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

The macroscopic state of a superconductor is described
by a complex field the order parameter. For low T$_c$ superconductors
it can be assumed that only the phase of the order parameter
varies. The coupling of two such superconductors
across a thin oxide layer is described
by the Josephson equations \cite{josephson}.
\begin{equation}\label{josephson}
V=\frac{d\Phi}{dt},~~~I=sJ_c\sin({\Phi \over \Phi_0})~,
\end{equation}
where $\Phi$ is the phase difference between the top and bottom
superconductor, $V$ and $I$ are, respectively, the voltage and current across
the barrier, $s$ is the contact surface, $J_c$ is the critical current
density and $\Phi_0=\hbar/2e$ is the reduced flux quantum.
These two Josephson relations together with Maxwell's equations imply
the modulation of DC current by an external magnetic field in the static
regime and the conversion of AC current into microwave radiation
\cite{Barone,Likharev}. Such Josephson junctions are then unique electronic
systems for applications like the detection of magnetic fields, ultra
fast electronics\cite{Likharev} and microwave sources and signal mixers
\cite{Salez},\cite{bsf09}.

For the applications the devices are often associated to form
arrays. The junctions can be in parallel or in series.
The series arrays can lead to synchronization\cite{vbsl02}.
and deliver more output power for some applications. Their
description is however more complex and we will not consider it
here. Parallel arrays where the junctions are embedded
between two superconducting planes are now relatively easy to prepare and the
junction is protected from the atmosphere. In addition one
can easily prepare an array with junctions of specific
sizes and positions. Such non uniform arrays have been produced
and analyzed in particular by Salez and co-workers at the Observatory of
Paris. For these systems, the phase difference $\Phi$
satisfies an inhomogeneous 2D damped driven sine Gordon equation \cite{cfv95}
resulting from Maxwell's equations and the Josephson constitutive relations
(\ref{josephson}). The damping is due to the normal electrons and the driving
through the boundary conditions with an external current or
magnetic field applied to the device.

To model such arrays authors have used lumped models where
the spatial dependence between the junctions is omitted. This
obliterates the wave features of the solution and does not
describe well the experiments. Solving numerically the full
two dimensional problem is of course possible, however it
does not lead to understand simply the role of the parameters.
Similar difficulties occur with global (hard) analysis. Consider
for example the problem of finding the maximum current giving
a static solution for a given magnetic field.
Using such global analysis we obtained bounds \cite{cftv03} on the
gradient of the solution that were independent of the area of the junctions
so that little information could be obtained from them.
To overcome these difficulties we recently introduced a continuous/discrete
model that preserves the
continuity of the phase and its normal gradient across the junction interface
and where the phase is assumed constant in the junctions. The relative
simplicity of the model allowed an unprecedented understanding of the
static problem\cite{cl06} and gave excellent agreement with the
complex static response of the array\cite{sblc07}. Additionally the model allows
to solve the inverse problem of building a device that produces a given static
behavior\cite{cl08}.

The dynamic behavior of Josephson junctions is characterized by
the current-voltage (I-V) characteristic curve.
To understand this one needs to analyze periodic solutions of the problem.
For a homogeneous long junction Kulik\cite{kulik} developed
a formalism using a high voltage ansatz and obtained average equations
describing the I-V curve.
This approach was extended by Cirillo et al \cite{cgssv98} who showed that
a magnetic field $\Gamma$ reinforces the cavity modes such that
$\Gamma=n \pi/l$. Using this approach these authors obtained excellent
quantitative agreement with their experimental results.
For arrays of equidistant junctions,
a recent study by Pfeiffer et
al\cite{pasu08} analyzed the fine features of the first resonant step
in terms of Cerenkov radiation between a sine-Gordon discrete kink (fluxon)
and a cavity mode.
For one junction in a cavity, our theoretical
study \cite{cl05}
revealed that the junction could stop waves across the cavity
or enhance them throughout. We also found kink like solutions for the
problem \cite{cl07} and explained some features of the current-voltage
characteristics. Here we analyze theoretically and
numerically the model in particular
when there is a capacity miss-match between the junctions and the cavity.
This capacity ratio is usually large in experiments because the
oxide layer in the junction is about 10 Angstroms while it is about
0.2 Micron in the strip. Taking this miss-match into account
we introduce an associated linear problem which
enables us to predict the position of the resonances in the current-voltage
curve for any array. This linear problem defines eigenvalues and eigenmodes
orthogonal with respect to an inner product that we establish.
At these resonances the solution just contains the Goldstone mode
and the corresponding eigenmode so the dynamics is described by two
simple amplitude equations that we present and analyze. \\
The article is organized as such. After introducing the model in section
2, we analyze it in section 3, establish the periodicity of the current
voltage curve as a function of the magnetic field and simplify the
model using the time averaged (high voltage) solution. The resonances
are studied in section 4 where we define the appropriate spectral problem
for any given array
and find its spectrum and associated inner product. Using the latter
we analyze numerically the current voltage curves in section 5. In particular
we establish and analyze the amplitude equations describing the system
at resonance and discuss the situation for arrays containing many junctions.

\section{The model}
\subsection{The $2D$ problem}
The device we
model (see Fig.\ref{f1}) is composed of two
overlapping superconducting layers, in which stand
small\footnote{compared to the Josephson coherence length $\lambda_J$}
Josephson junctions.
 Using the Josephson constitutive equations
(eq.\ref{josephson}) and Maxwell's equation, one obtains
the following inhomogeneous sine-Gordon equation for $\Phi$ in the
junction region $\Omega_j$ \cite{Barone,Likharev}
\begin{equation}\label{2D_sg}
C_j \Phi_{tt} - \frac{1}{L} \Delta \Phi + J_c
\sin\left(\frac{\Phi}{\Phi_0}\right)+\frac{1}{R} \Phi_t=0\;,
\end{equation}
\begin{figure}
\centerline{\epsfig{file=figs/crct_schm.eps,width=0.49\linewidth,angle=0}
\epsfig{file=figs/jj_fen_dessus.eps,width=0.49\linewidth,angle=0}}
\caption{The left panel shows the top view of a superconducting
microstrip line containing three Josephson junctions.
The parameters $H,I$ and $\phi$ are respectively the applied
magnetic field, the current and the
phase difference between the two superconducting layers.
The right panel
shows the associated 2D domain of size $l \times w$ containing $n=3$ junctions
placed at the positions $y=w/2$ and $x=a_i,~i=1,n$.}
\label{f1}
\end{figure}
where $C_j$ is the capacity of the junction
per unit area, $R$ the resistance per unit
area due to normal electrons. The branch inductance $L =\mu_0 d_0$ involves the
magnetic thickness $d_0$, a quantity used by experimentalists.
In the microstrip $\Omega-\Omega_j$, the Josephson and
quasi-particle currents are
absent so one obtains the wave equation,
\begin{equation}\label{2D_onde}
C_l \Phi_{tt} - \frac{1}{L_l} \Delta \Phi = 0~~,
\end{equation}
where the $l$ subscripts indicate that we are in the linear region
(i.e.: outside the junction).
For inhomogeneous circuits like the one of Fig. \ref{f1} these
two equations can be written as \cite{cl05,louk_th}
\begin{equation}\label{dimsgi}
{C_l} \Phi_{tt} -\nabla \left({1 \over L(x)}\nabla \Phi \right)
+ g(x,y)\left [ (C_j- C_l) \Phi_{tt}
+ J_c \sin \left({\Phi \over \Phi_0}\right) + {\Phi_t \over R} \right ] =0,\end{equation}
where $g=1$ in the junctions and 0 outside.
This formulation guarantees the continuity of the phase and its normal gradient
across the interfaces.
At this point we will assume the same surface inductance $L$ in the junctions
and linear region. This simplifies greatly the formulation and can be realized
in practical situations.
To normalize the equation, we introduce the units of length and time,
respectively the Josephson length $\lambda_J$ and plasma frequency $\omega_p$
\begin{equation}\label{ljop}
\lambda_J = \sqrt{\frac{\Phi_0}{J_c L_l}}\;,~~~\frac{1}{\omega_p}=
\sqrt{\frac{C_l \Phi_0}{J_c}}\;.
\end{equation}
Finally we normalize space, time and phase as
\begin{equation}\label{norma}
\tilde{x}=x/\lambda_J,~~
\tilde{y}=y/\lambda_J,~~\tilde{t}=t \omega_p,
~~\varphi = \Phi / \Phi_0,
\end{equation}
to get the normalized 2D inhomogeneous perturbed sine-Gordon equation
\begin{equation}\label{sg2d}
\varphi_{\tilde t \tilde t} -\Delta \varphi + g( \tilde x, \tilde y)(\kappa \varphi_{ \tilde t \tilde t} +
\alpha \varphi_{ \tilde t} + \sin\varphi  )= 0\end{equation}
where the coefficients $\alpha$
and $\kappa$ are
\begin{equation}\label{alphakappa}
\kappa = \frac{{C_j}}{{C_l}}-1,~~~\alpha=\frac{1}{{R}}
\sqrt{\frac{\Phi_0}{J_c {C_l}}}\;.
\end{equation}

The boundary conditions are
\begin{eqnarray} \label{bc2d}
\varphi_{\tilde x}|_{({\tilde x}=0)} = {\tilde H} + (1-\nu) \frac{\tilde I}{2\tilde w},~~~
\varphi_{\tilde x}|_{({\tilde x}={\tilde l})} = {\tilde H} - (1-\nu)
\frac{\tilde I}{2\tilde w}, \\
\varphi_{\tilde y}|_{({\tilde y}=0)} =  \nu \frac{\tilde I}{2\tilde l},~~~
\varphi_{\tilde y}|_{({\tilde y}={\tilde w})} =  -\nu \frac{\tilde I}{2\tilde l},
\end{eqnarray}
where
\begin{equation}\label{dimH_I}
\tilde H = H \frac{\lambda_J d_0}{\Phi_0}\;,~~
\tilde I = I \frac{\mu_0 d_0}{\Phi_0} =  \frac{L I}{\Phi_0}\;,
\end{equation}
and $l$ and $w$ are normalized by $\lambda_J$.
After this section all tildes will be omitted for simplicity.

\subsection{The $1D$ model}

This equation is difficult to analyze and its solutions can only be obtained
numerically. In addition most real devices have a width that is much smaller
than their length and the junctions are distributed symmetrically so that it
is reasonable to reduce the problem to one dimension. To do this we
expand $\varphi$ on transverse Fourier modes
\begin{equation}\label{2Dto1D}
\varphi(x,y,t) = \frac{\nu \gamma}{2 l}\left(y-\frac{w}{2}\right)^2
+ \sum^{\infty}_{n=0} \phi_n(x,t) \cos \left(\frac{n \pi y}{w}\right)\;,
\end{equation}
where $\gamma=I/w$ and
the first term takes care of the boundary condition (\ref{bc2d}).
After inserting (\ref{2Dto1D}) into (\ref{sg2d}) and integrating across
$y$ we get the evolution of $\phi_0$
\begin{equation}\label{phi0}
\phi_{tt} -\phi_{xx} + {w_j \over w} g(x,y=0)
(\kappa \phi_{tt} + \alpha \phi_{t} + \sin\phi  )= \nu {\gamma \over l}\;,
\end{equation}
where we omitted the $0$ in $\phi_0$ and terms in $\phi_i, i \ge 1$
which are small because
of the smallness of the current \cite{bcf02}.

The boundary conditions are
\begin{equation}\label{bc1d}
{\phi}_x|_{x=0}= H-(1-\nu){\gamma \over 2},~~~~
{\phi}_x|_{x=l}= H+(1-\nu){\gamma \over 2},
\end{equation}
and
$$g(x) = \left\{
\begin{array}{l r}
\frac{w_j}{w}, & a_j-\frac{l_j}{2}<x<a_j+\frac{l_j}{2}, \\
0, & {\rm elsewhere}. \\
\end{array} \right..$$
The factor
$w_j/w$ is exactly the "rescaling" of $\lambda_J (=1)$ into
$\lambda_{eff} = \sqrt{w \over w_j}>1 $ due to the presence of the
lateral passive region \cite{cfv96}.

