $C_1$ and $C_2$
and the value of $x$.

\subsection{The level 1 correlators}

For $x=1$, {\it i.e.} $k=1$, we are able to write the solutions of
(\ref{de1}) in a closed form:
\begin{equation}
F^-(z)=g_0 \frac{1}{z(1-z)} + g_1 \left(1+\frac{\log z}{z(1-z)}\right)
+ g_2 \left(1+\frac{\log(1-z)}{z(1-z)}\right)\;.
\end{equation}
We then want to construct the physical correlation functions which
must be single-valued on the whole Riemann sphere. It is therefore enough
to ensure this property at the two singular points 0 and 1.
It is easy to see that one can only have
\begin{eqnarray}
F_{AB}(z,\zb) &=& \alpha_0 F^{(0)}_A(z) F^{(0)}_B(\zb) + \alpha_1 (F^{(0)}_A(z)
F^{(1)}_B(\zb) +  F^{(1)}_A(z) F^{(0)}_B(\zb))\nonumber \\
&+& \alpha_2 (F^{(0)}_A(z)
F^{(2)}_B(\zb) +  F^{(2)}_A(z) F^{(0)}_B(\zb))\;,
\end{eqnarray}
where the constants $\alpha_i$ are independent of the indices $A, B$, and
\begin{eqnarray}
&F^{(0)}_1(z)=\frac{z-2}{4\epsilon\gamma (1-z)}\;,\;
F^{(0)}_2(z)=\frac{2 z-1}{4\epsilon\gamma z(1-z)}\;,\;
F^{(0)}_3(z)=\frac{1}{z}&\nonumber\\
&F^{(1)}_1(z)=\frac{1}{4\epsilon\gamma} \left(-1+\frac{z-2}{1-z}\log z\right)\;,\;
F^{(1)}_2(z)=\frac{1}{4\epsilon\gamma z(1-z)} \left(1-z+(2 z-1) \log z\right)\;,&
\nonumber\\
&F^{(1)}_3(z)=\frac{1}{2 z} \left( z-1 +2\log z\right)\;,\;
F^{(2)}_1(z)=\frac{1}{4\epsilon\gamma} \left( \frac{z}{z-1} +\frac{2-z}{1-z} \log(1-z)
\right)\;,&\nonumber\\
&F^{(2)}_2(z)=\frac{1}{4\epsilon\gamma z(1-z)}\left(z+(1-2 z)\log(1-z)\right)\;,\;
F^{(2)}_3(z)=\frac{1}{2 z}\left( \frac{z(z-2)}{1-z}-2\log(1-z)\right).&
\end{eqnarray}

This form of $F_{AB}$ differs from the usual diagonal one
one finds for minimal models. Here there are three so far
unconstrained constants (one constant is an overall
normalization), even  after requiring monodromy invariance around the
three singular points.

We still have to require crossing symmetry among the correlators:
\begin{equation}
F^{i j k l}(z,\zb)=F^{i k j l}(1-z,1-\zb) \;,\;\;
F^{i j k l}(z,\zb)=z^{-1/x} \zb^{-1/x} F^{i l k j}(1/z,1/\zb)\;.
\end{equation}
These conditions imply that $\alpha_1=\alpha_2=0$. These correlators are thus free
from logarithms.

At level one the free fields $(\gamma,\psi_-,\psi_+,\eta)$ provide a
representation $[0,\frac{1}{2}]$ of the affine algebra $osp(2/2)_1$.
The corresponding correlation functions are then easily
calculated by making use of Wick's theorem. They agree with the results
obtained from the above calculation, and are free of logarithms.

We then looked at $x=1/2$ ($k=3/2$) correlators for which we
explicitly found the functions $F^{(i)}_A$. Again the physical correlators
were free from logarithms. It is possible to show that such is the case
for all  non-vanishing integer values of  $x^{-1}$. The solutions
$F^{(i)}_A$ have the same form as those for $x=1$ and $x=1/2$,
namely a combination
of rational functions of $z$ and of $\log z$ and $\log (1-z)$.
We were not able to  determine
the connection matrices between  functions  with arguments
$z$, $1-z$ and $1/z$. The limiting values for those
obtained in \cite{dotfat} depend on the way the limit is taken.
But it is enough to know the form
of these matrices to implement the  crossing symmetries. In doing so
we also used the matrices connecting the tensors $I_a$ with transposed indices,
and which can be found in the appendix.  We found that $\alpha_1=\alpha_2=0$.
Note that the values of $k$ considered here are exactly those of equation
(\ref{restri}). We comment on this result in the conclusion.

From the four-point correlation functions we can obtain information
about the fusion rules for the $[0,1/2]$ representation. As the contribution
from the conformal blocks 1 and 2 vanishes we infer that the corresponding
operators do not appear in the OPE $\phi_{[0,1/2]}(z) \phi_{[0,1/2]}(w)$.
The only primary field appearing in their OPE corresponds to the operator
$\phi_t$, which is now the identity.

\subsection{Logarithms for generic $x$}

We now consider the generic case where $ 2 x^{-1}$ is not an integer.
Contrary to what happens in the preceding section, it does not
seem possible to isolate a solution which is regular at both $z=0$ and $z=1$.
A solution regular at 0 will have a logarithmic singularity at $z=1$
and vice-versa. This means that there is no way to get rid of logarithms
if the correlators are not vanishing.
We expect that  there should be logarithmic operators, and the knowledge
of the four-point functions allows one to find the OPE's of the fields
we consider, including the logarithmic ones.

