text:
\section{Introduction}

 The common practice
of  setting  experimental conditions resides on trial-error method
performed by experimentalists while the subsequent data processing
is not always taken into account.
It is not a rare case that large amount of data is routinely
generated without a clear idea about further  data processing.
Here, we suggest to analyze simultaneously both the data (i.e., the
processes hidden in data) and the experimental protocol, aiming to
establish the link between experimental conditions and the accuracy
of our results.
 The whole idea is presented in a simplified case study of Fluorescence Recovery
 After Photobleaching (FRAP) data processing. It serves as a paradigmatic
 example of the inverse problem of identifying the diffusion parameter
 from spatio-temporal concentration measurements.

 FRAP is a classical
method used to study the mobility of fluorescent mo\-le\-cules
in membranes of the living cells \cite{Florian10}.
The FRAP   technique is based on measuring the fluorescence
intensity, which is proportional to the non-bleached particles
concentration  in response to a high-intensity laser pulse.
We suppose  the laser pulse (the  \emph{bleach}) causes an
irreversible loss in fluorescence  of a certain amount of particles
originally in the bleached area.
The monitored region or the region of interest (ROI) where the data
are measured is usually an Euclidean 2D domain containing the
bleached area.
 After the {bleach},  the  change in
fluorescence intensity  in a monitored region is observed due to the
transport of  fluorescent compounds from the area outside the bleach
as well as bleached particles from the bleached region to originally
non-bleached regions. In general,
 we observe both recovery   and loss
in fluorescence  in different regions corresponding to FRAP or FLIP
(Fluorescence Loss in Photobleaching), respectively
\cite{Kana14,IWBBIO15}.
The natural question to be asked is: How does the bleach shape (and
topology) influence the accuracy of resulting parameter
estimates\footnote{The problem is partially solved in our paper
\cite{AAA15} where we use a relation similar to \eqref{conf}
assuring smaller confidence intervals of the parameter estimates for
the higher values of a sensitivity measure.}.
Therefore, the main focus of this study concerns the searching for
the optimal bleach shape (and its topology), or, from the more mathematical viewpoint,
to optimize  binary-valued initial conditions in a diffusion-parameter identification problem
with respect to sensitivity.

 The rest of this article is organized as follows: In Section 2 we provide the preliminaries for the
 further optimization problem
 formulation  as well as
  the background information concerning the FRAP method.
 In Section 3 we rigorously formulate the optimization problem, evaluate the key term in the objective function and announce the Proposition~3.2, which allows to solve the problem efficiently. Section 4   provides two numerical results: (i) set of solutions (depending on the characteristic time of diffusion) for an optimal
   setting of the  bleached region shape and topology when the bleach depth is the only constraint,
   and (ii) corresponding results as in (i) when the total bleach energy is restricted as well.
  %
The  novelty and  benefits of our approach as well as outlooks for
further research are resumed in the final Section 5.

\section{Preliminaries}
\subsection{Inverse problem of model parameter identification from FRAP data}

  Based on the spatio-temporal FRAP data,  the so-called \emph{effective diffusion coefficient}
 was  estimated using  a {closed form model}
\cite{Axel76,Moul97,Ellenberg} in past decades.
Nowadays, numerical simulations are preferred, cf.
\cite{IvoF,MCM2013,AAA15,Mai11,NatureSR15}, mainly because there is
no need for some unrealistic assumptions.

 Further, we consider a normal Fickian diffusion problem with a
{\em constant} diffusion coefficient $D>0$. In the setup of this
paper, we assume a sufficiently large domain such that we can treat
diffusion in the free space $\mathbb{R}^2.$ In FRAP, the simplest governing
equation for the spatio-temporal distribution of fluorescent
particle concentration $u(x,t)$ is a diffusion equation without
reaction term as follows:
\begin{align}
\pa{t} u(x,t) &= D \Delta u(x,t) & & x \in \mathbb{R}^2, t \in [0,T], \label{one} \\
u(x,0) &= u_0(x)& &  x \in \mathbb{R}^2.  \label{two}
\end{align}

The main issue in FRAP and related identification problems is to
find the value of the diffusion coefficient $D$ from spatio-temporal
measurement of the concentration $u(x,t)$. As another simplification, we assume a spatially radially
symmetric observation  domain, that is, we consider the data
\begin{equation}\label{data}
\begin{split} \text{data}&:= u(x,t) \qquad x \in \Omega  \times [0,T], \\
\Omega &=  \{ x \in \mathbb{R}^2\,|\, \|x\| \leq R \}.
\end{split}
\end{equation}
Hence, the data are observed on a cylinder with radius $R$ and
height $T$. The parameters $R,T>0$ are fixed for the further
analysis and not subject of optimization. Although, in practice, it
is often more convenient to consider an square spatial domain (i.e.,
an image), our cylindrical domain is a reasonable general model to
indicate the most important conclusions in our further analysis.

The data \eqref{data} are insofar idealized as
in experiments, they are given only at discrete points $u(x_i,t_i)$, $i, 1,\ldots N_{data}$.
However, for simplicity and as it was done in \cite{AAA15}, we assume a sufficiently
dense set of data points $(x_i,t_i)$ such that the model \eqref{data} is valid with
reasonable accuracy  and
such that certain sums over the data points can be approximately well by
the corresponding  integrals over  $\Omega  \times [0,T];$ see \cite{AAA15}.