As the area of the junction is reduced the total super-current is
reduced and tends to zero. Small junctions where the
phase variation can be neglected and that have a significant
supercurrent we introduce the
following Dirac distribution model. First define the function
$g_h(x)$,
$$g_h(x) = \left\{
\begin{array}{l r}
\frac{w_j}{ w h}, & a_j-\frac{hl_j}{2}<x<a_j+\frac{hl_j}{2}, \\
0, & {\rm elsewhere}, \\
\end{array} \right.$$
notice that $g_1(x)=g(x)$.
When the junction widths $w_j \ll \lambda_J$ we can further reduce the problem
by neglecting the variation of the phase inside the junctions.
We then obtain the Dirac delta function distributed model \cite{cl05,louk_th}
by making $h$ tend to $0$. For an $n$ junctions device,
$$\lim_{h \to 0}g_h(x) = \sum_{j=1}^{n}\frac{w_j l_j}{l}\delta(x-a_j).$$
Finally we obtain the $\delta-1D$ model for the device,
\begin{equation}\label{delta-1D}
{\phi}_{tt} -{\phi}_{xx}+ \sum_{j=1}^{n} d_j \delta(x-a_j)
( \kappa {\phi}_{tt} + \sin {\phi} + \alpha {\phi}_{t})
=\nu \frac{\gamma}{l},
\end{equation}
where
\begin{equation}\label{defdj}
d_j = \frac{w_j l_j}{w}
\end{equation}
and where the boundary conditions are given by eq.(\ref{bc1d}).
This is the main model of the article and we will analyze it in
detail.

\section{Preliminary analysis}

From the Josephson equation (\ref{josephson}) it can be seen that
${\phi}_{t}$ is a voltage. In experiments this instantaneous voltage
is of very high frequency ($\sim$ 500 GHz) and can only be detected
by making it beat with a well-known source. On the other hand the time
average voltage can be measured in a fairly standard way. This
current voltage relation (the $I-V$ curve) is a characterization of the device.
It is therefore important for the analysis to explain it.
We now give some important symmetries of the $I-V$ curve. The first one
is the periodicity with respect to the magnetic field $H$. This is similar
to the one obtained in the static case \cite{cl06}.

\subsection{Periodicity of the $I-V$ curve with $H$}

The $I-V$ curve of the device modeled by eq.(\ref{delta-1D}) and
(\ref{bc1d}) depends on the magnetic field $H$. We denote $I-V|_H$
the $I-V$ curve of the device for the magnetic field $H$.
Let us introduce $l_j=a_{j+1}-a_j$ the distance between two
consecutive junctions. Let $l_{min}$ be the smallest distance
$l_j$. We define a harmonic array as a circuit
where $l_i$ is a multiple of $l_{min}$ for all $i$.

\paragraph{Proposition, Periodicity of the device:}
For a harmonic circuit, the $I-V|_H$
curve is periodic with a period $2\pi/l_{min}$.

\paragraph{Proof:}
Let $\phi$ be a solution of (\ref{delta-1D}) for a current $\gamma$ and a magnetic field $H$. We introduce $f(x)=(2\pi/l_{min})(x-a_1)$ and
$\psi(x;t) = \phi(x;t) + f(x)$. So $\psi$ verifies
\begin{equation}
{\phi}_{tt} -{\phi}_{xx}+ \sum_{j=1}^{n}\frac{w_j l_j}{l}\delta(x-a_j)
( \kappa {\phi}_{tt} + \sin ({\psi}-f) + \alpha {\phi}_{t})
=\nu \frac{\gamma}{l},
\end{equation}
with $\psi_x(0;t) = H+2\pi/l_i - (1-\nu)\gamma/2$, and
$\psi_x(l;t) = H+2\pi/l_i + (1-\nu)\gamma/2$.
Since, $f(a_j)=2k\pi$, $\forall i\in \{1;...;n\}$, then $\psi$
is a solution of (\ref{delta-1D}) for $H+H_p \equiv H+2\pi/l_{min}$
and for the same $\gamma$.

Conversely, by subtracting $f$ from a solution associated to $H+H_p $
and a current $\gamma$, we obtain a solution for $H$ and the same
current $\gamma$. We have shown that for a given $\gamma$ and $H$,
if there is a solution, we can find another for $H+2k\pi/l_{min}$
($k \in {Z}$). We obtain the same $I-V$ curves.
\begin{equation}\label{Prdct}
I-V|_{H+H_p} = I-V|_H.
\end{equation}
with $H_p = 2\pi/l_{min}$.
\qquad

In the non harmonic case, if the junctions are set such that
$l_j=p_j/q_j$, where $p_j$ and $q_j$ are integers, prime with
each other, then  $I-V|_H$ is periodic with period $H_p$ such that
\begin{equation}\label{periodicity}
H_p = 2\pi \frac{LCM(q_1;...;q_{n-1})}{HCF(p_1;...;p_{n-1})},
\end{equation}
where $LCM$ is the Lowest Common Multiple and $HCF$ the
Highest Common Factor. To prove this write
$f(x) = H_p (x-a_1)$ and use again the previous argument.

\subsection{The high voltage approximation }

When the voltage $\phi_t$ is large, the phase $\phi$ is
rotating fast so that one can write
\begin{equation}\label{phi_hv} \phi(x,t) =
V t + \psi(x,t),\end{equation}
where the average
\begin{equation}\label{phiv} \left<\psi \right>
\equiv {1\over T}\int_t^{t+T} \psi(x,t') dt'
\equiv \phi_v(x).\end{equation}
Plugging the ansatz (\ref{phi_hv}) into (\ref{delta-1D}) and taking
the average we get
\begin{eqnarray}
\left<\psi_{tt} \right>-{\phi_v}_{xx} + \sum_{j=1}^n  d_j \delta(x-a_j)
&&\nonumber \\
[-1.5ex] \label{av_delta-1D} \\ [-1.5ex]
\left[  \kappa \left<\psi_{tt} \right>
+ \left< \sin(V t) \cos(\psi) \right>
+\left< \cos(V t) \sin(\psi)\right> +
\alpha \left< \psi_{t} \right> + \alpha V \right] &=&\gamma.
\nonumber
\end{eqnarray}
Then if we neglect the nonlinear terms we obtain
the static equation, we obtain a new equation such
that $\left<\psi_{tt} \right> =\left<\psi_{t} \right>=0$ is a
solution. Thus,
\begin{equation}\label{hvolt}
 -{\phi_v}_{xx} + \alpha V \sum_{j=1}^n  d_j \delta(x-a_j)
= \nu {\gamma \over l} , \end{equation}
together with the boundary conditions
\begin{equation}\label{bchvolt}
{\phi_v}_x|_{x=0} = H - (1-\nu) {\gamma \over 2},~~
{\phi_v}_x|_{x=l} = H + (1-\nu) {\gamma \over 2} .\end{equation}
By integrating the equation (\ref{hvolt}) one sees that
this problem has a solution if
\begin{equation}\label{iv}
V = {\gamma \over  \alpha \sum_j d_j}.
\end{equation}

Let us write this high voltage solution for a device with many junctions.
At each junction $x=a_j$, the phase must be continuous. Integrating equation
(\ref{hvolt}) over a neighborhood of $a_j$ and taking the limit of this neighborhood
going to zero one gets the jump condition
\begin{equation}\label{jump_phiv}
-\left[{\phi_v}_x\right]_{a_j^-}^{a_j^+}
+ \alpha V {d_j \over l}=0.
\end{equation}
Notice also that outside the junctions, the solution has to be of the form
$$\phi_v(x) = -\nu {\gamma \over 2l}x^2 + c_1 x + c_2.$$
It is then natural to build the solution by steps, using a sort of
shooting method. To simplify the discussion, let us consider a device with two
junctions placed respectively at the positions $a_1$, $a_2$. We write
\begin{equation}\label{shoot_phiv}
\phi_v(x) = \left\{ \begin{array}{ll}
\phi_0(x),&~0\le x\le a_1,\\
\phi_1(x),&~a_1\le x\le a_2,\\
\phi_2(x),&~a_2\le x\le l.
\end{array} \right.
\end{equation}
At each junction $a_j$ we have $\phi_{j+1}(a_j)=\phi_j(a_j)$ and
$${\phi_{j+1}}_x(a_j^+)- {\phi_j}_x(a_j^-) +\alpha V {d_j \over l}=0.$$
It is then natural to write
\begin{equation}\label{phiv0}
\phi_0(x) = -\nu {\gamma \over 2l}x^2 + \left[H-(1-\nu)\frac{\gamma}{2}
\right]x + C.
\end{equation}
From the relations at each junction, we infer that
\begin{eqnarray}
\phi_1(x) = \phi_0(x) + \gamma {d_1 \over d_1+d_2}(x-a_1),
\nonumber \\
[-1.5ex] \label{phiv12} \\ [-1.5ex]
\phi_2(x) = \phi_1(x) + \gamma {d_2 \over d_1 + d_2}(x-a_2).
\nonumber
\end{eqnarray}
It can be checked that the right boundary condition for $x=l$ can verified
by this formulation whatever the value of $H$ by choosing $\gamma$ thanks to
eq.(\ref{iv}).
The function $\phi_v(x)$ defined by the equations (\ref{phiv0}-\ref{phiv12})
is then the solution of the problem (\ref{hvolt},\ref{bchvolt}).

\subsection{Simplification of the $\delta-1D$ model}

The static part $\phi_v$ of the high voltage solution can be used
to simplify the formulation of the problem, in particular the boundary
conditions. For that we introduce
\begin{equation}\label{psi}
\psi(x,t) = \phi(x,t) -\phi_v(x),
\end{equation}
so that the sine-Gordon equation (\ref{delta-1D}) becomes
\begin{equation}\label{delpsi_phiv}
\psi_{tt} -\psi_{xx} + \sum_{j=1}^n  d_j \delta(x-a_j)
\left [ \kappa \psi_{tt}+ \alpha \psi_{t} + \sin(\psi + \phi_v)
- \frac{\gamma}{\sum_{k=1}^n d_k}
\right ]= 0\;,\end{equation}
with the homogeneous Neumann boundary conditions
\begin{equation}\label{bcpsi}
\psi_x|_{x=0} = 0,~~\psi_x|_{x=l} = 0 . \end{equation}
Equation (\ref{delpsi_phiv}) now contains for each junction
a sine term with an argument
that is shifted by $\phi_v(a_j)-\phi_v(a_1)$. When averaging
over time we get
$$ \left< \phi(a_j) -\phi(a_1) \right>_t
\approx \phi_v(a_j) - \phi_v(a_1)\;.$$
We define $\Psi_j$ as the phase-shift of the junction $j$
\begin{equation}\label{def_dephas}
\Psi_j=\phi_v(a_j) - \phi_v(a_1)\;.
\end{equation}
By a simple translation, equation (\ref{delpsi_phiv}) becomes
\begin{equation}\label{delpsi}
\psi_{tt} -\psi_{xx} + \sum_{j=1}^n  d_j \delta(x-a_j)
\left [\kappa \psi_{tt} + \alpha \psi_{t} + \sin(\psi + \Psi_j) -
\frac{\gamma}{\sum_{k=1}^n d_k}\right ]= 0\;.\end{equation}
The boundary conditions (\ref{bcpsi}) are unchanged and $\Psi_1=0$.

\section{Resonances for an array of point junctions}

\subsection{Spectral problem}

When the system is on a resonance, the solution is periodic so that
it can be written as $\psi(x,t) = e^{i \omega t} \varphi(x)$. In that
case the terms $\alpha \psi_{t} + \gamma/\sum_{k=1}^n d_k$
globally balance each other. The sine term can be averaged out. Therefore
to satisfy the equation (\ref{delpsi}) it is necessary that
\begin{equation}\label{linpsi}
\psi_{tt} -\psi_{xx} + \sum_{j=1}^n  d_j \delta(x-a_j)
\kappa \psi_{tt}  = 0\;,\end{equation}
together with the homogeneous Neumann boundary conditions (\ref{bcpsi}).