\subsection{Squared logarithms}

We now turn to the third and last possible scenario
where $2  x^{-1}$ is an integer
but $x^{-1}$ is not an integer: $x^{-1}=n+1/2$, $n\in {\Bbb Z}$.
The theory of fuchsian differential equations indicates that there
might be a solution with a $\log^2 z$ dependence. A careful analysis
shows that this is indeed the case. And again it seems impossible
to isolate a logarithm-free solution. However one has to modify the
form of $F_{AB}$. Write the solutions as
\begin{eqnarray}
&F_A^{(0)}(z)=F_A(z)\;,\;\;
F_A^{(1)}(z)=F_A(z) \log z + H_A(z)\;,&\nonumber \\
&F_A^{(2)}(z)=F_A(z) \log^2 z + 2 H_A(z) \log z  + G_A(z)\;.&
\end{eqnarray}
Then the monodromy invariant combinations around $z=0$
are
\begin{eqnarray}
F_{AB}(z,\zb)&=& \alpha_0 F_A^{(0)}(z) F_B^{(0)}(\zb) + \alpha_1 \left(F_A^{(0)}(z)
F_B^{(1)}(\zb) + F_A^{(1)}(z) F_B^{(0)}(\zb)\right) \nonumber\\
&+& \alpha_2 \left(F_A^{(0)}(z) F_B^{(2)}(\zb) + F_A^{(2)}(z) F_B^{(0)}(\zb)
+ 2 F_A^{(1)}(z) F_B^{(1)}(\zb) \right)\; .
\end{eqnarray}
We expect such a form to be also invariant around 1 without any additional
constraint on the $\alpha_i$. Again, because there is no solution free of
logarithms one expects a $\log^2$ dependence if the physical correlators
do not vanish.

\section{Conclusion}

The current superalgebra  approach to gaussian disordered systems provides
a unifying framework for determining exponents of the models and
correlation functions. The logarithmic dependence of correlation functions
at critical points
is traced back to indecomposable representations which are common in
superalgebras. {\it Chiral} correlation functions possess a logarithmic
dependence. However for values of the level $k$ for which the theories
have good locality properties, the physical operators do not have a logarithmic
dependence. This conclusion was not apparent  from the start; it came out
of imposing crossing symmetry. It is not clear at this stage what is the
significance of the logarithmic operators in the context of the disordered
systems. If they have vanishing dimension, as is the case here, they might be
related to the density of states.

The correlation functions with $\log$ and
$\log^2$ dependence provide examples of
correlators whose existence was predicted  in \cite{gurarie}.
The negative conformal weights are natural in this treatment of disorder
because one is considering non-unitary theories at $c=0$.
It should be also possible to obtain similar
results by considering $WZW$ models of cosets of Kac-Moody superalgebras.

\bigskip\ {\bf Acknowledgements:} It is a pleasure to thank D.~Bernard for
many suggestions. We also thank M. Bauer, P. Di Francesco for discussions.
Z.M. thanks R. Balian and J.M. Luck for discussions, and D.S. thanks
P. Sorba and L. Frappat. We are grateful to S. Nonnenmacher for
many discussions.

\section{Appendix}

The non-vanishing components of the three
invariant tensors are equal to the components of the three vectors:
\begin{eqnarray}
I_1&=& (1144)+(1234)4  \epsilon\gamma+(1324)4  \epsilon\gamma-(1414)+(2143)4  \epsilon\gamma
\nonumber\\&+&(2233)16 \epsilon^2\gamma^2+
(2323)16 \epsilon^2\gamma^2-(2413)4  \epsilon\gamma+(3142)4  \epsilon\gamma
+(3232)16 \epsilon^2\gamma^2 \nonumber \\
&+& (3322)16 \epsilon^2\gamma^2 -(3412)4  \epsilon\gamma-(4141)
-(4231)4  \epsilon\gamma-(4321)4  \epsilon\gamma+(4411)\\
I_2&=& (1234)4  \epsilon\gamma-(1243)4  \epsilon\gamma+(1324)4  \epsilon\gamma-(1342)4
\epsilon\gamma \nonumber\\
&-& (1414)+(1441)-(2134)4  \epsilon\gamma +(2143)4  \epsilon\gamma+(2233)32 \epsilon^2\gamma^2
+(2323)16 \epsilon^2\gamma^2 \nonumber\\
&+& (2332)16 \epsilon^2\gamma^2-(2413)4  \epsilon\gamma
+(2431)4  \epsilon\gamma-(3124)4  \epsilon\gamma+(3142)4  \epsilon\gamma \nonumber\\
&+& (3223)16 \epsilon^2\gamma^2+(3232)16 \epsilon^2\gamma^2
+(3322)32 \epsilon^2\gamma^2-(3412)4  \epsilon\gamma + (3421)4  \epsilon\gamma \nonumber\\
&+&(4114) - (4141)+(4213)4  \epsilon\gamma
-(4231)4  \epsilon\gamma+(4312)4  \epsilon\gamma-(4321)4  \epsilon\gamma\\
I_3&=&(1234)-(1243)+(1324)-(1342)+(1423)+(1432)-(2134)+(2143)
\nonumber\\
&+&(2233)8  \epsilon\gamma +(2314)+(2323)8  \epsilon\gamma+(2332)8  \epsilon\gamma
-(2341)-(2413)+(2431) \nonumber \\
&-& (3124)+(3142)+(3214)+(3223)8  \epsilon\gamma+(3232)8  \epsilon\gamma-(3241)
+(3322)8  \epsilon\gamma \nonumber \\
&-& (3412)+(3421)-(4123)-(4132)+(4213)-(4231)+(4312)-(4321)
\end{eqnarray}

The matrices $P$ and $Q$ are given by:
\begin{equation}
P=\left( \begin{array}{ccc}
1 & 0 & 0 \\
-2 & -3 & -\frac{1}{2\epsilon\gamma} \\
4\epsilon\gamma & 8\epsilon\gamma & 1    \end{array} \right) \quad ,\quad
Q=\left( \begin{array}{ccc}
-1 & 0 & -\frac{1}{2\epsilon\gamma} \\
2 & 1 & \frac{1}{2\epsilon\gamma} \\
-4\epsilon\gamma & -4\epsilon\gamma & -1 \end{array} \right) \;.
\end{equation}