The equations~\eqref{one}--\eqref{two} and the data \eqref{data} are the basis for all
the subsequent analysis, hence, although we mentioned FRAP as one important application,
our results are of course applicable to other diffusion parameter identification problem
which are governed by similar equations and data assimilation processes.

Based on the parameters $R,T$, it is convenient to introduce the following scaling of
the space and time coordinates:
\begin{equation} \label{scaling}
\begin{split}
&z:= \frac{x}{R}, \quad  \tau := \frac{t}{T}, \\
&v(z,\tau) := u(z R ,\tau t) \Leftrightarrow  u(x,t) =
v(\tfrac{x}{R},\tfrac{t}{T}) \quad v_0(z) :=u_0(z R).
\end{split}
\end{equation}
For later use we define a scaled version of the inverse
of the diffusion coefficient\footnote{For the characteristic
diffusion time  $ t_c = \frac{R^2}{4 D} $, we have $ \beta =
\frac{t_c}{T} $.}
\begin{equation}\label{beta} \beta := \frac{R^2}{4 T D} \, \cdot \end{equation}
The scaled concentration $v$ satisfies the equation
\begin{align}
 \pa{\tau} v(z,\tau) &=  \tfrac{T D}{R^2}   \Delta v(z,\tau)
 =\tfrac{1}{4 \beta}  \Delta v(z,\tau)
 & & z \in \mathbb{R}^2, \tau \in [0,1] ,\label{onesc} \\
 v(z,0) &= v_0(z)= u_0(z R) & &  z \in  \mathbb{R}^2.  \label{twosc}
\end{align}
In our case of constant coefficients and free-space diffusion in two dimensions, the solution to this problem can be
expressed by means of the Green's function $G(x,t;y)$ for the heat equation.
\begin{equation}\label{trans} u(x,t) =
 \int_{\mathbb{R}^2} G(x-y,t D) u_0(y) dy,
 \qquad v(z,t) =  \int_{\mathbb{R}^2} G(z-y,\tfrac{\tau}{4 \beta} ) v_0(y) dy
\end{equation}
with   the well-known two-dimensional heat kernel
\begin{align*}
      G(z,t) &:= \tfrac{1}{ 4 \pi t} e^{-\frac{\|z\|^2}{4t}}\, .
\end{align*}

\subsection{Sensitivity of the output observation with respect to model parameter}
Given the data as above, the diffusion coefficient $D$ can be
inferred by fitting the solution given in \eqref{trans} to the data
on $\Omega \times [0,T]$. Because of unavoidable  noise in the data,
one obtains an estimated value $\overline{D}$ which  reasonably well
approximates the true $D$.
It can be shown, see  \cite{AAA15,Bates} and references within
there, that for our case of single scalar parameter estimation and assuming white noise as
data error, the
expected relative error in $D$ depends on the data noise and a
factor, which we call relative global sensitivity $S_{GRS},$ as follows:
\begin{equation}\label{conf}
 \mathbb{E} \left(\left|\frac{\overline{D}-D}{D}\right|^2 \right)
 \sim \frac{1}{S_{GRS}} \frac{\sigma ^2}{{u_{{\rm ref}}}^2} ,
\end{equation}
where $\sigma^2$ denotes the noise variance (and
$\frac{\sigma}{u_{{\rm ref}}}$ is related to the coefficient of variation or
to the inverse of the signal to noise ratio), and
$u_{{\rm ref}}$ is some (chosen) reference value of the observed output.

The global relative squared sensitivity is given by
\begin{equation}\label{sensi}
 S_{GRS} := \frac{D^2}{u_{{\rm ref}}^2} \sum_{i=1}^{N_{data}} (\pa{D} u(x_i,t_i))^2,
\end{equation}
where
 $\tfrac{\partial u}{\partial D}(x_i,t_i)$ is the usual sensitivity of the output
 observation data point $(x_i,t_i)$ with respect to the parameter $D$ and
 $N_{data}$ is the number of data points in space-time domain.
It is obvious from this estimate that if the noise level is fixed,
the estimation can be only improved by switching to an experimental
design with a higher sensitivity. Thus, the thrive for good
estimators $\overline{D}$ leads to the problem of experimental
design optimization in order to maximize the sensitivity measure
$S_{GRS}$. The solution of this optimization problem, which is the
central aspect of this article, is rigorously formulated and
partially solved in the next Section 3.