When the capacities per unit area of the junctions and passive regions
are equal $\kappa=0$, so the array resonates on a cosine Fourier mode,
see \cite{cl05,cl07}.
$$\psi_n(x,t) = e^{i  {n\pi t \over l}}
\cos\left( {n\pi x \over l}\right).$$
In this description, we have neglected the higher harmonics which
decay exponentially as shown by numerical calculations.
When $\kappa\neq 0$, the eigenmodes of equation (\ref{linpsi})
differ from $n \pi /l$. To analyze them we substitute
$\psi(x,t) = e^{i \omega t} \varphi(x)$ into (\ref{linpsi}) and obtain
the following eigenvalue problem
\begin{equation}\label{eigvp}
\varphi_{xx} + \omega^2 \left [ 1+
\sum_{j=1}^n  d_j \delta(x-a_j) \kappa \right] \varphi =0,\end{equation}
together with homogeneous Neumann boundary conditions for $\varphi$.

We consider the two junctions case.
We will obtain results that can be generalized to an array with
$n>2$ junctions. We introduce for ease of notation
$\kappa_j = \kappa d_j$.
It is then natural to assume a solution
\begin{equation}
\varphi(x) = \left\{ \begin{array}{lll}
A_1 \cos(\omega x),&{\rm for}& 0\le x \le a_1,\\
A_2 \cos(\omega x) + B_2 \sin(\omega x),
& {\rm for} & a_1\le x \le a_2,\\
A_3 \cos(\omega x) + B_3 \sin(\omega x), & {\rm for} & a_2\le x \le l,
\end{array} \right .
\end{equation}
where the form of $\varphi$ in the first interval was chosen to satisfy the
boundary condition at $x=0$.
As usual $\varphi$ is continuous at the junctions
$x=a_j$ and one can see that the jump of the derivative is
\begin{equation}\label{jvpx}
[\varphi_x]_{a_j^-}^{a_j^+}+\kappa_j \omega^2 \varphi(a_j)=0.
\end{equation}
Writing these $2 n = 4 $ conditions at the junctions and the
boundary condition
at $x=l$, one obtains a 5th order homogeneous system in $A_1,A_2,B_2,
A_3,B_3$. The solution is non trivial if the following determinant is zero
\begin{equation}\label{disp}
\left \| \begin{array}{lcccr}
C_1   &  -C_1  & -S_1  & 0 & 0   \\
S_1   &  - S_1 +\kappa_1 \omega C_1 & C_1 + \kappa_1 \omega S_1  & 0 & 0   \\
0   & C_2  &   S_2      & -C_2  & -S_2    \\
0 & S_2 & -C_2 & -S_2 +\kappa_2 \omega C_2   & C_2+\kappa_2 \omega S_2    \\
0     & 0       &   0      & -S_l   & C_l
\end{array} \right \|  =0,
\end{equation}
where
$C_1=\cos(\omega a_1),~S_1=\sin (\omega a_1),
~C_2=\cos(\omega a_2),~S_2=\sin (\omega a_2),
~C_l = \cos(\omega l),~S_l = \sin (\omega l).$
When  more junctions are present in the device, the determinant giving the
dispersion relation can be generated by adding the elementary component
in rows 3 and 4 corresponding to each additional junction.

We now give the resonant frequencies when there is only one junction.
The determinant (\ref{disp}) becomes
\begin{equation}\label{detdisp1}
\left \| \begin{array}{lcr} C_1   &  -C_1  & -S_1     \\
S_1   &  - S_1 +\kappa_1 \omega C_1 & C_1 + \kappa_1 \omega S_1 \\
0     & -S_l   & C_l
\end{array} \right \|  =0,
\end{equation}
which implies
\begin{equation}\label{dispersion}
\sin(\omega l) + \kappa \omega d_1 \cos(\omega a_1) \cos(\omega (l-a_1)) =0.\end{equation}
Several remarks should be made on this relation. \begin{itemize}
\item First, when
$\kappa \ll 1$ we recover the usual $\sin(\omega l)=0$ dispersion
leading to harmonic frequencies.
\item The opposite limit $\kappa \gg 1$
is more interesting because it leads to a splitting of the oscillations
in the left and in the right side of the film. We obtain
$\cos(\omega a_1)=0$ or $\cos(\omega (l-a_1)) =0$ leading to
$\omega a_1 = (2n+1) \pi/2$ or $\omega (l-a_1)= (2m +1)\pi/2$ where
$m,n$ are integers.
\end{itemize}
\begin{figure}
\centerline{\epsfig{file=figs/1jj_frequency_a1.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/1jj_frequency_kappa.eps,width=0.5\linewidth,angle=0}}
\caption{Dependence of the eigenvalues (zeros of the dispersion relation
(\ref{dispersion}) ) on the position of the junction
$a_1$ in the microstrip (left panel) and on the capacity miss-match $\kappa$
(right panel). For the left panel, $\kappa=0.5$. For the right panel
$a_1=0.3 l = 0.3 \pi$.}
\label{f2}
\end{figure}

Notice that Larsen et al \cite{ldm91} considered
a centered junction $a_1=l/2$ in a microstrip. Their dispersion relation
$$ \tan \pi  {f \over f_e} = - \pi {C_j \over C_e} {f \over f_e},$$
is similar to the one we get with our approach except
that the coefficient is different. We obtain using (\ref{dispersion})
\begin{equation}\label{disp_larsen}
\tan (\omega l) = - \omega l  \left({ C_j \over C_l} -1\right)
{w_j^2 \over w l}.\end{equation}
This latter expression gives
the correct eigenmodes as shown in the current voltage characteristics
computed numerically shown in the next sections.

\subsection{Eigenvectors and inner product}

We introduce here the inner product associated with the dispersion
relation (\ref{dispersion}).
The eigenvectors $\varphi_i$ (respectively
$\varphi_j$) associated to the eigenvalue $\omega_i$ (resp.
$\omega_j$) satisfy
\begin{eqnarray}\label{varepsi_i}
{\varphi_i}_{xx}+\omega_i^2 \varphi_i
\left(1+\delta(x-a_1)\kappa_1\right)&=&0,\\
\label{varepsi_j}
{\varphi_j}_{xx}+ \omega_j^2 \varphi_j
\left(1+\delta(x-a_1)\kappa_1\right)&=&0,
\end{eqnarray}
together with the boundary conditions
\begin{equation}\label{bc_varpsi}
\left. {\varphi_i}_{x}\right|_{x\in \{0,l\}}=0,
~{\rm and}~~\left. {\varphi_j}_{x}\right|_{x\in \{0,l\}}=0.
\end{equation}
We assume the eigenvalues to be different.
As usual we multiply eq.(\ref{varepsi_i}) by $\varphi_j$ and
eq.(\ref{varepsi_j}) by $\varphi_i$, substract the second
from the first and integrate on the domain. We obtain
\begin{equation}
\int_0^l ({\varphi_i}_{xx} \varphi_j-
{\varphi_j}_{xx} \varphi_i) dx
+(\omega_i^2-\omega_j^2)
\left [
\int_0^l\varphi_i\varphi_jdx
+ \kappa_1\varphi_i(a_1) \varphi_j(a_1)
\right ] =0.\end{equation}
The first integral is zero because of the boundary conditions
(\ref{bc_varpsi}). Since $\omega_i\neq \omega_j$ we get
\begin{equation} \int_0^l (1+ \kappa_1\delta(x-a_1))
\varphi_i\varphi_j dx =0. \label{subs_lin}
\end{equation}
Thus we obtain the orthogonality of the eigenvectors $\varphi_i$,
$i$ is an integer with the associated inner product defined by
\begin{equation}\label{scl}
\left< f;g\right> \equiv  \int_0^l \left(1+ \kappa_1\delta(x-a_1)
\right)fg dx.
\end{equation}
It is then possible to normalize the eigenvectors so that they form
an orthonormal basis. When there are $n$ junctions in the array, the
inner product can be generalized easily to
\begin{equation}\label{scln}
\left< f;g\right> \equiv
\int_0^l \left(1+ \sum_{i=1}^n\kappa_i\delta(x-a_i)\right)fg dx.
\end{equation}

In the case of a single junction,
the normalized eigenvectors $\varphi_n(x) $ are given by
\begin{equation} \label{phin}
\varphi_n(x) = \left\{\begin{array}{lll}
A_n \cos(\omega_n x),& {\rm for}& 0\le x \le a_1,\\
A_n {\cos(\omega_n a_1) \over\cos(\omega_n(l- a_1))}
\cos(\omega_n(l- x)), & {\rm for} & a_1\le x \le l,
\end{array} \right.
\end{equation}
where $\omega_n$ satisfies the dispersion relation (\ref{dispersion})
and
\begin{equation}\label{an}
A_n = \frac{1}{\sqrt{{l \over 2} +{\sin 2 \omega_n a_1 \over 4 \omega_n }
+{\cos^2 \omega_n a_1 \over \cos^2 \omega_n (l-a_1)}
{\sin 2 \omega_n (l-a_1) \over 4 \omega_n }
+\kappa_1 \cos^2 \omega_n a_1 }}.
\end{equation}
Fig. \ref{f3a} shows the 4 non trivial eigenmodes $\phi_i,~i=1-4$ for
a large capacity miss-match. Notice how the modes are almost zero
on one side of the cavity.
\begin{figure}
\centerline{
\epsfig{file=figs/ka30_f1_lpi_a103.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/ka30_f2_lpi_a103.eps,width=0.5\linewidth,angle=0}}
\centerline{
\epsfig{file=figs/ka30_f3_lpi_a103.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/ka30_f4_lpi_a103.eps,width=0.5\linewidth,angle=0}}
\caption{Normalized eigenmodes $\varphi_i(x), i=1,2 $
(top panels from left to right)
and $\phi_i(x), i=3,4 $ (top panels from left to right) for a
large capacity miss-match $\kappa=30$. The other parameters are
$d_1=1$, $l=\pi$ and
$a_1=0.3\pi$.}
\label{f3a}
\end{figure}