The following tensors are needed in order to implement the
crossing-symmetries:
\begin{equation}
J^{\alpha_1\alpha_2\alpha_3\alpha_4}_A = (-1)^{\varepsilon_{\alpha_2} \varepsilon_{\alpha_3}}
I^{\alpha_1\alpha_3\alpha_2\alpha_4}_A\;,\;\;
K^{\alpha_1\alpha_2\alpha_3\alpha_4}_A = (-1)^{\varepsilon_{\alpha_2} ( \varepsilon_{\alpha_3} +\varepsilon_{\alpha_4})
+\varepsilon_{\alpha_3}\varepsilon_{\alpha_4}} I^{\alpha_1\alpha_4\alpha_3\alpha_2}_A\; .
\end{equation}
The $\varepsilon_{\alpha_i}$ refers to the parity of the index $\alpha_i$.
One has the following decomposition:
\begin{eqnarray}
& J_1 = - I_1\;,\;\; J_2 = -I_1+I_2 -4\epsilon\gamma I_3\;,\;\; J_3 = -I_3 & \\
& K_1 = I_2 -4\epsilon\gamma I_3 \;,\;\; K_2 = I_1 -4\epsilon\gamma I_3 \;,\;\;
K_3 = - I_3 & \; .
\end{eqnarray}
We rewrite this compactly as $J=C_1\, I$ and $K=C_2\, I$ where $C_{1,2}$ are
two matrices which are used to implement the crossing symmetry constraints.

redpajama_set_name: RedPajamaArXiv


text:
Angry Bordeaux fans force match to be halted, Marseille wins
Nigerian forward Josh Maja scored a hat trick as Bordeaux routed Nimes 6-0 Tuesday in a French league game that was temporarily halted in the first half after dozens of home fans stormed onto the field to protest against the club's board
Angry Bordeaux fans force match to be halted, Marseille wins Nigerian forward Josh Maja scored a hat trick as Bordeaux routed Nimes 6-0 Tuesday in a French league game that was temporarily halted in the first half after dozens of home fans stormed onto the field to protest against the club's board Check out this story on app.com: https://www.usatoday.com/story/sports/soccer/2019/12/03/angry-bordeaux-fans-force-french-league-match-to-be-halted/40752683/
AP Published 2:11 p.m. ET Dec. 3, 2019 | Updated 5:19 p.m. ET Dec. 3, 2019
Nigerian forward Josh Maja scored a hat trick as Bordeaux routed Nimes 6-0 Tuesday in a French league game that was temporarily halted in the first half after dozens of home fans stormed onto the field to protest against the club's board.
Bordeaux's group of hardcore fans Ultramarines came down from the stands 11 minutes into the game and staged a protest on the sideline, causing referee Clement Turpin to stop the game and send both teams back into the dressing room.
The game resumed after about 25 minutes and the 20-year-old Maja put the hosts 2-0 up by halftime and completed his hat trick in the 53rd minute. He then set up Nicolas de Preville for Bordeaux's fourth goal and Otavio, who had never scored in the French league, also netted two goals to finish off the rout.
Bordeaux is third in the league standings but the club's angry fans are asking for the club's longtime president Frédéric Longuépée to step down.
The Ultramarines group says Longuépée is prioritizing commercial interests over fans. Longuépée said after the match he has no intention to step down.
The protests came amid a period of uncertainty at Bordeaux, with the club's two main shareholders currently planning to part ways.
FIVE IN A ROW
Dimitri Payet set up Marseille's opening goal then scored from the penalty spot as the nine-time champion posted a fifth consecutive league win by beating Angers 2-0.
After a tough start to the match at Angers, Marseille scored against the run of the play when Payet caught the hosts off guard on a set piece in the 17th minute. With Angers defenders looking elsewhere, Payet quickly played a free kick to set up the unmarked Morgan Sanson, who coolly beat goalkeeper Ludovic Butelle with a low shot from a tight angle.
Payet made it 2-0 from the spot before the interval by sending Butelle the wrong way after Romain Thomas fouled Hiroki Sakai in the box.
The result lifted Marseille within two points of league leader Paris Saint-Germain, which has two games in hand and hosts Nantes on Wednesday.
After failing to qualify for the Champions League last season, Marseille has found a renewed confidence under new coach Andre Villas-Boas, making it five league victories in a row for the first time since October 2014.
BATTOCHIO HAT TRICK
Midfielder Cristian Battocchio scored a hat trick in the space of 30 minutes as Brest routed Strasbourg 5-0 for its biggest win this season.
Battocchio, an Argentine-born Italian midfielder, made the most of a rare start to open his scoring account in the French league.
A former Under-21s Italy international, Battocchio returned to Brest at the start of 2019 after a spell at Maccabi Tel Aviv.
EFFICIENT LILLE
Jonathan Ikone scored the winner as Lille secured its first away win of the season by beating Lyon 1-0 in a scrappy game.
Lyon slumped to a third home defeat after missing seven attempts on goal after the interval. Already lagging 11 points behind PSG after 16 rounds, Rudi Garcia's players were whistled off the pitch by their fans.
In contrast, Lille scored from its only shot in the second half and moved into fourth place.
More AP soccer: https://apnews.com/Soccer and https://twitter.com/AP_Sports

redpajama_set_name: RedPajamaCommonCrawl


Dawson Design Associates is a full service Hospitality Design Consultancy with a distinctive difference. Our team is trained to create hotel experiences by amplifying through our collaborative approach, a product that goes beyond your guest's expectations. We play the role of 'Guest Advocate' taking into considerations all aspects of the hotel design and project needs. We fuse this expertise with Branding and Strategic Positioning in an impactful and fashion forward manner. Our unique and distinctive approach provides our clients with the edge they need to be competitive in their respective markets.


The Carlow University Art Gallery, located on the 2nd floor of the University Commons, will be having its first national exhibit starting this month. The exhibit is the first all-ceramics show to be held in the gallery and it will feature in the broader exhibition that NCECA (the National Council on Education for the Ceramic Arts) will be hosting in mid-March.
The exhibit serves to celebrate the work of lesser-known artists and the work of their predecessors. The collection has a variety of styles in it, from more modern styles and techniques, like 3D printing ceramic, to more classical ceramic techniques, like hand-thrown pieces. The juxtaposition of the more classically styled pieces and the more modern-themed pieces are carefully arranged to add a powerful underlying dialogue to the exhibition said Zahner. Adding to this dialogue is the fact that, while it wasn't an intentional choice, 12 out of the 20 artists in the exhibition are women artists, many from diverse backgrounds. Dr. Sylvia Rhor, art professor at Carlow and director of the art gallery, spoke about the importance of "…disrupt[ing] the artistic space," by way of featuring so many great female artists and also the general importance of having female voices within the traditionally male field of art. Professor Zahner also added to that point by saying that it is "…very important that [women artists] have the same support system that has helped men." Not only are many of the artists featured women, but much of the work of curating and setting up this exhibition has been completed by women as well. Included in this group of women is Britney Smith, who is a functional ceramic artist and a student at Carlow. She assisted greatly with the setup of the gallery and the pieces in it. While her work is not featured in this exhibition, it is noteworthy that she is one of these women artists providing a support system for other women, by way of her work on the gallery.
The gallery will be fully complete by mid-February, and the official gallery opening ceremony will be held late in February. Updates on gallery events and upcoming projects can be found on their Facebook page (Carlow University Art Gallery) and their Instagram (@carlow_gallery).
Excellent article, thank you so much for giving your time and energy to this exhibition, and representing us so well.