\section{Optimization of the bleach design}
\subsection{Problem formulation}
The sensitivity measure \eqref{sensi} involves several design
parameters. Note that $R$ and $T$ are involved implicitly because  the number of data
points  $N_{data} \approx \frac{\pi R^2 T}{\Delta x \Delta t}$, where
$\Delta x$ is defined by the pixel size and $\Delta t$ corresponds
to the time interval between two consecutive measurements.
However, if all the above parameters $R,T, \Delta x, \Delta t$ are fixed,
there is only one way to maximize the sensitivity measure $S_{GRS}$:
 considering the {\em initial bleach} $u_0$ in \eqref{two} as the
experimental design parameter. By optimizing the bleach design, we
mean to select the initial conditions in such a way that $S_{GRS}$
is maximized and hence the expected error in $D$ minimized. In order
to do so, we have to chose the class of designs out of which we take
the initial conditions. In FRAP, it is the usual  case that the
initial bleach $u_0$ is a binary-valued function with fixed given
values $u_0(x) \in \{u_{00},0\},$ $u_{00}>0$. Without loss of
generality, we assume $u_0(x)$ being  a
$\{1,0\}$-function, which also cancels out the term $u_{{\rm ref}}$ in $S_{GRS}$\footnote{When we choose
$u_{{\rm ref}} = u_{00}$, then the
normalization of the signal, i.e. $0 \leq u(x) \leq 1$ is actually
done due to the division by the maximal value which signal $u$ can
reach, cf. \eqref{sensi}.}:
\[ u_0(x) = \begin{cases} 1  & x \in B \\ 0, & \text{ else, }\end{cases}\]
with some bounded open set $B$, the {\em bleach shape}.
Moreover, if the sum in
the term $S_{GRS}$ is furthermore  approximated by an integral
\[
 \sum_{i=1}^{N_{data}} (\pa{D} u(x_i,t_i))^2  \sim \frac{N_{data}}{\pi R^2 T}
 \int_{\|x\|\leq R} \int_0^T  |\pa{D} u(x,t)|^2 dx dt,
 \]
 then we have a final sensitivity term $S_{int}$ to be maximized:
\begin{equation}\label{sensii}
S_{int} := \int_{\|x\|\leq R} \int_0^T |\tfrac{\partial u}{\partial D}(x,t)|^2 dx d t.
\end{equation}

Depending on the different restrictions imposed on the initial bleach,
we study the following two setups:

\begin{itemize}
\item Problem 1: optimization with fixed bleach depth
\[  S_{int} \to \max_{B \subset \mathbb{R}^2}  \qquad \text{with}
\quad u_0(x) = \begin{cases} 1  & x \in B \\ 0 & \text{ else }\end{cases}, \]
over the set of bounded open sets $B \subset \mathbb{R}^2.$
\item Problem 2: optimization with fixed bleach depth and fixed energy
\[   S_{int} \to \max_{B \subset \mathbb{R}^2} \qquad \text{ with}
\quad  u_0(x) = \begin{cases} 1  & x \in B \\ 0 & \text{ else }\end{cases}
 \ \text{ and }  \ \int_{\mathbb{R}^2} u_0(x) dx = c_1,
\]
$c_1>0$ given,  over the set of bounded open sets $B \subset \mathbb{R}^2.$
\end{itemize}
Note that these problems are  shape (and topology) optimization
problems, i.e., the unknown bleach shape $B$ is the independent
variable.

However, the problems can be simplified by restricting $B$ to a radially symmetric shapes; the
reason for that is the following lemma.
\begin{lemma}\label{hh}
If solutions to the Problems 1 and 2 exists, then there also exists
 radially symmetric solutions. In particular, if the solutions are unique, they must be radially symmetric.
\end{lemma}
\begin{proof}
It is easy to see that the sensitivity measure $S_{int}$ is a convex
functional of the initial conditions.
From convex analysis, we conclude that
the optimization problems can be relaxed and expressed as problems with box constraints.
In either case, the set of initial conditions can be  relaxed to
\begin{equation}\label{eq:rel}  u_0 \in \{ w(x) \,| \, 0 \leq w(x) \leq 1
\},
\end{equation}
without altering the solution. Indeed, this holds because a convex functional attains its maximum at the
boundary, hence, by considering the relaxed constraints \eqref{eq:rel}, a solution will be at the boundary of
the admissible domain in \eqref{eq:rel} and hence a $\{0,1\}$-function. Thus, solutions of the relaxed
problem are also solutions of the original problem (and vice versa).

Due to our choice of an  observation domain, the
problem is invariant with respect to coordinate rotations, i.e., the
sensitivity does not change if the initial bleach is rotated. If
$u_0(x)$ is a solution to the optimization problem, then so is
$u_{0}(Q_\theta x)$ for any rotation $Q_\theta$  around the origin
with angle $\theta$. It is easy to see that taking the
angle-averaged initial condition $\frac{1}{2 \pi} \int_0^{2 \pi}
u_0(Q_\theta z) d \theta$ yields  a solution to the relaxed
optimization problem which is radially symmetric. Because it must
also be  a solution to the original problem, the set $B$ must be
radially symmetric. $\blacksquare$
\end{proof}

Note that Lemma~\ref{hh} is not valid if the observation domain is not radially symmetric.
When speaking to experimentalists, the result of Lemma~\ref{hh} is usually intuitively
clear to them. However, a first guess for the optimal shape that is often uttered by them, is
that it must  be a ``disk'' (or a ball). It is maybe the main conclusion of our work that
this guess not always hits the truth  as the optimal shape can be an annulus or multiple annuli as well; see below.

\begin{remark}\label{rem2.2}
{\rm Let us mention  several related design optimization problems.
In a previous paper \cite{AAA15}, the authors have considered the
problem of optimizing the sensitivity with respect to the
observation domain $R,T$ and also the problem of selecting reduced
data that only little reduce the sensitivity $S_{GRS}$.

In view of
the initial conditions, there are other classes of designs possible,
for instance, if not binary-valued functions are used, then it is
natural to optimize sensitivity with respect to a fixed energy
(i.e., the $L^1$-norm) of the initial conditions. That is, the
problem
 \[  S_{int} \to \max_{u_0} \qquad \text{ with}   \quad \int_{\mathbb{R}^2} u_0(x) dx = c_1  \]
with  $c_1>0$ given.  However, when analyzing this problem, one
finds out that it does not have a solution. A sequence of almost
optimal solutions will tend to a $\delta$-distribution. Thus a
$\delta$-peak is the ``optimal'' solution in this case. The
sensitivity is infinite for this case, which is caused by the fact
that the solutions are not square-integrable any more. Thus, for a
$\delta$-peak, one cannot use the formula above for the expected
error but have to used different, weighted norms (for the data noise
and the sensitivity).