\subsection{Eigenvalues and $I-V$ curves}

The eigenvalues calculated in the previous section appear as
resonances in the current-voltage characteristics of the device.
We computed the $I-V$ curves from the numerical solution of
equation (\ref{delta-1D}). The singular partial differential
equation is integrated over reference volumes and the time
advance is done by an ordinary differential equation solver
(see Appendix \ref{appendix_num} for more details). The average voltage
$V\equiv \left<\phi_t\right>$ is computed over a time interval
$10/\alpha$ after waiting a time $100/\alpha$ for the solution
to stabilize. When the system is locked on a resonance, it oscillates
periodically on an eigenfrequency solution of the dispersion relation
eq.(\ref{dispersion}). We will show that the numerical solution
follows closely the corresponding eigenvector.
\begin{figure}
\centerline{ \epsfig{file=figs/d1lph05kp0.ps,height=0.5\linewidth,angle=270}
\epsfig{file=figs/d1lph05kp1.ps,height=0.5\linewidth,angle=270}}
\centerline{ \epsfig{file=figs/d1lph05kp30.ps,height=0.5\linewidth,angle=270}}
\caption{$I-V$ curves for an array with one junction,
$d_1=1$, $\alpha=0.5$, $l=\pi$,
$a_1=0.3\pi$. Top panel, $\kappa=0$, middle panel $\kappa=1$ and
bottom panel $\kappa=30$.}
\label{f4}
\end{figure}
We choose a device with $l=\pi,~ a_1=l/3,~d_1=1$ and take three limiting
cases $\kappa=0$  $\kappa=1$ and $\kappa=30$. When there is no capacity
miss-match ($\kappa=0$) we showed in
\cite{cl05} that the resonances of the $I-V$ curve are positioned at
$V=k\pi/l$ with $k$ integer and are
bounded by
$$\gamma l= d_1 \alpha V,~{\rm and}~~\gamma l=d_1 \sqrt{(\alpha V)^2+1}.$$
This is shown in the top panel of Fig. \ref{f4}. In the middle
panel we show the $I-V$ curve for $\kappa=1$. Notice how the resonances
are not equally spaced and get sharper. The resonances are very close
to the ones given by the dispersion relation, they are reported in
table (\ref{disp_1j_k1}). Finally on the bottom panel
we computed the $I-V$ curve for $\kappa=30$. This large value is
typical in experiments where the oxide layer in the junction is
about 100 times thinner than in the passive region. This together
with the ratio $d_1 = w_j^2/w$ gives $\kappa_1 \approx 50$.
Notice how the
resonances are vertical indicating that the system is almost
linear. This is to be expected because $\kappa \gg 1$ so that the
sine term plays little role.
The position of the resonances is exactly given by the
dispersion relation (\ref{dispersion}). They are reported in the table
(\ref{disp_1j}) together with the coupling coefficient
$\phi_n(a_1)$. The eigenvalues are clearly not harmonic.
Notice how some
resonances do not go all the way to the junction curve. Another observation
is that
we have a resonance for $\omega_2$ even though the coupling coefficient
is almost zero. We will come back to this point in the next section.
Also it is interesting that there is no resonance for $\omega_4$.
\begin{table}\label{disp_1j_k1}
$$\begin{array}{|c|c|c|c|} \hline
i & \omega_i & \cos(\omega_i a) & {\rm Normalization} ~ A_i\\ \hline
0 &  0.      &       1.           &     0.491    \\ \hline
1 &  0.917   &  0.649             &     0.562  \\\hline
2 &  1.931   & -0.247             &    1.094    \\ \hline
3 & 2.438    & -0.665              &    0.704   \\ \hline
4 & 3.682    & -0.946              &    0.237   \\ \hline
\end{array}$$
\caption{Eigenvalues for the array of Fig. \ref{f4} with $\kappa=1$
(middle panel of Fig. \ref{f4}).}
\end{table}
\begin{table}\label{disp_1j}
$$\begin{array}{|c|c|c|c|} \hline
i & \omega_i & \cos(\omega_i a) & {\rm Normalization} ~ A_i\\ \hline
0 &  0.      &       1.           &     0.1737    \\ \hline
1 &  0.734   &  0.770             &     5.341 ~10^{-2}\\\hline
2 &  1.687   & -1.950~10^{-2}      &    1.447       \\ \hline
3 & 2.150    & -0.440              &    3.462~10^{-2}\\ \hline
4 & 3.575    & -0.974              &    9.102~10^{-3}\\ \hline
\end{array}$$
\caption{Eigenvalues for the array of Fig. \ref{f4} with $\kappa=30$
(bottom panel of Fig. \ref{f4}).}
\end{table}

Fig. \ref{f4a} shows the eigenvectors in dashed line (green online) for
a realistic value $\kappa_1=30$ corresponding to a capacity
miss-match $\kappa=30$ and a surface ratio $d_1=w_1~l_1 /w= 1$.
On the same plot are indicated in continuous line (red online)
the instantaneous voltages
$\varphi_t$  for different successive times. From top to bottom
panels we show respectively the first, second and third resonance.
One can see that the voltages follows very well the eigenvectors.
\begin{figure}
\centerline{ \epsfig{file=figs/res1.ps,height=0.5\linewidth,angle= 270}
\epsfig{file=figs/res2.ps,height=12 cm,height=0.5\linewidth,angle= 270}}
\centerline{\epsfig{file=figs/res3.ps,height=0.5\linewidth,angle= 270} }
\caption{Plot of the instantaneous voltage $\phi_t(x)$ for successive times
for the first, second and third resonances in the $I-V$ curve
for an array with one junction and with a large
capacity miss-match $\kappa_1=30$. The corresponding eigenmodes
(\ref{phin}) are plotted in dashed lines.}
\label{f4a}
\end{figure}

\subsection{Projection on the normal modes}

The normal modes that we have exhibited can be used to project the
solution. This enables to analyze the dynamical behavior in a
simple way.
The eigenvectors
solutions of the problem (\ref{varepsi_i})
are orthonormal for the inner product defined above.
We will assume that
all solutions $\phi$ belong to the vector space generated by the
eigenvectors $\varphi_i$. This is difficult to justify theoretically
but we will come back to this using the numerical results in the
last section.
Then the solution $\psi$ of equation (\ref{delpsi_phiv})
$\phi$ can be written as
\begin{equation}\label{vctr_spc}
\psi(x;t)=\sum_{i=0}^{+\infty}\beta_i(t)\varphi_i(x).
\end{equation}
We assume the uniform convergence of the series (\ref{vctr_spc})
so that we can permute integrals and sums.

We will present the calculations for a single junction for
simplicity. The results can be generalized to arrays with
multiple junctions.
We replace $\psi$ in eq.(\ref{delpsi_phiv}) for one junction,
\begin{eqnarray}
\sum_{i=1}^{+\infty}\left(\beta_i^{\prime \prime}\varphi_i -
\beta_i\varphi_i^{\prime \prime}\right) + \nonumber \\
[-1.5ex] \label{dlt1D_vct} \\ [-1.5ex]
d_1 \delta(x-a_1)\left( \sum_{i=1}^{+\infty}
\left(\kappa \beta_i^{\prime \prime}
\varphi_i + \alpha  \beta_i^{\prime}\varphi_i\right)+
\sin \left(\psi + \Psi_1 \right)-
\frac{\gamma }{d_1 }
\right)&=&0, \nonumber
\end{eqnarray}
We multiply eq.(\ref{dlt1D_vct}) by $\varphi_j(x)$
and integrate it on its domain
\begin{eqnarray}
\sum_{i=1}^{+\infty}\beta_i^{\prime \prime}
\underbrace{\left(\int_0^l \varphi_i \varphi_j dx +
\kappa_1 \varphi_i(a_1)\varphi_j(a_1) \right)}_{=0~{\rm if}~i\neq j,~
=1~{\rm else}}
- \beta_i \int_0^l \varphi_i^{\prime \prime} \varphi_j dx
+d_1 \alpha\beta_i^{\prime}\varphi_i(a_1)\varphi_j(a_1)
 && \nonumber \\
\label{proj1}
+ d_1 \sin \left(\psi(a_1) + \Psi_1 \right) \varphi_j(a_1)
-\gamma \varphi_j(a_1)
=0. &&
\end{eqnarray}
and from eq.(\ref{eigvp}) we know that,
$$\varphi_i^{\prime \prime}=-\left(1+
\kappa_1 \delta(x-a_1)\right)\omega_i^2 \varphi_i.$$
Eq.(\ref{proj1}) becomes
\begin{eqnarray}
\beta_j^{\prime \prime}+ \sum_{i=1}^{+\infty}
\beta_i \omega_i^2 \underbrace{ \int_0^l (1+\kappa_1 \delta(x-a_1))
\varphi_i \varphi_j dx}_{=0~{\rm if}~i\neq j,~ =1~{\rm else}}
+\frac{d_1 \alpha}{l}\beta_i^{\prime}\varphi_i(a_1)\varphi_j(a_1)
&& \nonumber \\
+ \frac{d_1 }{l} \sin \left(\psi(a_1)+ \Psi_1\right)
\varphi_j(a_1)-\gamma \varphi_j(a_1) =0. &&
\end{eqnarray}
We obtain the final equation giving the evolution of $\beta_j$
in terms of $\phi$
\begin{equation} \label{rltn2}
\beta_j^{\prime \prime}+ \omega_j^2 \beta_j+
c_j \left( \alpha \phi_t(a_1)
+\sin \left(\phi(a_1) \right) -\frac{\gamma }{d_1}\right)=0,
\end{equation}
where we have returned to the usual field $\phi$ and
where  the coupling coefficient $c_j$ is
\begin{equation}\label{cj}  c_j=d_1 \varphi_j(a_1).\end{equation}
This coefficient does not depend directly on $\kappa$.
Also notice that all equations are coupled by the same term
\begin{equation}\label{brack}
F \equiv  \alpha \phi_t(a_1) +\sin (\phi(a_1))-\frac{\gamma }{d_1 }
,\end{equation}
where the coefficient $c_j$ regulates the forcing for each mode.
When $n$ junctions are present in the device, the modal
equations can be generalized to
\begin{equation} \label{bjt}
\beta_j^{\prime \prime}+ \omega_j^2 \beta_j+
\sum_{k=1}^n c_j^k \left( \alpha \phi_t(a_k)
+\sin \left(\phi(a_k) \right) -\frac{\gamma }
{\sum_{l=1}^n d_l}\right)=0,
\end{equation}
where the generalized coupling coefficient $c_j^k$ is
\begin{equation}\label{cjk}  c_j^k=d_k \varphi_j(a_k).\end{equation}

\section{Numerical analysis of the IV curves}

To analyze the mechanism leading to a resonance in the IV curves
shown in Fig. \ref{f4} we project the numerical
solution onto the normal modes that we defined in section 4. Following
the definition of the inner product (\ref{scl}) we have
$$\beta_i= \left <\phi | \phi_i\right>
\equiv \int_0^l \phi \phi_i dx + \kappa d_1
\phi(a_1) \phi_i(a_1) .$$
The integral on the right hand side is computed using the trapeze
method. Fig. \ref{f5b} shows a plot of the amplitudes $\beta_i,~i=1-4$
for the second resonance with $\kappa=1$ (middle panel for
Fig. \ref{f4}). Clearly $\beta_2$ is dominant.
In the left panel we did not subtract the high voltage solution
$\phi_v(x)$ (\ref{phiv0}-\ref{phiv12}) so that the other modes
appear as parasites. If the high voltage solution is taken out then
we have a clear dominance of $\beta_2$, all the other modes being
close to 0. This shows that we have to a
good approximation
\begin{equation}\label{phi_res}
\phi(x,t) \equiv \psi(x,t) +\phi_v(x) \approx
\beta_0(t) \varphi_0(x) + \beta_i(t) \varphi_i(x) + \phi_v(x),\end{equation}
where $i=2$ and where we have included the 0 mode that is always present.
We recover the results suggested by the plots of Fig. \ref{f4a}. We have
observed this for all the resonances in the $I-V$ curve.
\begin{figure}
\centerline{\epsfig{file=figs/ka1_p002.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/ka1_p002_hv.eps,width=0.5\linewidth,angle=0}}
\caption{Plot of the amplitudes $\beta_i(t),~i=1-4$ obtained by projecting the
numerical solution for $\kappa=1$ at the top of the second resonance
onto the normal mode $\varphi_i$. The modes are $\beta_1$ continuous line,
(red online), $\beta_2$ dashed line (green online), $\beta_3$ short dash (blue
online) and $\beta_4$ dotted line (pink online). In the left panel the
projection is done directly on the solution $\phi(x,t)$. In the right panel
we have subtracted the high voltage solution $\phi_v(x)$ for the given
current $\gamma=1.36$. The time unit is the period $T_2 = 2 \pi / \omega_2
\approx 3.25$.
}
\label{f5b}
\end{figure}
For example for $\kappa=30$ (bottom panel of Fig. \ref{f4})
we show in Fig. \ref{f5c} the amplitudes
$\beta_i$ from top to bottom for the 1st , 2nd and 3rd
resonance. Again the dominant amplitudes are from top to bottom $\beta_1$,
$\beta_2$ and $\beta_3$.