text:
Marmosedio (Mammusé in dialetto locale) è una frazione del comune di Fiamignano, in provincia di Rieti. Si trova nel Cicolano a circa 850  Ha una popolazione di circa 25 abitanti.

Geografia fisica

Territorio
Il borgo, circondato dai monti del Cicolano, si trova a breve distanza dall'altopiano di Rascino, situato a circa 17 chilometri. Dista 36 chilometri da Rieti e tre dal capoluogo comunale di Fiamignano.

Storia
Qui sorgeva uno dei santuari dell'antica città equicola di Cliternia, con mura in opera poligonali della fine del II secolo a.C. In seguito nello stesso luogo venne fondata la chiesa di San Lorenzo in Fano.

Monumenti e luoghi d'interesse

Architetture religiose
 Chiesa di San Lorenzo in Fano.

Società

Tradizioni e folclore
 10 agosto, festa di san Lorenzo.
 Fino a circa il 1980 veniva festeggiata anche santa Barbara, il 4 dicembre.

Geografia antropica

Rioni
Il paese è suddiviso in tre rioni:
 Capolavilla: situato nella parte alta del paese.
 Perdesco: (in dialetto locale "u Perdiscu"), costituito successivamente agli altri rioni.
 Calenne: è il rione più a valle, è il fulcro della vita sociale del villaggio in quanto vi si trova piazza Calenne.

Infrastrutture e trasporti

Strade
Il paese è raggiungibile dall'uscita Valle del Salto dell'autostrada A24, da cui dista 16 chilometri, attraverso la strada statale 578 Salto Cicolana.

Note

Frazioni della provincia di Rieti
Fiamignano

redpajama_set_name: RedPajamaWikipedia


Excellent Service provided by Mr.Nazbul Khan and service members are so curitious for guests and feel like a home and food is also good , Housekeeping & reception person are too helpful , overall hotel is very nice .
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Excellent place with peaceful environment. Perfect spot for leisure outing.had a great time at this place rooms were clean and spacious with all the amenities. Food was amazing. Hospitality of Mr. Dinesh Hamal was really commendable. He made sure we have a great time resort and we did have.we would love to travel again at grand o7 resort.
Nice food n wonderful hospitality. They r really cordial. Very good food options available. It is not that far away as suggested by some. Besides it is giving a clean atmosphere to breathe fresh air. Value for money.


Max Von Sydow's height is 6 feet and 4 inches. That's 76 inches tall.
Or in metric units, Max Von Sydow is 194 centimetres. That's 1 metre and 94 centimetres.
Max Von Sydow is 23 centimetres (9.25 inches) taller than the average celebrity (the average is 171 centimetres, 5 feet 7 inches or 67 inches tall).
How tall is Max Von Sydow compared to the average person?


My family loves bread. I bake it from scratch almost daily - in the summer, this used to mean a really hot kitchen! Fortunately, my solar oven does as good a job with bread as the electric oven, without heating up the house.
I swap some of the white flour for whole wheat, add a little extra water to compensate for the drier flour & sometimes bake in bread pans. The recipe is quick, easy & forgiving!
Cover dough & let rise about an hour - it should just about double in bulk. The more wheat flour you add, the longer the rise may take - that's OK!
Cover & let rise again for another hour, until very puffy.
Focus your solar oven during the second rise, aiming for 400F. (On a clear New England fall day, this is no problem!) If your temperatures are running lower, you'll just need to add a little extra time baking. Not to worry, this loaf will bake even at 250-275F!
Brush the top of the loaves with milk, then sprinkle liberally with sesame seeds. If you're really feeling Italian, slash the loaves diagonally for an "authentic" look. Transfer the pans immediately to the oven to bake.
Bake for about 25 minutes, or until the top is golden brown & the internal temperature of the bread reaches about 200F using an instant read thermometer. If your oven temperatures are running lower, don't fret. It'll finish eventually! If you don't have an instant read thermometer, the bread should sound hollow if you tap on it lightly. (But consider picking up an instant read gadget...you'll use it a lot with solar cooking!) On poor-sun days, I've had to bake this well over an hour...it still tastes delicious!
Feel free to tinker with this recipe - change the ratio of white/wheat flour, use white/red wheat, top with different seeds (sunflower is yummy!), use milk instead of water...this recipe adapts well to what you have on hand. Keep in mind that the higher the percentage of whole wheat flour, the less rise you'll get & the more liquid you'll need to make a smooth dough. If you have technical baking questions, stop by the KAF website above - they run an online baker chat service to answer your questions about ingredients & bread techniques.
Thank you for the recipes and the great instruction. This should be very helpful to others.
Wow, i don't bake, but I wish I did. Your bread recipe sounds delicious!