A similar problem is that one with a fixed $L^2$-norm.
 \[  S_{int} \to \max_{u_0} \qquad \text{ with}  \int_{\mathbb{R}^2} u_0(x)^2 dx = c_2 \]
This one has an appealing interpretation as eigenvalue problem:
indeed, let $K$ be the linear operator $L^2(\mathbb{R}^2) \to L^2(\Omega
\times [0,T])$ that maps the initial conditions to $\tfrac{\partial u}{\partial D}$ on the
observation domain. It can be shown that this is a bounded linear
and compact operator. The problem can be rephrased in functional
analytic language as
\[ \|K u_0\|_{L^2(\Omega \times [0,T])}^2  \to \max \qquad \text{with } \|u_0\|_{L^2(\mathbb{R}^2)} \leq c_2. \]
It is well known that a solution exists and it is given by the
right singular function  associated
to the largest singular value of $K$. The associated optimization problem thus can
be (approximately) solved by applying a standard  singular valued decomposition to the
discretized operator $K$.
}
\end{remark}

Going back to the optimization problems, Problems~1 and 2, it can be observed
that the sensitivity $S_{int}$ is a quadratic functional with
respect to $u_0$. For the optimization it is important to have an
efficient formula for $S_{int}$ available, since it has to be
evaluated many times (for different initial conditions). Solving the
PDE \eqref{one}--\eqref{two} is not convenient in that respect. We
rather stick to the integral representation by the Green's function
and derive a formula for the sensitivity $S_{int}$ in the next
section.

\subsection{Evaluating the sensitivity measure \eqref{sensii}}

The formula for  $S_{int}$ involves the derivative of the observation
$u(x,t)$ with respect to the diffusion parameter $D$. Either by differentiating
\eqref{one}--\eqref{two} or \eqref{trans}, we find that
\begin{align*} \tfrac{\partial u}{\partial D}(x,t) &= \tfrac{t}{D} \pa{t}u(x,t)  =
 t \Delta u(x,t)    =  t \Delta_x \int_{\mathbb{R}^2} G(x-y,t D) u_0(y) dy.
\end{align*}
Because a convolution integral commutes with differentiation, we
also find that
\begin{align*} \tfrac{\partial u}{\partial D}(x,t) &
= t   \int_{\mathbb{R}^2} G(x-y,t D) \Delta u_0(y) dy.
\end{align*}
In a similar way, we can derive a formula using the scaled
variables:
 \begin{align}\label{udsc}  \tfrac{\partial u}{\partial D}(z R, \tau T) &=
 \tfrac{T}{R^2} \tau  \int_{\mathbb{R}^2} G(z-y,\tfrac{\tau}{4 \beta}) \Delta v_0(y) dy =
  \tfrac{T}{R^2} \tau \Delta v(z,\tau)
 \end{align}
with $v_0(y) = u_0(R y).$

 Now, in order to solve the above  optimization
problems (Problems 1 and 2), we have to evaluate \eqref{sensi} for
many different initial conditions. Therefore, in the sequel we
derive the kernel of the quadratic form associated to the
sensitivity measure $S_{int}$.

Consider two solutions $u^{(1)}$ and $u^{(2)}$ of
\eqref{one}--\eqref{two} with the respective initial conditions
$u_0^{(1)}$ and $u_0^{(2)}.$  We associate to them the scaled
functions $v^{(1)}$ and $v^{(2)}$ as in \eqref{scaling}. We
calculate the following pairing of $u^{(1)}$ and $u^{(2)}:$
\begin{align*}
&\int_{\|x\|\leq R} \int_0^T  \tfrac{\partial u^{(1)}}{\partial D}(x,t)
 \tfrac{\partial u^{(2)}}{\partial D}(x,t) dx dt  \\
& = T R^2 \int_{\|z\|\leq 1} \int_0^1
 \tfrac{\partial u^{(1)}}{\partial D}(z R,\tau T)
 \tfrac{\partial u^{(2)}}{\partial D}(z R,\tau T) dz d \tau  \\
& = T R^2 \int_{\|z\| \leq 1} \int_0^1  \tau^2
\Delta v^{(1)}(z  ,\tau ) \Delta v^{(2)}(z ,\tau ) dz d \tau  \\
& =
\frac{T^3}{R^2}
\int_{\|z\| \leq 1} \int_0^1
   \int_{\mathbb{R}^n}  \int_{\mathbb{R}^n}    \tau^2 G(z-y, \tfrac{\tau}{4 \beta})  G(z-w,\tfrac{\tau}{4 \beta}) \Delta v_0^{(1)}(y)
   \Delta v_0^{(2)}(w)  dw dy d \tau d z \\
& =
\frac{T^3}{R^2}
 \frac{\beta^2}{ \pi^2}
 \int_{\|z\| \leq 1} \int_0^1
   \int_{\mathbb{R}^n}  \int_{\mathbb{R}^n}
 e^{ - \beta \frac{\|z - y\|^2 + \|z - w\|^2 }{\tau}}
 \Delta v_0^{(1)}(y)
   \Delta v_0^{(2)}(w)  dw dy d \tau d z.
\end{align*}