Therefore the solution at the top of the resonances is
given by (\ref{phi_res}) to a good approximation. To analyze how
we reach this state we have projected the solution on the
normal modes for increasing values of the current $\gamma$ all the
way to the top of the resonance. The calculations were done over
a long time interval (about 200 periods). Projecting the solution
$\phi$ we observed a drift in the amplitudes $\beta_i$ due
to the rapid increase of $\phi$ and the finite precision
of the evaluation of the integrals. To avoid these technical problems
we have projected the time derivative $\phi_t$. The qualitative
conclusions are the same as for $\phi$ except that we will look at
${\beta_i}_t$. Fig. \ref{f5c} shows three amplitudes as a function of time
for $\gamma=1.12,~1.3,~1.42$ and $1.56$ and a voltage $V \approx \omega_3$
near the 3rd resonance for $\kappa=1$. Only three periods
$T_3= 2\pi / \omega_3$ have been represented for clarity, the rest
of the time evolution is the same. In the top left panel
for $\gamma=1.12$, the amplitude of the mode 3 is about 1 with small
components in the modes 2 and 1. When the current is increased the amplitude
of the 3rd mode increases and becomes periodic of period $T_3$.
The other modes tend rapidly to 0.
\begin{figure}
\centerline{\epsfig{file=figs/ka30_p001_hv.eps,width =0.5\linewidth,angle=0}
\epsfig{file=figs/ka30_p002_hv.eps,width =0.5\linewidth,angle=0}}
\centerline{\epsfig{file=figs/ka30_p003_hv.eps,width =0.5\linewidth,angle=0}}
\caption{Plot of the amplitudes $\beta_i(t),~i=1-3$ obtained by projecting the
numerical solution at the top of the second resonance
onto the normal mode $\varphi_i$ for $\kappa=30$. The dominant mode is
resp. $i=1,2,3$ for resp. the 1st resonance (top panel), the second resonance
(middle panel) and the third resonance (bottom panel).
}
\label{f5c}
\end{figure}
\begin{figure}
\centerline{
\epsfig{file=figs/ka1_p3_03l.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/ka1_p3_12l.eps,width=0.5\linewidth,angle=0}}
\centerline{
\epsfig{file=figs/ka1_p3_18l.eps,width=0.5\linewidth,angle=0}
\epsfig{file=figs/ka1_p3_htl.eps,width=0.5\linewidth,angle=0}}
\caption{Plot of the amplitudes of $\phi_t(x,t)$ as a
function of time for four different values of the current, from top left to bottom
right $\gamma = 1.12, 1.3, 1.42, 1.56$,
on the third resonance $V=\omega_3$ for $\kappa=1$. The index of the modes are
$i=1 $ continuous (red online), $i=2$ dashed line (green online),
$i=3$ short dash (blue online) and $i=4$ dotted
line (pink online). The time interval is the period $T_3=2\pi/\omega_3
\approx 2.58$.
}
\label{f5d}
\end{figure}

It is instructive to compute numerically the forcing term $F$
as one progresses up the resonance. Fig. \ref{f5e} shows $F(t)$
for three periods $T_3$ for the four values of $\gamma$ analyzed
in Fig. \ref{f5d}. The amplitude of $F$ decreases for increasing
$\gamma$ and $F$ becomes periodic of period $T_3$. This explains
why we obtain the correct resonant modes using the
spectral problem (\ref{eigvp}). Note that when $\kappa=0$ the
forcing term $F$ of the amplitude equations tends to 0 when one
gets close to the top of the resonances. Then one can solve the
differential equation for $\phi(a_1;t)$
\begin{equation}\label{odephi}
\alpha \phi^{\prime}(a_1;t) +\sin \left(\phi(a_1;t)\right)
-\frac{\gamma }{d_1 }=0,
\end{equation}
and close the system by expanding this solution using the standard
Fourier modes\cite{cl05}. This is not the case when $\kappa \neq 0$ as shown
by these numerical results.
\begin{figure}
\centerline{\epsfig{file=figs/bra.eps,width=0.5\linewidth,angle=0}}
\caption{Plot of the forcing term $F $ from (\ref{brack}) as a
function of time for four different values of the current on the third
resonance $V=\omega_3$ for $\kappa=1$. The values are
$\gamma=1.12$ continuous (red online), $\gamma=1.3$ dashed line (green online),
$\gamma=1.42$ short dash (blue online) and $\gamma=1.56$ dotted
line (pink online). The time interval is the period $T_3=2\pi/\omega_3
\approx 2.58$.
}
\label{f5e}
\end{figure}
To analyze the resonances, we assume as in \cite{kulik} that when in
resonance, the solution  has the spatial structure of the corresponding
eigenmode.
\begin{equation}\label{reson_anstaz}
\psi(x,t) = \beta_0(t) \varphi_0 + \beta_n(t) \varphi_n(x) ,\end{equation}
where the first term on the right corresponds to the
zero mode.
The evolution of $\beta_0,~\beta_n$ is then given by equation (\ref{rltn2})
\begin{eqnarray}\label{betat}
\beta_n^{\prime \prime}+ \omega_n^2 \beta_n+ c_n F =0, \\
\beta_0^{\prime \prime}+ c_0 F =0,
\end{eqnarray}
where the forcing term is
\begin{equation}\label{force}
F = \alpha \beta_n^{\prime} \varphi_n
+ \alpha \beta_0^{\prime} \varphi_0   +
\sin \left( \beta_n \varphi_n  + \beta_0 \varphi_0 \right)
-\frac{\gamma }{d_1 }
\end{equation}
To understand the specific shape of the resonances, the fact that
we cannot obtain in the IV curve the right part of the resonance
curve, one could carry out a bifurcation analysis similar to the
one of \cite{bfmp94}. However this is out of the scope of this
article.

The situation is more complex when there are more junctions in the
device. As an example we consider a two junction device
with $l=\pi, ~a_1=0.3 \pi,~a_2=0.5\pi,~ d_1=d_2=0.1$. The eigenmodes
$\omega_n$ are plotted in Fig. \ref{f5} as a function of $\kappa$ and
one can see them shift from integer values even for small $\kappa$.
\begin{figure}
\centerline{\epsfig{file=figs/frqc_kpp1-5_modif.eps,width=0.5\linewidth,angle=0}}
\caption{Plot of the frequency $\omega$ vs. $\kappa_i$ for a
device with two junctions where
$a_1=0.3 \pi,~a_2=0.5\pi,~ d_1=d_2=0.1$.}
\label{f5}
\end{figure}
In Fig. \ref{f6} we plot the $I-V$ curves obtained for this device
for $\kappa =0,~4$ and 8 with $d_1=0.1$ so that
$\kappa_1 = d_1 \kappa = 0,~0.4$ and 0.8. The resonances observed correspond
to the eigenfrequencies obtained.
For $\kappa=0$ we have explained the height of the resonances using
an approximate theory \cite{cl07} based on the amplitude
of oscillation of $\phi_t(a_j)$ for each junction. This is the
envelope function plotted in dashed line in the top left panel of
Fig. \ref{f6}.  When $\kappa \geq 0$ it is more difficult. We think
that this amplitude of oscillation, which can be obtain by the eigenvectors
does not determine completely the height of the resonances.
\begin{figure}
\centerline{ \epsfig{file=figs/d01lph025k0_bis.ps,height=0.5\linewidth,angle=270}
\epsfig{file=figs/d01lph025k4_bis.ps,height=0.5\linewidth,angle=270} }
\centerline{\epsfig{file=figs/d01lph025k8_bis.ps,height=0.5\linewidth,angle=270} }
\caption{IV curves for the two junction circuit studied in Fig. \ref{f5}.
The parameters are
$l=\pi, d_1=d_2=0.1,~\alpha=0.25,~\kappa=0$. From top left to bottom $\kappa =0,4$
and 8. In the top panel the envelope function (see text)
is plotted as a dashed line (pink online).}
\label{f6}
\end{figure}
When the length of the device is larger more resonances can be accommodated
in the $I-V$ curve. An interesting effect we found is that
for large values of $\kappa$ the system locks to linear combinations
of the eigenfrequencies. Such an example is shown in Fig. \ref{f10}
for a two junction device in a microstrip of length $l=10$.
For $\kappa \ge 5$ there appears in the $I-V$ curve a resonance
for $\omega_1+\omega_6$. For $\kappa=10$ (bottom left panel) we see
in addition
resonances for $2 \omega_1, ~\omega_1+\omega_9$. This is typical
of a linear system. When we observe numerically
the evolution of $\phi_t$ at this resonance we see the two eigenvectors
1 and 9. But we cannot explain this linear behavior: how the system
can sum two non linear solution?

\begin{figure}
\begin{minipage}{0.49\linewidth}
\epsfig{file=figs/a10o3b25o3.ps,height=\linewidth,angle=270}
\end{minipage}
\begin{minipage}{0.49\linewidth}
\epsfig{file=figs/a10o3b25o3beta5.ps,height=\linewidth,angle=270}
\end{minipage}
\begin{minipage}{0.49\linewidth}
\epsfig{file=figs/a10o3b25o3beta10.ps,height=\linewidth,angle=270}
\end{minipage}
\begin{minipage}{0.49\linewidth}
\epsfig{file=figs/a10o3b25o3beta22.ps,height=\linewidth,angle=270}
\end{minipage}
\label{f10}
\caption{\it $I-V$ curves of a two junction device, $a_1=10/3$, $a_2=25/3$,
$l=10$, $d_i=0.1$, $\alpha=0.3$ and different values of
$\kappa$. Top panels, $\kappa = 0$ (left) and $\kappa = 5$ (right).
Bottom panels $\kappa = 10$ (left) and $\kappa = 22 $ (right).}
\end{figure}

\subsection{Conclusion}
To summarize we have analyzed a long wave model describing a parallel
array of Josephson junctions. We defined an appropriate spectral
problem whose spectrum gives the resonances of any array with
junctions of arbitrary (small) sizes and positions. This task
would not have been possible without the complexity reduction
provided by asymptotic analysis.

The adapted spectral problem leads to an inner product so that
it becomes possible to project the dynamics of the system and
describe arrays with more junctions.

It may now be possible to solve the inverse problem of finding the
device yielding a given I-V curve. Another open question is the
study of the amplitude equations (\ref{betat}) to analyze the stability
of the resonances.

{\bf Acknowledgements}

J.G.C. and L. L. thank Faouzi Boussaha and Morvan Salez for helpful discussions
and for their experimental results.
The computations were done at the Centre de Ressources
Informatiques de Haute-Normandie (CRIHAN).