Extensive research points to the positive impact of high-quality early childhood education. Unfortunately, demand continues to outpace investment: Today, 64% of Pennsylvania children eligible for publicly-funded slots remain unable to access them due to lack of funding, while community-based child care centers operate on razor-thin margins, struggle to adequately compensate staff, and have limited administrative capacity.
To address these problems, communities across the country are seeking to combine the benefits of existing early childhood programs with the resources and capacities of public school districts. This report examines implementation of one such model in Pittsburgh Public Schools. We conclude that the benefits outweigh the challenges–and the city is well-positioned to scale up.


text:
The Most Conservative Congress Ever Recorded
Paul Breer,
In the last few weeks, ThinkProgress has been documenting studies by political science professors Keith Poole and Howard Rosenthal that show Republicans are both beholden to the 1 percent and responsible for the hyper-polarization of Congress. A new study by Poole has found that Republicans have moved so far to the right that the House is now the most conservative it has even been in the last 133 years.
Note on the graph: The closer to 1.0, the more conservative the party's votes were that year. The black line in the middle represents the median location of the overall chamber.
Poole's graph shows that when the Republican Party gained a majority in 2010, they brought the median of the House into a realm of ideological extremism not seen before. For comparison, when the Democratic Party held a majority from 2006 to 2010, the House median made a small move to the left (-0.2) — equal to the level of a moderate Democrat. But when the Republican Party took over in 2010, the House median more than tripled (.43) along the ideological scale.
Considering approval ratings for Congress are hovering around 11% — lower ratings than the IRS, lawyers, and even Nixon during Watergate — a recent poll shows that Americans still disapprove of congressional Republicans more than Democrats. In fact, under Democratic majorities the House median hasn't passed the level equal to a moderate Democrat (-0.2) in the last 100 years. Meanwhile, just in the last 20 years alone, the House median under Republican control has been above that moderate level on the conservative side (0.2) for approximately 10 years.
Paul Breer


We have been working for several years on a TV series, entitled "The Dolphin People". We began principle filming in 2005, and have, to date, completed the first hour of a projected four-part series.
Our Director/Producer/Cinematographer/Editor friend, Richard Mordaunt, has recently returned from a journey to New York, where he attended an international congress on documentary films, meeting with commissioning editors from many companies around the world. We await their responses, confident that we will secure a production deal to complete the series.
To obtain copies of "The Dolphin People" one-hour TV program on DVD, write to dolphin@dolphintale.com.
Copies sell for $30 + $6 postage and handling.
JAPAN has apparently agreed not to kill humpback whales during its current Antarctic hunt, the US ambassador to Tokyo said today, a move that could help ease criticism of its controversial whaling program.
Japan's whaling fleet set sail last month with plans to catch more than 1000 whales, including 50 humpbacks, which are popular among whale-watchers for their distinctive silhouettes and acrobatic leaps, before returning to port early next year.
Humpbacks were hunted to near extinction until the International Whaling Commission ordered their protection in 1966 and the planned hunt had sparked a loud outcry from activists.
"I think we had an agreement ... between the United States and Japan that humpback whales would not be harvested, I think, until maybe the International Whaling Conference in June,'' US ambassador to Japan Thomas Schieffer said.
Because of migration patterns, the delay would mean it would be "a while before they are at risk again,'' Mr Schieffer said.
Australia yesterday announced that it would send a fisheries patrol ship to shadow Japan's whaling fleet near Antarctica and gather evidence for a possible international court challenge to halt the yearly hunt.
Separately, Greenpeace sent a ship yesterday to try to stop the Japanese fleet hunting whales.
Japan has long resisted pressure to stop what it calls scientific whaling, insisting that whaling is a cherished cultural tradition.
"Japan's whaling is being conducted in line with international treaties and for the purpose of scientific research. We would like to win the understanding of others,'' a Japanese foreign ministry spokesman said in Tokyo.
The "cherished cultural tradition" mentioned should be seen like the "cherished tradition" of early English (Saxon) people*, the people of Easter Island**, the native people of the California coast***, who were also dolphin and whale eaters. It proved unsustainable in each case, wreaking havoc on local ecologies. With the extremely high levels of toxic chemicals now found in the meat of dolphins and whales (they are at the top of the chain of "bio-accumulation" and concentrate the toxins in their blubber and the milk of nursing mothers, passing it on from generation to generation, each time gaining in concentrated killing power) the tradition of "cetaceans on the menu" is now deadly. It is time to end this "tradition".
Since the actual tradition of the Japanese whalers and dolphin hunters was to hunt and kill and eat the cetaceans from the waters around Japan (which still goes on in the tens of thousands each year, now), what does killing the whales of the Southern Ocean Sanctuary have to do with tradition?
* "Continuing study of the huge middle and late-Saxon vertebrate assemblage from Flixborough, North Lincolnshire, has shown that the inhabitants of this important settlement were consuming whale and dolphin meat. More than twenty fragments of these marine mammals have been recovered from only 10% of the material so far recorded. These animals may have been caught at sea or, more likely, in the nearby Humber estuary where they may have been stranded. It is possible that they represent trade with the nearby east coast fisheries, although the almost negligible quantities of marine fish from the site appear not to support this theory. More specific identification of the species of whale and dolphin will provide valuable information about the past distribution of cetaceans since all are rare or uncommon visitors to these waters today."
** "During the later centuries of the Mayan civilization, a new society was evolving on faraway Easter Island, some 166 square kilometers of land in the South Pacific roughly 3,200 kilometers west of South America and 2,200 kilometers from Pitcairn Island, the nearest habitation. Settled around ad 400, this civilization flourished on a volcanic island with rich soils and lush vegetation, including trees that grew 25 meters tall with trunks 2 meters in diameter. Archeological records indicate that the islanders ate mainly seafood, principally dolphins—a mammal that could only be caught by harpoon from large sea-going canoes. The Easter Island society flourished for several centuries, reaching an estimated population of 20,000. As its human numbers gradually increased, tree cutting exceeded the sustainable yield of forests. Eventually the large trees that were needed to build the sturdy canoes disappeared, depriving islanders of access to the dolphins and dramatically shrinking their food supply. The archeological record shows that at some point human bones became intermingled with the dolphin bones, suggesting a desperate society that had resorted to cannibalism. Today the island has fewer than 4,000 residents."
Synthesis of faunal collections from several archaeological sites on the three southernmost California Channel Islands and one in the Cape Region of Baja California reveals a distinctive maritime adaptation more heavily reliant on the capture of pelagic dolphins than on near-shore pinnipeds. Previous reports from other Southern California coastal sites suggest that dolphin hunting may have occurred there but to a lesser extent. While these findings may represent localized adaptations to special conditions on these islands and the Cape Region, they call for reassessment of the conventionally held concept that pinnipeds were invariably the primary mammalian food resource for coastal peoples. Evidence of the intensive use of small cetaceans is antithetical to the accepted models of maritime optimal foraging which assume that shore-based or near-shore marine mammals (i.e., pinnipeds) would be the highest-ranked prey because they were readily encountered and captured. While methods of dolphin hunting remain archaeologically invisible, several island cultures in which dolphin were intensively exploited by people using primitive watercraft and little or no weaponry are presented as possible analogs to a prehistoric Southern California dolphin-hunting technique. These findings also indicate that dolphin hunting was probably a cooperative endeavor among various members of the prehistoric community."
We had a woman attend who is a therapist, whose specialty is Cranial-Sacral Therapy. Her skill in her work -- using sensitivity to detect and redirect the pulses of energy that move up and down our spines -- was evident in her approach to the dolphins.
Another guest was a "bird lady". She traveled to Coffs Harbour from western Victoria with her birds. She loves her birds, and her sensitive understanding of them was expressed in her hands -- and they were fascinating to Bella. She approached the woman repeatedly, requesting contact, rubbing against her, spinning around and returning for more touch, more strokes, and especially, having her tail rubbed and massaged.
Our time among the dolphins -- swimming, diving, dreaming, listening and learning -- is about to take a short break. We will be away from them for about six weeks, during the busy season of the year, when they entertain thousands of visitors. We, and the PPP staff, are careful to not ask too much of them. The dolphins are very busy keeping so many people happy at this time of year.
Our Dolphin EDventures Wellness Program will recommence in February. For details, go to www.dolphintale.com.