In the next step we assume  radially symmetric initial conditions and introduce polar coordinates:
\begin{align*}
 y &= r (\cos(\theta_1),\sin(\theta_1)) \quad   w = s (\cos(\theta_2),\sin(\theta_2))
\quad   z = q (\cos(\phi),\sin(\phi) \\
v_0^{(1)}(y) &=:  g^{(1)}(\|y\|) = g^{(1)}(r) \quad v_0^{(2)}(w) =:  g^{(2)}(\|w\|) = g^{(2)}(s)
\end{align*}
such that
\begin{align*}
    \Delta v_0^{(1)}(y) &=
\tfrac{1}{r} \tfrac{\partial}{\partial r} \left( r \tfrac{\partial}{\partial r}  g^{(1)}(r) \right) \qquad r = \|y\|,  \\
    \Delta v_0^{(2)}(w) &=
\tfrac{1}{s} \tfrac{\partial}{\partial s} \left( s \tfrac{\partial}{\partial s}  g^{(2)}(s) \right) \qquad s = \|w\|.
\end{align*}
Thus,
\begin{align*}
&\int_{\|x\|\leq R} \int_0^T  \tfrac{\partial u^{(1)}}{\partial D}(x,t)
 \tfrac{\partial u^{(2)}}{\partial D}(x,t) dx dt \\
& = \frac{T^3}{R^2}
 \frac{\beta^2}{ \pi^2}
 \int_0^1 q  \int_0^1
   \int_{\mathbb{R}}  \int_{\mathbb{R}}
   \int_0^{2 \pi} \int_0^{2\pi} \int_0^{2\pi}
 e^{ -\frac{\beta}{\tau}\left( q^2 + r^2 - 2 q r \cos(\phi-\theta_1) +   q^2 + s^2 -
 2 q s \cos(\phi-\theta_2)
 \right)} \\
 & \qquad \qquad {rs }
 \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial}{\partial r}  g^{(1)}(r) \right)
   \frac{1}{s} \frac{\partial}{\partial s} \left( s \frac{\partial}{\partial s}  g^{(2)}(s) \right)
   d \theta_1 d \theta_2  d\phi dr  ds d \tau d q\, .
\end{align*}
By a substitution of $\phi-\theta_1$ by $\theta_1$ and
$\phi-\theta_2$ by $\theta_2$, the integrand does not depend on $\phi$. Thus, the
corresponding integral gives a contribution of $2 \pi.$
The integrals over $\theta_i$ can be calculated explicitly using the formula
\begin{equation}\label{bessel}  \int_0^{2 \pi} e^{a \cos(t)} d t = 2 \pi I_0(a),  \end{equation}
with the modified Bessel function $I_k(z).$
\begin{align*}
&\int_{\|x\|\leq R} \int_0^T  \tfrac{\partial u^{(1)}}{\partial D}(x,t)
 \tfrac{\partial u^{(2)}}{\partial D}(x,t) dx dt \\
& =   \frac{T^3}{R^2}
 8 \pi \beta^2
 \int_0^1 q  \int_0^1
   \int_{\mathbb{R}}  \int_{\mathbb{R}}
 e^{ -\frac{\beta}{\tau}\left( q^2 + r^2 +   q^2 + s^2 \right)} I_0(\tfrac{2 \beta q r}{\tau})
 I_0(\tfrac{2 \beta q s}{\tau})
 \\
 & \qquad \qquad
\frac{\partial}{\partial r} \left( r \frac{\partial}{\partial r}  g^{(1)}(r) \right)
 \frac{\partial}{\partial s} \left( s \frac{\partial}{\partial s}  g^{(2)}(s) \right)
   d \theta_1 d \theta_2   dr  ds d \tau d q \, .
\end{align*}