redpajama_set_name: RedPajamaArXiv


Today, Microsoft released its latest revision to the Windows Live SkyDrive file hosting and sharing platform. As a result of the change, the user storage limits have been reduced from 25GB to 7GB. If you have an existing Windows Live ID, then as a loyalty reward, you can get a free upgrade to 25GB, allowing you to keep your existing storage limit.
To do this, simply login to SkyDrive at http://skydrive.live.com, login as normal and select the Upgrade banner near the top of page. Hurry though as this is being reported as a limited time offer.
Last week, when converting our existing ICT internal dogfood trial of App-V to a highly available production capable App-V solution, we came to a decision to utilize the RTSPS (Real Time Streaming Protocol Secure) protocol for streaming our applications.
The specified Application Virtualization Server has shut down the connection. Try again in a few minutes. If the problem persists, report the following error code to your System Administrator.
We initially discovered from an App-V blog article (http://blogs.technet.com/b/appv/archive/2010/03/09/troubleshooting-common-rtsps-issues-with-app-v.aspx) that this issue occurs when the server lacks permissions for the NETWORK SERVICE account to access the certificate store machine keys.
Following the advise of the article for Windows Server 2008 R2 systems, this was quickly resolved by using a Certificate Management based Microsoft Management Console to grant Read permission for the NETWORK SERVICE account to the certificate which is being used to sign the RTSPS protocol in App-V.
The Application Virtualization Client could not update publishing information from the server App-V Server. The server will not allow a connection without valid NTLM credentials. Report the following error code to your System Administrator.
Leaving us puzzled. We were unable to find a solution initially, so we turned to Bing for some assistance, unearthing an interesting but niche blog post.
According to the source of our findings (http://blogs.ethz.ch/jlaville/2011/08/25/app-v-error-00002002/) machines with components from the Windows Live Essentials suite of applications cannot run the RTSPS protocol due to a registry key added to the LSA Security Packages key.
After removing the livessp value from the multi-value string in the registry and restarting the system we were successfully able to refresh the server and also stream the applications.
Today, I was investigating the management and control of the Outlook Social Connector via Group Policy, using the Office 2010 ADM/ADMX files from Microsoft.
Two of the settings of interest for the Outlook Social Connector are the ability to control which social connectors are displayed, and which are automatically loaded without user interaction. Whilst looking online, a Microsoft Forum thread appeared in my results with the ProgID for some of the available connectors, however they were missing a big one – Facebook.
Looking in the HKEY_CLASSES_ROOT registry hive on my machine, where I have the Facebook connector installed, I found it, so here is a list of the currently available Outlook Social Connector ProgIDs which can be used (semi-colon seperated) in the Group Policy Management Console to configure the behaviour.
Earlier in the year, I got my hands on a leaked build of Home Server Vail. I didn't bother to upload or post any of my findings because it was a very early build and a lot of the features didn't work, and in addition because a lot of other people posted the stuff too.
This week Microsoft released the Public Beta of Windows Home Server Codename Vail which is to be the second release of Windows Home Server, this time based on Windows Server 2008 R2 Standard Edition. I went through the installation process and then tried to get a few things configured in Vail so here is what I found.
I'm adding the break early on this post to stop the screenshots spoiling the view, but jump to the full post for all the screenshots and information so far on Vail.
So in case reading my recent blogs you hadn't noticed, I'm posting a lot of my blogs with Polaroid style images which hasn't appeared before.
Well it's all part of the all new Baby-Green. Last weekend I was up until 3:30am migrating the site to a new web host to give us some more functionality. The new host lets me use Silverlight and ASP.NET code which I'm going to be taking advantage of soon starting with the wedding subdomain (http://wedding.baby-green.co.uk).
The problem previously was that TextDrive did not permit FTP access to the server where as the new host does which allows me to directly add content via Windows Live Writer.
On the cards are a recode of the PHP scripts for the wedding site into ASP.NET and using Silverlight Adaptive Streaming and Silverlight Slideshows to show off the media from the wedding.
Over time I'm going to be redesigning the Baby-Green site in this way also which will be version 3 of the website. My plan is to make it not only better looking and give a much better user experience for the users of the site but to make it easier for me to manage and generate content. That way you get a better looking, better working and more current website for viewing.
The forum and the blog will also get updates to the latest code base and the same redesign. The blog is already running the latest code base with the latest updates to the add-on's in use which I did as part of the migration.
– How stupid this person is.
The article is about a girl who got sacked three weeks after starting a job because she wrote on Facebook about how boring the job is. Now fine the job may have been boring and if you want to tell people on Facebook about it to share your life then great: But if your going to do that make sure first nobody in your friends works for said company or set your permissions properly so that lurkers (like your boss for example) can't see that message.


text:
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TechCabal Daily, 860 – The "Cannibalistic" Nature of the Gig Economy is Facing a Major Challenge
Akindare Okunola 11th September 2019
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Access Bank, Axa Mansard, among other financial industry players, have signed up to VerifyMe's proprietary Know-Your-Customer (KYC) solution. VerifyMe offers real-time ID verification that is compliant with CBN tier 3 and international Anti-Money Laundering (AML) standards. Organizations can access up to 200 verifications per minute for faster decision making. Get started now.
Transsion Holdings has reactivated its initial public offering (IPO) efforts and will set a share price or range by September 17. According to Bloomberg, the Africa-focused smartphone company is planning to sell as much as 80 million class A-shares when it lists on the Star Board of the Shanghai Stock Exchange. Transsion is the biggest phone maker in Africa with a market share of 48.7%. Last year, the continent absorbed nearly 80% of the company's 124 million phone sales, contributing a large portion of its $3.2 billion revenue. With the IPO, Transsion wants to expand its manufacturing bases as well as its research and development initiatives.
MTN Group is making executive management changes in some African countries. The company announced that Freddy Tchala and Philip van Dalsen, the CEOs of MTN Zambia and MTN Côte d'Ivoire respectively, are leaving the company at the end of the month. However, while Tchala's replacement has not been announced Dalsen will be replaced by Bert Hofker, the CEO of MTN Rwanda. While Mitwa Kaemba Ng'ambi, the immediate past CEO of Airtel Tigo Ghana, will take up the reign as CEO of MTN Rwanda. Ng'ambi's appointment marks the start of her second stint with MTN. She had formerly worked with the telco in Zambia and Benin Republic before joining rival company Airtel Tigo where she was CEO for Tigo Senegal and later CEO of Airtel Tigo Ghana.
What is the future of Africa's mobility industry? Aanu Adeoye provides an overview of industry trends in recent times ahead of our TC:Mobility townhall which holds on September 27.
Jack Ma has ended his reign at Alibaba Group. The former English teacher has stepped down as Executive Chairman of the Chinese conglomerate after 20 years of leadership. He first announced the decision last year, promising to step aside on his 55th birthday. Under his leadership, Ma transformed Alibaba into one of the world's biggest technology company, with one Bloomberg analyst calling it "one of the most spectacular creations of wealth the world has ever seen." Himself a billionaire, Ma's Alibaba also created 10 other billionaires.
Applications are open for the Tech Meets Renewable Energy Hackathon organised by Clean Technology Hub. The competition is open to innovators who are solving problems in Nigeria power sector. The top three solutions will receive cash prizes of N150,000, N100,000 and N50,000 respectively and three months of business advisory. Follow here to apply.
Endeavor Nigeria is launching our flagship annual scaleup entrepreneurship forum, Catalysing Conversations on Friday, September 13, 2019, in Lagos. Catalysing Conversations 2019 will be chaired by H.E. Prof. Yemi Osinbajo (SAN, GCON), Vice-President of the Federal Republic of Nigeria, and features guest speaker, Efosa Ojomo, co-author of 'The Prosperity Paradox: How Innovation Can Lift Nations Out of Poverty'.
Other speakers include Endeavor Entrepreneurs – Bukky George (HealthPlus), Etop Ikpe (Cars45), GB Agboola (Flutterwave) and Kene Okwuosa (FilmHouse).
Catalysing Conversations is an invite-only event, but we are opening up 10 guest slots to TechCabal Daily readers. If you're interested in attending, please register your interest through this link.
Internationally, a few US states led by California are contemplating or proposing a legislature that could impact how we think about the gig economy. For the uninitiated, the gig economy is freelancing or independent work, and it is characterised by short-term contracts. The rise of the gig economy in Africa and globally has paralleled the growth of billion-dollar companies like Uber, Bolt, Fiverr, Freelancer.com, UpWork, among many others. Each of these services allows people to earn money while doing short term jobs on an independent basis, and these companies say their platforms help people work in a "flexible" way.
However, California lawmakers believe that these companies are growing their revenues off the labour of gig workers, but have provided no support for these workers in terms of social security, insurance or minimum wage. Mbugua Njihia explained on iAfrikan that freelancing started out as supplemental work, but it has now evolved into full-time jobs for many people. "Our economy has stopped working for working people," said California governor Gavin Newsom in an op-ed. The state is close to completing a law that will change the definition of who should be called "gig workers", and could force gig companies to increase welfare and other benefits to these workers. This is the strongest governmental effort to control what many see as the dark side of the gig economy. A 2017 New Yorker article opined that gig companies are promoting a "cannibalistic" work culture that makes it "acceptable to applaud an individual for working himself to death."
On Quartz, Alison Griswold writes that while many gig companies like Uber believe that the degree of independence of workers determines their classification, the proposed law holds that a worker is not a freelancer if they perform work that is the normal business of their clients. Andrew Hill writing on the Financial Times explains that user-based rating systems on these platforms show they truly control workers and can "deactivate" them if they fall below a threshold.
But, the proposed law is restricted to California, and there's little chance that it will get federal support. Regardless, it is an interesting conversation and could become even more important if foreign governments including the European Union (EU) and Africa get interested in it. Already in Canada, gig workers working for Foodara, a food delivery app, are trying to unionise. In Europe, the reappointment of Margrethe Vestager as the EU's competition chief and the expansion of her work to include Europe's digital policy could also prop up moves in this direction. And in Africa, more governments are showing greater interests in the activities of digital platforms in recent times. This could mean conversations about the gig economy may not be so remote from the continent either.
Tomorrow, Wednesday, September 11th at 1 pm, TechCabal is holding a Twitter chat with Ridwan Olalere, Director of Operations, ORide by OPay. Ridwan is one of the speakers at TC Townhall: Mobility, a forum about the future of Africa's transport and logistics tech sector. Got any questions for Ridwan? Please email us via team@techcabal.com or fill this form. Learn more about TC Townhall: Mobility and get tickets by clicking here.
Vodacom Tanzania and Smart Lab have launched the Vodacom Digital Accelerator, an accelerator programme that will support early-stage and growth-stage tech startups in the country. The new tech hub will offer a three-month programme and allow startups a chance to pitch to investors. It will also provide extended mentoring and network support for startups lasting up to six months. Vodacom has dedicated $150,000 to the tech hub and it targets startups in areas such as mobile services, telecom, fintech, media, health, education, and e-commerce.
Understanding the Challenges Confronting Digital Natives Learners and Digital Immigrant Educators
Digital technologies are changing how African businesses operate and they are also creating new venture opportunities across the continent. However, Elvira Bolat and Nasiru Taura explain that there are still challenges that have, so far, limited the benefits of technology for entrepreneurs.
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TechCabal Daily, 859 – DHL's e-commerce platform for Africa is expanding rapidly, posing a threat to existing players
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redpajama_set_name: RedPajamaCommonCrawl


starting your Spanish lessons and if you have previous knowledge of Spanish, you will have to undergo our level test.
After the test, the participant will be allocated to a group based on their written language skills and overall knowledge of the language.
Oral language ability will be evaluated through conversation with their classmates in the classroom.
student in two different groups according to their linguistic skills and abilities.
It is therefore possible that groups might be changed around, if the teacher and Coordinator of Studies think it is appropriate.


Non-optimal diets were behind 1 in 5 deaths, a global report found.
About 11 million people eat themselves into early graves each year — but the riskiest diet habits may come as a surprise.
The new Global Burden of Disease Study, which analyzed dietary consumption in 195 countries between 1990 and 2017, concluded that poor (or "non-optimal") diets are responsible for 1 in 5 deaths — more than any other risk factors, including tobacco smoking. (Tobacco kills 8 million people globally, in comparison.) After all, unhealthy diets are associated with cardiovascular disease, type 2 diabetes and many forms of cancer.
But its definition of a "non-optimal" diet may shock consumers who have been fed countless meal-plan fads that call for cutting carbs, sugar or dairy to lose weight and stay healthy as part of the $66 billion weight loss market. Turns out, not getting enough milk, whole grains, fruits, vegetables, nuts and seeds (as well as consuming too much sodium) was associated with more deaths than eating too much unhealthy food, like red meat, processed meat, trans fats and sugar-sweetened drinks, according to the report. And eating insufficient amounts of whole grains and fruits, and taking in too much sodium, accounted for more than half of diet-related deaths across the globe.
"While sodium, sugar and fat have been the focus of policy debates over the past two decades, our assessment suggests the leading dietary risk factors are high intake of sodium, or low intake of healthy foods, such as whole grains, fruit, nuts and seeds, and vegetables," concluded study author Dr. Christopher Murray from the University of Washington. The study was funded by the Bill & Melinda Gates Foundation.
Uzbekistan saw the highest rate of diet-related deaths — 892 deaths per 100,000 people — among the 195 countries studied in this report, followed by Afghanistan and the Marshall Islands. Israel had the lowest rate of diet-related deaths, with just 89 deaths per 100,000. The U.S. ranked 43rd with 171 deaths per 100,000.
Ten million of 2017's diet-related deaths were from cardiovascular disease; cancer accounted for 913,000 deaths, and Type 2 diabetes was responsible for 339,000 deaths. But the diet gaps differed by country.
Not getting enough whole grains was the greatest risk factor for mortality in the U.S., India, Brazil, Pakistan, Nigeria, Russia, Egypt, Germany, Iran and Turkey. Indeed, a recent study has linked going low-carb with a higher risk of atrial fibrillation (aka AFib), the most common heart rhythm disorder. And a report published in Lancet last year suggested that both low-carb and high-carb diets were linked with a higher risk of death. Yet Americans are keto-crazy, helping build the low-carb diet into a $9.08 billion industry expected to hit $12.35 billion in 2024, according to Mordor Intelligence.
Residents of China, Japan, Indonesia and Thailand are ingesting too much sodium, according to the report, which could stem from many traditional Asian dishes being made with salty sauces, pastes and rice vinegars. Consuming salt raises blood pressure, which can lead to heart disease.
Razo also noted that while Mexico produces fresh fruits and vegetables, like avocados, those health crops tend to be distributed more to countries like the U.S. than they are distributed in local cities. The U.S. gets almost 90% of its avocados (essential for guacamole and avocado toast) from Mexico.
The report highlights the need for food system interventions and dietary policies to help individual countries identify which ingredients their residents are missing in order to have a well-balanced diet, and to then produce and distribute these healthy foods in those areas.
But health experts say the main takeaway for most people should be that adding in healthy foods to their diets is more important that cutting "bad" things out.
"We have to look at diet quality as a whole over time. No one food is that bad and terrible that you have to avoid it forever," said Hogan. "It really is looking at the big picture: Is the majority of your diet plant foods that maybe you're not eating enough of now? If it's not, then work toward that, knowing there is still for other things that you might enjoy as well." Just eat those in moderation.