Although Charlie Chaplin's "Little Tramp" character remains so iconic, people all over the world are still familiar with it, even if they have never seen a Chaplin film. That had been the case with me up until 1993, when I saw the restored, original version of The Gold Rush (1925) during a PBS pledge drive. To put it mildly, I was blown away. The storytelling, the inventiveness of the shots, and above all, the charisma of Chaplin himself just knocked me out. For a silent film to retain such remarkable power all those years later clearly indicated that the legendary status Chaplin had long been accorded was well-deserved.
So when I heard that The Gold Rush was to be given The Criterion Collection treatment, I was excited to say the least. Nobody does a better job with classic films than Criterion. With The Gold Rush they have once again come through in spectacular form. The first surprise for me was that this edition contains two versions of The Gold Rush. Call me uninformed, but I was not familiar with the "updated" 1942 adaptation that Chaplin had created for re-release.
The original 1925 film was one of the most successful silent films of all time. United Artists was a studio founded by Chaplin, D. W. Griffith, Mary Pickford, and Douglas Fairbanks in 1919. In 1942, the studio was in trouble, and in a move to raise some cash, Chaplin hit upon the idea of re-releasing The Gold Rush. He knew that in 1942, a reissue of a 17-year-old silent film would not really fly, and so he did what we would today call a "re-boot" of the film. He wrote and read a narrative, composed a new score, and edited the movie from its initial 88 minutes down to 72.
Chaplin considered his 1942 version of The Gold Rush to be definitive. So much so that he ordered the 1925 edition to be destroyed. In 1942, the idea of film history, and of archiving material was not yet a consideration. This is what made the eventual restoration of the 1925 film so difficult, and such an event when it was finally accomplished. Maybe if I had seen the narrated 1942 update first, I may have a different feeling towards The Gold Rush. But for me, the definitive edition is the silent one.
In my slow process of becoming a "cinema geek," seeing the original 1925 The Gold Rush was something of a watershed moment. It helped me to understand the art of silent film, and to move past my former bias towards them. One has to "work" a bit with silent films. It is such a different experience to allow the images to (for the most part) speak for themselves. This extra bit of effort also helped me in more fully embracing foreign films. They too force the viewer to more deeply engaged - by watching the picture more closely, and reading the dialogue.
Thankfully, the Criterion Collection of The Gold Rush includes both the 1925, and 1942 versions. No matter which one you prefer though, The Gold Rush is deservedly one of Chaplin's most acclaimed films ever. For one thing, it contains a number of scenes which remain remarkable examples of filmmaking at its highest level.
One is the famous bit where Chaplin boils his boot for dinner. He and his friend Big Jim (Mack Swain) are starving, and it is all he can come up with to cook. After this unsatisfying meal, Big Jim hallucinates Chaplin as a giant chicken, and almost kills him. The technical expertise in making our hero become a chicken then switch back is a marvel, and was state of the art at the time. Rather than simply cutting away, and having The Little Tramp re-appear in the next frame as a chicken, Chaplin actually appears to transform right before our very eyes.
Another brilliant bit is when the cabin that Chaplin and Big Jim occupy gets pushed to the precipice of a mountain during a storm. Again, the techniques employed to create this illusion were incredible for their day. Then there is one of the most magical moments in film history; "The Dance of the Rolls." Robert Downey Jr. did a nice re-creation of this in Chaplin (1992), but nothing can compare to the original.
The two versions of The Gold Rush comprise the first DVD, while the second is devoted to supplemental material. Of these I found the 27-minute "Chaplin Today: The Gold Rush" (2002) to be the most illuminating. The piece includes interviews with Georgia Hale, Mary Pickford, and filmmaker Idrissa Ouedraogo, and offers a thorough account of the making of the film. "Presenting The Gold Rush" is a 16-minute piece featuring filmmaker Kevin Brownlow and Chaplin biographer Jeffery Vance discussing the film. Rounding out the second DVD are the self-explanatory "A Time of Innovation: Visual Effects in The Gold Rush," and Music by Charles Chaplin."
Criterion always includes a booklet with their releases. These generally include essays by noted film scholars, in the case of The Gold Rush, the essay "As Good As Gold" was written by Luc Sante. The real treasure here though is the reprint of the Time magazine review of the 1942 reissue of The Gold Rush, written by none other than the famous author James Agee.
The Gold Rush is a masterpiece, without question. Some fans prefer City Lights (1931), and some swear by Modern Times (1936), among others. But this is just a matter of personal preference, academic in the end. But for the sake of argument, I stubbornly stick by the restored 1925 silent The Gold Rush. I think that all fans can agree upon one thing though, this Criterion Collection edition is the best home-video presentation of The Gold Rush ever.