We next assume that $g^{(1)}$ and $g^{(2)}$ have compact support and
we integrate by parts with respect to the derivatives
$\frac{\partial}{\partial r}$ and $\frac{\partial}{\partial s}$
observing that by radial symmetry the terms at $r = 0$ and $s = 0$
vanish. Thus,
\begin{align*}
&\int_{\|x\|\leq R} \int_0^T  \tfrac{\partial u^{(1)}}{\partial D}(x,t)
 \tfrac{\partial u^{(2)}}{\partial D}(x,t) dx dt \\
& = \frac{T^3}{R^2}
  8 \pi \beta^2
 \int_0^1 q  \int_0^1
   \int_{\mathbb{R}}  \int_{\mathbb{R}}
 \frac{\partial^2}{\partial r \partial s}
 \left[
  e^{ -\frac{\beta}{\tau}\left( q^2 + r^2 +   q^2 + s^2 \right)} I_0(\tfrac{2 \beta q r}{\tau})
 I_0(\tfrac{2 \beta q s}{\tau})
 \right] \\
 & \qquad \qquad
 r s \frac{\partial}{\partial r}  g^{(1)}(r)
\frac{\partial}{\partial s}  g^{(2)}(s)
  dr  ds d \tau d q  .
\end{align*}
The $r,s$-derivatives can be calculated using the fact for modified Bessel functions that $I_0' = I_1.$
\begin{align*}
 &\frac{\partial^2}{\partial r \partial s}
 \left[
  e^{ -\frac{\beta}{\tau}\left( q^2 + r^2 +   q^2 + s^2 \right)} I_0(\tfrac{2 \beta q r}{\tau})
 I_0(\tfrac{2 \beta q s}{\tau})
 \right] \\
 & =   e^{ -\frac{\beta}{\tau}\left( q^2 + r^2 +   q^2 + s^2 \right)} \times \\
& \qquad  \left\{\tfrac{- 2 r \beta}{\tau} I_0(\tfrac{2 \beta q r}{\tau})  + \tfrac{2 q \beta}{\tau}I_1(\tfrac{2 \beta q r}{\tau})
 \right\}
  \left\{\tfrac{- 2 s \beta}{\tau} I_0(\tfrac{2 \beta q s}{\tau})  + \tfrac{2 q \beta}{\tau}I_1(\tfrac{2 \beta q s}{\tau})
 \right\},
 \end{align*}
which with $\rho = \frac{\beta}{\tau}$  finally leads to
\begin{equation}\label{final1}
\begin{split}
&\int_{\|x\|\leq R} \int_0^T  \tfrac{\partial u^{(1)}}{\partial D}(x,t)
 \tfrac{\partial u^{(2)}}{\partial D}(x,t) dx dt \\
&   = \frac{T^3}{R^2}
  32 \pi \beta^3
 \int_0^1 q  \int_\beta^\infty
   \int_{\mathbb{R}}  \int_{\mathbb{R}}    e^{ - \rho \left( q^2 + r^2 +   q^2 + s^2 \right)}  \times   \\
  & \qquad \qquad
 \left\{ r  I_0(2 q r \rho )  -  q  I_1(2  q r \rho )
 \right\}
  \left\{ s   I_0(2 q s \rho )  -  q  I_1(2  q s \rho )
 \right\} \\
 & \qquad \qquad
 {r s} \frac{\partial}{\partial r}  g^{(1)}(r)
\frac{\partial}{\partial s}  g^{(2)}(s)
  dr  ds d \rho  d q.   \\
\end{split}
  \end{equation}

For later reference we define a kernel as follows:
\begin{equation}\label{defker}
\begin{split}
k(r,s) &:=
 \int_0^1 q  \int_\beta^\infty    e^{ - \rho \left( q^2 + r^2 +   q^2 + s^2 \right)}  \times   \\
  & \qquad \qquad
 \left\{ r  I_0(2 q r \rho )  -  q  I_1(2  q r \rho )
 \right\}
  \left\{ s   I_0(2 q s \rho )  -  q  I_1(2  q s \rho ) \right\}  {r s} d \rho  d q.
 \end{split}
\end{equation}

For the numerical calculations, we also require the kernel for the case of
$\beta = 0$. We present another  formula which avoids integration over the
infinite domain with respect to $\rho$.

\begin{lemma}
For $\beta = 0$ the kernel $k(r,s)$ can be written as
\begin{equation}\label{fff}
\begin{split} &k(r,s) = \\
&\frac{1}{4 \pi^2}
\int_0^1  \int_0^{2 \pi} \int_0^{2 \pi}  q r s
\frac{ \left( r  -  q   \cos(\theta_1) \right) \left( s -  q   \cos(\theta_2) \right)}
{q^2 + r^2  - 2 q r \cos(\theta_1) +   q^2 + s^2
- 2 q s \cos(\theta_2) }
d\theta_1 d \theta_2  d q
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Setting $\beta = 0$ and \eqref{bessel}
and
\[  I_1(a) = \frac{1}{2 \pi}  \int_0^{2 \pi} e^{a \cos(\theta)} \cos(\theta) d \theta, \]
we observe  that
\begin{align*}
&k(r,s) = \frac{1}{4 \pi^2}
\int_0^1 q  \int_0^\infty  \int_0^{2 \pi} \int_0^{2 \pi}   e^{ - \rho \left( q^2 + r^2  - 2 q r \cos(\theta_1) +   q^2 + s^2
- 2 q s \cos(\theta_2) \right)}  \times   \\
  & \qquad \qquad
 \left( r  -  q   \cos(\theta_1) \right) \left( s -  q   \cos(\theta_2) \right) r s  d\theta_1 d \theta_2 d \rho  d q \\
 & = \frac{1}{4 \pi^2}
\int_0^1  \int_0^{2 \pi} \int_0^{2 \pi}  q r s
\frac{ \left( r  -  q   \cos(\theta_1) \right) \left( s -  q   \cos(\theta_2) \right)}
{q^2 + r^2  - 2 q r \cos(\theta_1) +   q^2 + s^2
- 2 q s \cos(\theta_2) }
d\theta_1 d \theta_2  d q  \, .
\end{align*}
$\blacksquare$
\end{proof}

Next we use Lemma~3.1 according to which  the optimal initial  conditions are
radially symmetric $\{0,1\}$-valued with compact support.  This
means that in this case, the functions $g^{(i)}$ attain the value
$1$ on a  number of intervals and $0$ else, and the associated
initial condition is supported on  a  union of annuli and possibly a
disk.  We moreover assume that the function $g$, which
defines the initial conditions, has only jumps at $N \in \mathbb{N}$ places:
\begin{equation}\label{gequ}  g^{(i)}(r) = \begin{cases} 1 &  r \in [r_j,r_{j+1}], j = 1,\ldots N, 0 \leq r_1 <r_{j-1} < r_{j} < r_{j-1},  \\
 0 &\text{ else } \end{cases} \end{equation}
It follows in the sense of distributions that
\begin{equation}\label{deri}
 \frac{\partial}{\partial r}  g^{(i)}(r)
= \sum_{j= 1, r_1 \not= 0}^{N} (-1)^{N-j+1} \delta_{r_j}(r),
 \end{equation}
 where $\delta$ denotes the Dirac-distribution. The case that $r_1 = 0$ means that $u_0$ has  support in
 a disk around $0$ and clearly, there is no jump in the derivative there, hence this must be excluded from the sum.
 Note that the signs are such that the
 outer radius has $(-1)$ and the following have alternating signs, which becomes  clear by drawing
 the graph of such a function.