At the moment, decide up on this Small Business Proprietor Advertising tip and begin applying it to your marketing materials to see a rise in sales, ultimately maximizing your profits and gross numbers. seventy three. Use Twitter successfully to create rapid ROI. 34. Optimize your website and content material for cell or loose out on potential clients. Having a deep understanding of your purchaser persona(s) is essential for content material creation, product development, sales and actually something that relates to customer acquisition and retention.
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Since nearly every one that lives in the area is aware of about that restaurant and drives on that avenue, it was a perfect example of bringing collectively previous and new marketing ideas into an effective strategy. When you may create content material, both online and offline, that is aesthetically pleasing and beautifully designed.


What are the risk factors for spondylolisthesis?
What types of doctors treat spondylolisthesis?
How do doctors diagnose spondylolisthesis? How is the grading determined for spondylolisthesis?
What is the treatment for spondylolisthesis? What type of surgery treats spondylolisthesis?
Is it possible to prevent spondylolisthesis?
Spondylolisthesis is a forward or backward slippage of one vertebra on an adjacent vertebra.
Causes of spondylolisthesis include trauma, degenerative, tumor, and birth defects.
The diagnosis of spondylolisthesis is based on imaging.
Most people with spondylolisthesis can be treated conservatively, without the need for surgery.
Patients who fail to improve with conservative treatment may be a candidate for surgery.
Spondylolisthesis (spon + dee + lo + lis + thee + sis) is a condition of the spine whereby one of the vertebra slips forward or backward in relation to the next vertebra. Forward slippage of an upper vertebra on a lower vertebra is referred to as anterolisthesis, while backward slippage is referred to as retrolisthesis. Spondylolisthesis can lead to a deformity of the spine as well as a narrowing of the spinal canal (central spinal stenosis) or compression of the exiting nerve roots (foraminal stenosis). Spondylolisthesis is most common in the low back (lumbar spine) but can also occur in the mid to upper back (thoracic spine) and neck (cervical spine).
Isthmic spondylolisthesis: In isthmic spondylolisthesis, there is a defect in a portion of the vertebra called the pars interarticularis. If there is a defect without a slip, the condition is called spondylolysis. Isthmic spondylolisthesis can be caused by repetitive trauma and is more common in athletes exposed to hyperextension motions, including gymnasts and football linemen.
Degenerative spondylolisthesis: Degenerative spondylolisthesis occurs due to arthritic changes in the joints of the vertebrae due to cartilage degeneration and is acquired later in life. Degenerative spondylolisthesis is more common in older patients.
Traumatic spondylolisthesis: Traumatic spondylolisthesis is due to direct trauma or injury to the vertebrae. This can be caused by a fracture of the pedicle, lamina, or facet joints that allows the front portion of the vertebra to slip forward with respect to the back portion of the vertebra.
Risk factors for spondylolisthesis include a family history of back problems. People who are born with a defect in the pars interarticularis bone in the spine (a condition called spondylolysis) are at increased risk of isthmic spondylolisthesis. Other risk factors include a history of repetitive trauma or hyperextension of the lower back or lumbar spine. Athletes such as gymnasts, weight lifters, and football linemen who have large forces applied to the spine during extension are at greater risk for developing isthmic spondylolisthesis.
The most common symptom of spondylolisthesis is lower back pain. This is often worse after exercise especially with extension of the lumbar spine. Other symptoms include tightness of the hamstrings, and stiffness and decreased range of motion of the lower back. Pain in the legs, thighs, and buttocks may worsen with standing. Some patients can develop pain, numbness, tingling, or weakness in the legs due to nerve compression. Severe compression of the nerves can cause loss of control of bowel or bladder function, or cauda equina syndrome.
Radiologists diagnose spondylolisthesis using X-rays, CAT scans, or MRI scans. Orthopedic surgeons, neurosurgeons specializing in the spine, physical medicine and rehabilitation doctors, neurologists, internists, primary-care providers, and rheumatologists commonly treat spondylolisthesis. Chiropractors and physical therapists also frequently treat patients with spondylolisthesis.
Usually, it is not possible to see visible signs of spondylolisthesis by examining a patient. Patients typically have complaints of pain in the back with intermittent pain to the legs. Spondylolisthesis can often cause muscle spasms, or tightness in the hamstrings.
The diagnosis of spondylolisthesis is easily made using plain radiographs. A lateral X-ray (from the side) will show if one of the vertebra has slipped forward compared to the adjacent vertebrae. Spondylolisthesis is graded according the percentage of slip of the vertebra compared to the neighboring vertebra.
Grade I is a slip of up to 25%.
Grade II is between 26%-50%.
Grade III is between 51%-75%.
Grade IV is between 76%-100%.
Grade V, or spondyloptosis, occurs when the vertebra has completely fallen off the next vertebra.
If the patient has complaints of pain, numbness, tingling, or weakness in the legs, additional studies may be ordered. These symptoms could be caused by stenosis, or narrowing of the space for the nerve roots to the legs. A CT scan or MRI scan can help identify compression of the nerves associated with spondylolisthesis. Occasionally, a PET scan can help determine if the bone at the site of the defect is active. This can play a role in treatment options for spondylolisthesis as described below.
Patients with pain, numbness, and tingling in the legs may benefit from an epidural steroid (cortisone) injection.
Patients with isthmic spondylolisthesis may benefit from a hyperextension brace. This extends the lumbar spine, bringing the two portions of the bone at the defect closer together, and may allow for healing to occur.
For those whose symptoms fail to improve with conservative treatment, surgery may be an option. The type of surgery is based on the type of spondylolisthesis. Patients with isthmic spondylolisthesis may benefit from a repair of the defective portion of the vertebra, or a pars repair. If an MRI scan or PET scan shows that the bone is active at the site of the defect, it is more likely to heal with a pars repair. This involves removing any scar tissue from the defect and placing some bone graft in the area followed by placement of screws across the defect.
If there are symptoms in the legs, the surgery may include a decompression to create more room for the exiting nerve roots. This is often combined with a fusion that may be performed either with or without screws to hold the bone together. In some cases, the vertebrae are moved back to the normal position prior to performing the fusion, and in others the vertebrae are fused where they are after the slip. There is some increased risk of injury to the nerve with moving the vertebra back to the normal position. Outcomes and recovery after surgery are improved with physical therapy rehabilitation.
The prognosis for patients with spondylolisthesis is good. Most patients respond well to a conservative treatment plan. For those with continued severe symptoms, surgery can help alleviate the leg symptoms by creating more space for the nerve roots. The back pain can be helped through a lumbar fusion.
Spondylolisthesis cannot be completely prevented. Certain activities, such as gymnastics, weight-lifting, and football, are known to increase the stress on the vertebrae and increase the risk of developing spondylolisthesis.
Wheeler, Stephanie G., Joyce E. Wipf, Thomas O. Staiger, and Richard A. Deyo. "Approach to the diagnosis and evaluation of low back pain in adults." UpToDate.com. Nov. 20, 2014. <http://www.uptodate.com/contents/approach-to-the-diagnosis-and-evaluation-of-low-back-pain-in-adults>.


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One of the great innovators in modern training, Charles R. Poliquin has helped numerous pro athletes and bodybuilders, as well as Olympic medalists in over a dozen sports, reach their full potential. Along the way, he also helped popularize the German volume training (GVT) protocol, thick-bar training, and many other tried-and-true training, nutrition, and supplementation techniques.
An accomplished author, Poliquin has written for several publications and websites, including Muscle Media 2000, Bodybuilding.com, Iron Man, Huffington Post, Flex, Muscular Development, T-Nation, Muscle and Fitness, and Sports Review, and has produced many manuals for his international training certifications and several books. His works have been translated into 24 different languages.
Adam Nelson, Olympic champion in shot put and most-medalled man in the history of shot put
Dwight Phillips, Olympic gold medalist in long jump
Al MacInnis, Olympic gold medalist in ice hockey
Cathy Millen, World champion in powerlifting IPF
You can read more of Charles R. Poliquin's articles on his website at www.strengthsensei.com.
8 Ways To Find The Will To Train
Sometimes, it's not as simple as sucking it up. Learn the motivation secrets one of the world's great coaches uses to get athletes to push through the doldrums!
Published on: Feb 11, 2016
The Perfect Rep Scheme For Strength And Size
Power training isn't just for competitive athletes: It can do wonders for your physique as well! Cast a bigger shadow with this world-class program from renowned coach Charles Poliquin.
Hamstring Paradigm!
Understanding the functions of each of the hamstring muscles is the first step towards devising an effective hamstring workout. Learn why!
An Interview With Dr. Sherry Rogers M.D.
Recently I had the pleasure and honor to interview cutting edge health expert Dr. Sherry Rogers M.D. Her latest book, Detoxify or Die is a must for any health conscious individual's library. Here are just 3 of the facts revealed in her book.
Published on: Dec 8, 2003
Frequency Of Training!
Frequency of training is measured as the number of training sessions for a given muscle group or lift per unit of time. Learn what the three main schools of frequency are! BONUS: Charles' German Volume training program!
German Volume Training
It's brutally hard, but I've found it to be an effective way to pack on muscle fast! In strength-coaching circles, this method is often called the Ten Sets Method.
German Volume Training!


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Obłęd moralny (głuptactwo moralne, zespół anetyczny, ang. moral insanity; łac. mania sine delirio, insania ex morum defectione; fr. folie raisonnante, folie lucide raisonnante, folie morale, monomanie affective; niem. moralisches Irresein) – historyczny termin wprowadzony przez angielskiego lekarza J. C. Pricharda w 1835 w dziele A Treatise on Insanity and Other Disorders Affecting the Mind na określenie szczególnej postaci rozstroju psychicznego. Prichard opisał obłęd moralny jako "szaleństwo, charakteryzujące się chorobliwym wypaczeniem naturalnych odczuć, uczuć, skłonności, temperamentu, postawy moralnej i naturalnych odruchów, bez zauważalnych zaburzeń wiedzy czy zdolności rozumowania, a zwłaszcza bez obłąkańczych złudzeń czy omamów".

Koncept obłędu moralnego wiele zawdzięcza pracom Philippe Pinela, cytowanego przez Pricharda, i jego opisowi chorób umysłowych częściowych i afektywnych. Manie sans délire, później znana jako folie raissonante uległa z czasem przekształceniu w postać częściowego obłędu, co oznaczało, że dotknięty nią chory był uważany za obłąkanego jedynie w części swoich funkcji umysłowych, jego osobowość mogła ulec zmianie, natomiast funkcje intelektualne pozostały nienaruszone.

Współczesne niezrozumienie tego pojęcia wiąże się z różnicą znaczenia pojęcia moralności w XIX wieku i obecnie. Wówczas przez moralność rozumiano raczej sferę psychologii niż etyki. Dlatego obłęd moralny odnosił się do zmian afektu, dla odróżnienia od zmian intelektu.