The concept and construction of signed real measure of regular languages have been recently developed in literature. This article formulates an analytical ap proach to determine the language measure parameters, namely a state transition cost matrix and a state cost vector. The proposed approach is based on the the ory of Markov stochastic processes. Kronecker algebra is used to facilitate the description of large systems, while minimizing memory requirements. It is shown, through a manufacturing example, how the formulated method may provide a systematic way for quantita tive evaluation and comparison of the discrete-event dynamic behaviour of the supervised plants.


is available as 16 or 32 shafts in each width.
We can assemble your loom for you, or you can assemble it yourself with the instructions that are included. It is best to assemble it with a friend.
Read the Blog page on Megado as it gives more information on the Megado from the Louet website..


The Kyles Allotment Group runs regular Producers' Markets in Kames Hall, where, along with lots of other local producers, KAG sell their surplus veg. There is normally a spring market, a couple of summer markets, an autumn market and a fabulously festive Christmas market. Keep en eye online for the exact dates for 2019.


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ALL POSTS Car Rental Franchise U-SAVE Franchise
U-SAVE Car & Truck Rental 2017 Franchise Disclosure Document is Not Pretty
April 13, 2017 April 13, 2017 ADMIN 1 Comment FSNA, car rental franchise opportunity, Franchise Disclosure Document, Franchise Services of North America Inc., Thomas McDonnell III, Tom McDonnell, U-Save, U-Save Car and Truck Rental, U-Save Car and Truck Rental franchise, U-Save complaints, U-Save FDD, U-Save franchise, U-Save franchise complaints, U-Save Holdings Inc., unhappy franchisee, USave, Usave franchise, USave rental
U-SAVE Car & Truck Rental Franchise Disclosure Document for 2017 doesn't paint a pretty picture of the struggling Ridgeland, Mississippi-based franchisor. Franchise and employee lawsuits, a 25% franchisee turnover rate and a working capital deficit of $12,374,676 are a few of the red flags that stand out.
(UnhappyFranchisee.Com) U-Save Auto Rental has released its updated franchise disclosure document (fdd) for 2017.
You can read all 182 elecrifying pages here:
2017 U-Save Car & Truck Rental Franchise Disclosure Document (FDD)
We've included some highlights (lowlights?) below.
U-SAVE Car & Truck Rental Franchise: Business Experience
Item 2 of the U-Save Auto Rental FDD discloses the following executives as the U-Save management team:
Thomas P. McDonnell III Chief Executive Officer, President, Director and Treasurer
O. Kendall Moore Vice President, General Counsel and Secretary
Ashley M. Chambliss, CPA Chief Financial Officer
Jason T. Brown Director. Franchise Sales
Doug L. Foster Franchise Service Manager
Gerry Foster Franchise Service Manager
Carrie Kelly Director of Marketing
Bradley Kjar Director of Operations
Paul Wood Franchise Service Manager/Regional Sales Director
U-SAVE Car & Truck Rental Franchise Litigation
Item 3 of the 2017 U-Save Auto Rental FDD lists the following litigation:
Trek Auto Sales, Inc, v. U-Save Auto Rental of America, Inc.* (U.S. Dist. Ct. for the District of New Jersey, Case No. 3:15-cv-08323-FLW-TJB).
U-Save Auto Rental of America. Inc, v. RAC Enterprises L.L.C.. and David Rapier (U.S. Dist. Ct. for the S.D. of Miss., Civil Action No. 3:12CV464 WHM-LRA).
U-Save Auto Rental of America. Inc, vs Sanford Miller (U.S. Dist. Ct. for the S.D. of Miss., Case No. 3:13-CV-127 HTW-LRA).
Sanford Miller and Narcoossee Place LLC v. Macquarie Group Limited. Macquarie Capital (USA), Inc., MIHI. LLC. Boketo. LLC. Adreca Holdings Corp,. and Bruce Donaldson (In the Supreme Court of the State of New York, County of New York, Case No. 651 612/13, Filed 3 May 2013).
Bob Barton v. U-Save Auto Rental of America, Inc.** AAA #69-166-57-14.
* Trek Auto Sales, Inc. franchisee lawsuit against U-Save is currently in arbitration
** Former U-Save executive Bob Barton sued U-Save after his employment was terminated for "cause." On May 6, 2015 "the arbitrator issued an award, in Barton's favor, in the amount of $596,645.44."
U-SAVE Car & Truck Rental Franchisee Turnover: 25%
According to Item 20 of the U-Save Auto Rental Franchise Disclosure Document (FDD), U-Save had a total of 123 franchises between 2014 and 2016.
U-Save terminated 31 franchises during that time.
U-Save had a 25% turnover of franchisees during that period.
The U-Save domestic franchise count declined from 104 operating units at the start of 2014 to just 92 units at the end of 2016, a 16% decline.
Franchise Services of North America (FSNA) Has a Working Capital Deficit of $12M
The audited financial statements for franchisor Franchise Services of North America (FSNA) include this statement under "Plan of Operations":
The Company's future operations are dependent upon the Company's ability to maintain profitable operations and FSNA's ability to obtain additional capital to meet its cash flow needs. In addition, FSNA has a working capital deficit of $12,374,676 as of September 30, 2016 primarily associated with liabilities incurred in connection with its acquisition and reconsolidation of Simply Wheelz. Management believes the Company has sufficient capital to meets its obligations in the ordinary course of business through fiscal 2017, but may require additional capital in subsequent periods. There can be no assurance that FSNA or the Company will raise additional capital on favorable terms or at all.
Read the 2017 U-Save Car & Truck Rental Franchise Disclosure Document (FDD)
U-SAVE Franchise Complaints
U-SAVE Franchise Lawsuits
American Car Rental Association (ACRA) Board Member Bradley Kjar Wins Award
U-SAVE FRANCHISE, Brad Kjar Earn Pinocchio Award
MACQUARIE CAPITAL vs. FRANCHISE SERVICES OF NORTH AMERICA
ARE YOU FAMILIAR WITH THE U-SAVE FRANCHISE OPPORTUNITY? SHARE A COMMENT BELOW.
Contact UnhappyFranchisee.com
TAGS: U-Save franchise complaints, U-Save, U-Save franchise, U-Save complaints, U-Save Car and Truck Rental, U-Save Car and Truck Rental franchise, car rental franchise opportunity, Tom McDonnell, Thomas McDonnell III, U-Save Holdings Inc., Franchise Services of North America Inc., FSNA, U-Save FDD, Franchise disclosure document, unhappy franchisee
← JIMMY JOHN'S Franchise Complaints (Part 2)
Christine Johnson, General Counsel, Dickey's Barbecue Restaurants, Inc. (DBRI) →
DELI DELICIOUS Prohibited from Selling Franchises in California
REVIVE ENERGY MINTS 15 Action Steps for Victims
October 2, 2011 October 2, 2011 ADMIN 20
Robyn & Corey Rivera – Java Jo'z / Cuppy's Coffee
April 13, 2008 July 26, 2008 ADMIN 13
One thought on "U-SAVE Car & Truck Rental 2017 Franchise Disclosure Document is Not Pretty"
Arman Arman
This franchise, in my opinion, has very little to offer. I have been told that the Usave franchisor shut down the reservation system on a West Coast licensee, over a dispute, that operates over 350 cars. Usave has shut off the flow of business to more than one licensee and in my opinion has violated the Franchise Practices Act. Buyer beware.