By plugging in the formula \eqref{deri} in place of $g^{(i)}$
we obtain the following proposition.

\begin{proposition}\label{pr:one}
Suppose that the initial conditions $v_0(z) = g(|z|)$ are radially symmetric $\{0,1\}$-valued
with $g$ having finitely many jumps as in \eqref{gequ}.
Then the sensitivity is given by
\begin{equation}
S_{int} =  \frac{T^3}{R^2}
  32 \pi \beta^3 \sum_{j,k=1}^N (-1)^{k + j} k(r_j,r_k)
  \end{equation}
 with the kernel $k$ given by \eqref{defker}.
\end{proposition}
Note that we do not have to take into account if $r_1 = 0$ or not, because the kernel is $0$ if one of the
arguments is $0$. We remark that Proposition~\ref{pr:one} is also true for the case of $g$ having
infinitely many jumps; in this case we have to set $N=\infty$.

By Proposition~\ref{pr:one} we find that
\begin{itemize}
\item Problem 1 is equivalent to
\[ \max_{r_i, N}  \sum_{j,k=1}^N (-1)^{k + j} k(r_j,r_k) , \qquad 0 < r_1 < r_2 \leq \ldots r_N, N \geq 1. \]
\item Problem 2 is equivalent to
\begin{align*}  &\max_{r_i, N}  \sum_{j,k=1}^N (-1)^{k + j} k(r_j,r_k) , \qquad 0 < r_1 < r_1 \leq \ldots r_N, N \geq 1 \\
\intertext{under the constraint that}
&\sum_{k=N}^{2}  r_{k}^2 - r_{k-1}^2 = \frac{c_1}{\pi} \quad \text{ or }
r_1^2 = \frac{c_1}{\pi} \quad \text{ if } N = 1.
\end{align*}
\end{itemize}

\subsection{Shape (and topology) optimization of the bleached region}
\label{shape} We try to solve the above problems numerically.
Note that we have the radii of the annuli/disks as
variables and the number $N$ of them. As optimization routine we use
the simplest and least sophisticated one, namely grid search over
the radii restricting us to a finite number of annuli. In fact, we
managed to search for the case $N \leq 4$ and for radii restricted
to $r_i \leq 5$ on a sufficiently fine grid. We could not think of a
smarter algorithm, but there are good reasons to believe that there
is no fast algorithm available.
In fact, by considering the relaxed problem and introducing  the initial-to-sensitivity operator $K,$
Problem~1 is equivalent to
\[ \|K u_0\|_{L^2(\Omega) \times [0,T]}^2 \to \max   \qquad  \text{under } 0 \leq u_0 \leq 1. \]
This is a convex {\em maximization problem} and it can be regarded as the problem of calculating the
operator norm $K: L^\infty \to L^2$ (together with an additional positivity constraint $u_0 \geq 0$).
It is known that calculating this operator norm is an NP-hard problem \cite{Ro,Tro}. This makes us believe
that our optimization problems are hard problems indeed and that there is no simple efficient algorithm
possible.

Let us mention that a classical method to tackle such shape optimization methods is the
{\em level set method} \cite{OsFe}. This could be applied as well here, but the problem is
that this and related methods can only find {\em local} maxima. Thus, as we want to find
 {\em global} maxima, the level set method does not help much here.

\section{Numerical calculations and results}
As mentioned in Section~\ref{shape}, we perform a grid search to
solve Problems~1 and~2. For this we have to calculate the kernel
$k(r,s)$ for different values of $r$ and $s$ and also, to understand the
parameter dependence of the results, for different values of
$\beta$. The results are calculated using Matlab. Depending on the
grid for $r,s,$ and $\beta,$ according to \eqref{defker}, we have to
calculate a double integral for each point on a 3-dimensional grid
on $(r,s,\beta)$. No doubt, that for this tasks the evaluation has
to be done efficiently in order to have a reasonable runtime. When
calculating the double integral in \eqref{defker}, we observe that
Matlab does not have an ``ArrayValued'' option for multidimensional
integrals and hence the code cannot be vectorized. This faces us
with the problem that the 2-dimensional integral has to be
calculated by for-loops over the grid, which, as is well-known, is
usually extremely slow. As a remedy, we rewrite the problem as
iterated integral and apply Matlab's ODE-solver (which can be
vectorized).