Powstanie tego pojęcia wiązane jest ściśle z niewystarczającą definicją obłędu przedstawioną przez Johna Locke′a, w której do stwierdzenia obłąkania wymagano obecności objawów urojeniowych.

Pogląd, jakoby koncept obłędu moralnego wyprzedzał koncept psychopatii (osobowości dyssocjalnej, antyspołecznej), nie spotkał się z uznaniem historyków psychiatrii. F.A. Whitlock uważa, że "nie ma najmniejszego podobieństwa między tymi przykładami [Pinela i Pricharda] a tym, co dziś uznalibyśmy za osobowość psychopatyczną". Obłęd moralny Pricharda był określeniem zbiorczym na wiele zaburzeń zachowania, których jedyną cechą wspólną był brak urojeń. Tym samym, mimo nazwy, nie jest to odpowiednik współczesnej kategorii diagnostycznej osobowości dyssocjalnej (antyspołecznej).

Przypisy

Historia medycyny
Zaburzenia psychiczne

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From city streets to TV screens: alumna Miz Cracker's drag journey
Carter Ross
Inglemoor graduate Miz Cracker shares her journey from performing drag on the streets to starring in RuPaul's Drag Race Season 10.
Kath Shelden, Copy Editor
Former Viking Miz Cracker graduated from performing on the Little Theater Stage in 2002 and appeared 16 years later on the mainstage of RuPaul's Drag Race in 2018. Drag Race is a TV series where American drag queens compete against one another in drag performances. The show is one of the most popular shows with queer representation to air in the United States to date, with 738,000 viewers for the season 14 premiere.
Miz Cracker started performing drag on the streets of New York City during marriage equality protests in 2011 and eventually made her way to the top five in RuPaul's Drag Race Season Ten. She later reached the top three in RuPaul's Drag Race All Stars Season Five. She said that she applied for Drag Race because she was urged to by her drag mom, Bob the Drag Queen. Drag moms act as mentors to their drag daughters and teach them about drag culture. Miz Cracker is now touring the US and Canada with her one woman show: "She's a Woman!"
According to the MasterClass, "drag is a gender-bending art form in which a person dresses in clothing and makeup meant to exaggerate a specific gender identity, usually of the opposite sex. While drag's main purpose has been for performance and entertainment, it is also used as self-expression and a celebration of LGBTQ+ pride. A typical drag show will include lip-syncing or dance, and performers often have elaborate clothing, hair and makeup."
Q: Where did the name Miz Cracker come from?
A: It comes from my original drag name: Brianna Cracker, which is a play on my favorite snack: brie on a cracker.
Q: What is your favorite type of cracker?
A: Water crackers. I love to have them with brie. They taste like luxury treats.
Q: What inspired you to start doing drag?
A: I was nagged to try it by Bob the Drag Queen. And when I finally tried it, I loved it.
Q: Why do you do drag?
A: It makes me feel that I can make other people happy. Making people smile makes me smile.
Q: Out of all of the drag looks you've done, what has been your favorite?
A: My favorite one was the look that I wore for the finale of All Stars Five where I dressed up like a Russian doll. It made me really happy.
Q: How has performing drag changed during the pandemic?
A: Performing has not changed during the pandemic. Performing is still alive. But during the pandemic, I was forced to do Zoom presentations, and I'm glad that that is over and that we're back to actual performance. Real performance is on site, everything else is a fabrication.
Q: Where in your queer journey were you in high school?
A: I was closeted. But I was also in drama and French, [which] gave me a place where I could be myself.
Q: What advice would you give to young queer kids at Inglemoor?
A: Take your time finding out who you are. There's no rush.
Q: Do you have any final advice?
A: Do what makes you happy. That's the only way that you're going to make other people happy. Drag makes me happy. And because it makes me happy, it allows me to make other people happy, too.
Carter Ross, Design and Cheeky Editor
Senior Robyn Ross is returning as both the design and Cheeky Editor for Nordic News for the 2021-2022 school year. Her goal is to do as good a job...


A GREAT ESCAPE - ENJOY YOUR VIEW - this epic home stands out from the crowd and is ultra efficient (a smart home) - the dramatic architecture with concrete steel construction creates a simplistic, unique and modern living space. Features include stunning grand foyer, high ceilings, , alluring bathrooms, walls of windows, in-floor heating throughout the home and a fabulous trend setting professional kitchen. Lower level is legal private fully equipped and well appointed guest suite. Marina side lifestyle and entertainment paradise with huge decks overlooking the harbor. This is as good as it gets.


**** Original Silk Painting Size: 30"X 30"X 1.5"
My ORIGINAL paintings are done on SILK.
So I list my paintings under "ACRYLIC Category".
Though I am using a special acrylic for fiber to enrich paintings texture.
When drawing a women gentle and mysterious image I experience joy, peace, and harmony. Harmony to me is a poetry of lines and colors that has a life of its own.
and the full sheen of the silk is restored. Now you can wash it, the color won' t come off.


Every week seems to bring the Internet's apocalypse a step closer. Denial of server attacks crush popular commercial sites. Pro-Bin Laden hackers try to assault American and Israeli web servers. Most recently, it was a computer virus.
I don't think any recipients feel all warm and fuzzy. The Goner virus – actually called a "worm" – searches for and deletes anti-virus and firewall protections. Users of Microsoft (MS) Outlook and the Internet chat application ICQ found that Goner then sends itself to anyone in theiraddress book, rapidly spreading trouble worldwide in a matter of minutes.
The Goner virus, like most, had to be activated by the recipient. The attached file, "goner.src," had to be clicked on to launch the virus. People who received the previous week's email baddie, "BadTrans," had no such luck, as it was capable of launching itself without the user moving a muscle. Thankfully, viruses like that are rare.
Interestingly, if you don't use MS programs, you are usuallyreasonably safe. Why? Not to slam tech-giant Microsoft–for better orworse, I'm a regular user–but most e-mail worms and viruses single out Bill Gates' applications. Their sheer size and complexity inevitably creates dozens of security holes. Internet mischief-makers, who tend to demonize Gates anyway, are only too happy to take advantage of his company's mistakes.
Let's face reality; no system can ever be truly secure. No matter how hard techno-geeks work to immunize our networks from harm, there will always be others who can find a way to break through. That does not mean that we should just give up and not try to protect our electronic selves. There is no sense making it easier for any malingering hacker to crash your computer. The evolution of technology and know-how in computer security is an ongoing race.
We certainly ought to do a better job of keeping our anti-virus and firewall protections running and up to date. In addition to this kind of expense, we also need a more vigilant approach to our own responsibilities.
While it is advisable to bulk up your anti-virus and firewall protections, the simplest and most effective safeguard available cannot be purchased, downloaded or installed. It will definitely save you from a lot of on-line headaches.
It is called common sense. Try it the next time you feel the urge to point and click.
—Howard Fienberg is senior analyst with the Statistical Assessment Service (STATS), a nonprofit nonpartisan research organization in Washington, D.C.


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If you're like many people who have recently purchased your first home, you're excited about beginning your journey with home ownership. Perhaps you're imagining raising your family in the home, enjoying backyard barbecues with friends, relatives, and neighbors, and coming home at the end of a long day at work to a warm, comfortable home of your own. What you may not be thinking about, however, is the eventual resale value of the property at some point in the future.
Here's what you should begin doing now in order to help ensure that your home has optimal market appeal when the time comes to list it for sale and more on to your next adventure.
Inspecting your roof on a regular basis helps ensure that minor problems don't evolve into major issues. You don't necessarily have to climb up on your roof to perform an inspection -- in fact, you should probably avoid this approach if you have a pitched roof. If you choose to access your roof via a ladder in order to inspect it, always have another person on the ground in the event of a mishap.
Unless your roof is completely flat, however, you should be able to thoroughly inspect it using a good pair of binoculars. Making roof inspections part of your spring and fall maintenance strategy will help keep your roof in great shape for years to come.
Don't be afraid to call on the experts if you feel you aren't up to a particular job -- perhaps one of the the most important aspects of proper roof maintenance is knowing when to contact a qualified roof repair and maintenance professional from companies like K&R Roofing and Gutters instead of trying to tackle certain tasks yourself.
Although regular inspections provides the opportunity to catch and correct torn shingles, moss and fungal infestations, and other small issues, regular maintenance helps keep these types of problems from occurring in the first place. For instance, using a roof rake to remove vegetative debris from the surface of your roof decreases the chances that moss, lichen, and algae populations will gain a foothold.
It is also important to keep tree limbs and branches pruned back to prevent roof overhang. Not only do they create a shaded, damp environment conducive to the formation of fungal and moss colonies, they pose a potential fire hazard.
Depending on what material your roof is made out of, it may need to be sealed on a regular basis. It is also essential that the gutters and downspouts be kept clear of debris and are inspected often for leaks and other malfunctions.
Trees offer shade to household residents and visitors on warm summer days, shelter homes from wind, and, in some cases, provide fresh nuts and fruit for the family table and to help sustain local birds and other wildlife. Mature trees can also increase the value of a home as much as 19 percent. If your new home doesn't have an existing trees, be sure to plant some after doing some research on which varieties are most likely to add value to your property as well as which ones are likely to last for a number of years.
Keep in mind that you can always replant lawns and flower beds prior to putting your home on the market to increase its curb appeal, but you won't be able to conjure up mature trees in a hurry, so begin your tree care program as soon as possible after buying a new home.
Although it may be decades before you're ready to put your home up for sale, you'll be glad you protected your investment by practicing smart maintenance strategies.


PURPOSE-BUILT EXCAVATORS FOR THE WASTE, RECYCLING AND DEMOLITION ENVIRONMENTS.
Hyundai have developed their hydraulic elevating cab to provide the comfort of operating from a raised position ensuring an improved vision and safety. Larger working range and exceptional stability offer high productivity in a challenging environment.


As a clinical specialist you will get to utilize your medical skills to truly make a difference in the lives of your patients and the community. You will get to care for an underserved population while embracing your passion for helping and healing others. El Rio is home for professionals committed to excellence in their clinical practice. We want people who are focused on community service and making a difference in the lives of their patients. We are a family, with providers and staff who are committed to the mission and vision of El Rio: excellent, accessible and affordable Health care for all who need it. We invite you to become part of the El Rio family as a clinical specialist in one of the following disciplines.
Join a prominent Internal Medicine service that offers the rare opportunity to practice across the entire continuum: from primary care to continuity inpatient care, while preserving nights and most weekends free. Each physician in our 14 plus member group has their own primary patient panel for whom they provide clinic-based care. Approximately every 5-6 weeks, they rotate onto the inpatient service at Carondelet St. Mary's hospital for one week where they exclusively provide general inpatient care as hospitalist to the El Rio group of adult patients. This unique opportunity also affords them the benefits of collegiality with Tucson's specialists and positions of medical leadership within the community.
Join a vibrant Family Medicine group that is committed to excellence in patient care and service, delivered collaboratively in a team-based model. With sites throughout the Tucson area and on the Pascua Yaqui reservation, El Rio's Family Medicine providers serve newborns through seniors by providing acute and chronic disease management and preventive care. Family Physicians may opt to participate in the adult inpatient rotation or to work solely in the outpatient setting without call. We provide well-women and gynecology care but not maternity services.
Join 20 plus board certified/board eligible pediatricians who are dedicated to providing high-quality care for a patient-centered medical home. Care is provided to children from birth through childhood and adolescence. Our services include: sick child care, well-child visits, chronic disease management and in-hospital care.
Our Special Programs allow for demonstrated improved care within our NCQA Patient Centered Medical Home Level III recognized facilities. These programs include our Asthma Program, Integrated Behavioral Health program, Clinical Pharmacy Program, Immunization Program, Reach Out and Read, Hepatitis C program, HIV care program, and our Hospital discharge coordinated care program.
* J1 opportunities are available for physicians coming from outside the U.S. Due to visa restrictions, a J-1 sponsored physician must work 40 hours (5 clinic days) per week.
** Ask for more details.