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Frank William Crilley (September 13, 1883 – November 23, 1947) was a United States Navy diver and a recipient of the United States military's highest decoration, the Medal of Honor.

Biography
Crilley was born in Trenton, New Jersey, on September 13, 1883. After enlisting in the U.S. Navy in March 1900, he became a gunner's mate and received additional training as a diver.

On April 17, 1915, Crilley, by then a chief gunner's mate, participated in a dive to inspect the wreck of the submarine  in preparation for a salvage operation. The F-4 had sunk weeks earlier in  of water off the coast of Honolulu, Hawaii. Another diver, Chief Gunner's Mate William F. Loughman, became trapped underwater while returning to the surface after examining one of the F-4's hawsers. Loughman's lifeline and air hose became tangled in the hawser, preventing him from either ascending or descending. Crilley voluntarily dove down and untangled the lines, allowing Loughman to be pulled to safety. For these actions, he was awarded the Silver Lifesaving Medal in April 1916 and the Medal of Honor on November 19, 1928, thirteen years after the rescue.  The latter award was formally presented to him on the White House grounds by President Calvin Coolidge on February 15, 1929.

In 1917, Crilley was appointed to the warrant officer rank of Gunner(T), and in February 1918 became an ensign in the Naval Reserve. He commanded  in 1919 and left active duty in July of that year. In the mid-1920s, he was involved with salvaging , and returned to active naval service in 1927-28 to work on the recovery of . He was awarded the Navy's second-highest decoration, the Navy Cross, for his actions as a diver during that operation.

In 1931, Crilley served as second officer and master diver during the Arctic expedition of the civilian submarine Nautilus (previously known as ). Also in 1931, he assisted with the salvage of . Transferred to the retired list in May 1932, he was again employed on Navy work in 1939, during the salvage of .

Crilley died at the Naval Hospital in Brooklyn, New York, on November 23, 1947. He was buried at Arlington National Cemetery.

Medal of Honor citation

Citation:

For display of extraordinary heroism in the line of his profession above and beyond the call of duty during the diving operations in connection with the sinking in a depth of water 304 feet, of the U.S.S. F-4 with all on board, as a result of loss of depth control, which occurred off Honolulu, T.H., on 25 March 1915. On 17 April 1915, William F. Loughman, chief gunner's mate, U.S. Navy, who had descended to the wreck and had examined one of the wire hawsers attached to it, upon starting his ascent, and when at a depth of 250 feet beneath the surface of the water, had his lifeline and air hose so badly fouled by this hawser that he was unable to free himself; he could neither ascend nor descend. On account of the length of time that Loughman had already been subjected to the great pressure due to the depth of water, and of the uncertainty of the additional time he would have to be subjected to this pressure before he could be brought to the surface, it was imperative that steps be taken at once to clear him. Instantly, realizing the desperate case of his comrade, Crilley volunteered to go to his aid, immediately donned a diving suit and descended. After a lapse of time of 2 hours and 11 minutes, Crilley was brought to the surface, having by a superb exhibition of skill, coolness, endurance and fortitude, untangled the snarl of lines and cleared his imperiled comrade, so that he was brought, still alive, to the surface.

Navy Cross citation

Citation:

The President of the United States of America takes pleasure in presenting the Navy Cross to Chief Gunner's Mate Frank William Crilley, United States Navy, for extraordinary heroism and fearless devotion to duty during the diving operations in connection with the salvage of the U.S.S. S-4, sunk as a result of a collision off Provincetown, Massachusetts, 17 December 1927. During the period 17 December 1927 to 17 March 1928, on which latter date the ill-fated vessel was raised, Chief Gunner's Mate Crilley, under the most adverse weather conditions, at the risk of his life, descended many times into the icy waters and displayed throughout that period fortitude, skill, determination and courage which characterizes conduct above and beyond the call of duty.

See also

List of Medal of Honor recipients
List of Medal of Honor recipients during Peacetime

References

1883 births
1947 deaths
People from Trenton, New Jersey
American underwater divers
United States Navy Medal of Honor recipients
Recipients of the Navy Cross (United States)
United States Navy officers
Non-combat recipients of the Medal of Honor
Burials at Arlington National Cemetery
Military personnel from New Jersey


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Jennifer Lopez JLO turns 450 Comments
Posted on 24 Jul 2014 at 10:18am
Today, American actress Jennifer Lopez became 45 years old. Jennifer Lopez is now known as a famous writer, actress, fashion designer, dancer, singer and producer while she started her career with a small role in movie "My Little Girl" in 1986. The main role of American singer 'Selena' in the movie with same name was played by Jennifer in 1997. Jennifer started her singing career in 1999 with her debut music album "On the 6" that became instantly popular. Total 8,000,000 copies of this album sold worldwide while 3,000,000 copies sold in USA only. In 2001, Jennifer's second album "J.Lo" and movie "The Wedding Planner" released simultaneously and within a week, both came at 1st spot. Her 8th music album "A.K.A" has released in June 2014. Jennifer was also a participant of 2014 FIFA world cup official song.
Jennifer Lopez JLO turns 45
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