Indeed, from  formula \eqref{defker}, it follows that the kernel satisfies
\begin{equation}\label{trick}
\begin{split}
\frac{\partial}{\partial \beta} k(r,s) &=
- \int_0^1 q e^{ - \rho \left( q^2 + r^2 +   q^2 + s^2 \right)}  \times   \\
  & \qquad \qquad
 \left\{ r  I_0(2 q r \rho )  -  q  I_1(2  q r \rho )
 \right\}
  \left\{ s   I_0(2 q s \rho )  -  q  I_1(2  q s \rho ) \right\}  {r s}  d q.
\end{split}
  \end{equation}
with initial condition
\[ k(r,s)|_{\beta = 0} = \eqref{fff} \]
In this way we can use the ODE-solver to handle the $\beta$-dependence and only have to
calculate the one-dimensional parameter-dependent integral \eqref{trick}, where both
steps can be vectorized. This gives a speed-up of a factor of about 10 compared to
the for-loop and double integral approach. The only bottleneck is the calculation of the
initial conditions in form of a 3-dimensional integral. However, this only has to be
done on a 2-dimensional $(r,s)$ parameter grid (compared to the initially 3-dimensional
parameter domain including $\beta$).

\subsection{Problem 1}
We try to calculate the (numerically) optimal configuration for
Problem 1. As explained above, we only have to consider radially
symmetric initial conditions. We cannot calculate all configurations
but we restrict us to the case of at most $N=4$ radii. This means we
consider initial shapes in the form of a disk ($N = 1$), an
annulus ($N = 2$),  an annulus with a disk inside ($N =
3$), or a double annulus ($N = 4$). Other configurations with higher
$N$ are excluded because of runtime considerations. However, the
solutions indicate a pattern, which makes us believe that we have
found the optimal configurations in the respective parameter setting
of $\beta$.

We also note that we only have to evaluate the kernel $k$ because
the sensitivity is only multiplied by a constant which only depends
on the parameters $R,T,D$. Thus, with these values fixed, optimizing
$k$ is the same as optimizing $S_{int}$.

Let us come to the details of our calculations:
We set up a grid for $r$ and $s$ in the interval $[0,5]$ with 100 equal-sized subintervals
(of length $0.05$). For $\beta$ we set up a grid in the interval  $[0,20]$ with 200 subintervals
of length $0.1$. Then the kernel is calculated on these gridpoints (exploiting symmetry in $r$ and $s$).
With the calculated kernel, we evaluate the functional for $r,s$ on the grid
for Problem 1 as given in Proposition~\ref{pr:one}
for the cases $N = 1,2,3,4$ and find the  configuration that has maximal value.
 Note that this testing requires
up to 4 for-loops and is thus quite slow.
The results are depicted in Figure~\ref{fig1}.

\begin{figure}[p]
\centering
\includegraphics[width=0.7\textwidth]{figfirst}
\caption{Optimal configurations for different values of $\beta$. Tested were all configurations
with $N\leq 4$. Thick lines: values of the log of the optimal sensitivity. Dashed lines: values of the
log of the optimal sensitivity within each configurations (e.g. $N = 1, N=2. N= 3$).}\label{fig1}
\end{figure}

Let us explain the figure. The x-axis indicates  the values of $\beta$ while the $y$-axis corresponds
to the logarithm of the optimal value of the sensitivity for a fixed $\beta$.
The full lines in the figure display the log of the optimal value  over $\beta$  and the
associated  colors encode which of
the four possibilities ($N = 1,\ldots,4$) are the optimal configuration
(violet: $N=1$, green: $N = 2$, blue: $N = 3$, red: $N = 4$).

It can be observed that there are clear intervals for $\beta$  where
one specific configuration is always optimal. A value of $\beta$
where the shape of the optimal configuration changes (e.g., from
disk to annulus)  is indicated by a vertical line. Thus, we
observe  that from $\beta \in [0,1.8]$ the disk $(N = 1)$
is the optimal configuration, whereas for higher $\beta$ in the
interval $[1.8,6.1]$ the annulus is optimal. An even higher $\beta$
gives a annulus with a disk inside and for $\beta > 13.8$
we find the double annulus as optimal. Clearly a pattern can be
observed, namely that for increasing $\beta$ the number of
components of the optimal shape increases. We expect that for higher
$\beta$ not in the range of our calculations, more complicated
shapes (e.g. with $N = 5$) are optimal.  Although we tested only to
$N = 4$ the given pattern indicates that in the low range of $\beta,$
the shapes with few components are indeed optimal because in the
first interval $\beta \in [0,1.8]$ we never found  a better
structure with $N>1$.

In Figure~\ref{fig1} we also indicated the log-values of the optimal
sensitivity for each shape with fixed $N$ by a dotted line. For
instance, the dotted continuation of the violet line from $\beta =
1.8$ on corresponds to the optimal value within all disks
(i.e. $N = 1.$)  The continuation of the green line corresponds to
the optimal value within  all annuli and so on. It can be observed
that, for instance, in the range where the annulus is optimal, the
value of the sensitivity is significantly higher than the best
 disk. Roughly, the sensitivity of the best annulus is about
twice that of the best disk. This indicates that in
practice and for certain parameter setups, there can be a
significant gain of confidence when using an annulus instead of a
disk as initial bleach.

In the second figure, Figure~\ref{fig2}, we plotted the values of the radii of the optimal configuration
in each  interval of $\beta$. On the x-axis we put again the value of
$\beta$ and on the y-axis we have the scaled radius.
For instance in $\beta \in [0, 1.8]$ the disk is optimal
and only one radius is required for its parameterization. The value
of the optimal radius is given by the curve in the interval $[0,
1.8]$ In the next interval, the annulus is optimal which has an
outer and an inner radius as given by the two curves and so on, up
to the double annulus with 4 radii. Note that the jumps in this
graphs inside of the intervals are artefacts due to the discrete
grid for the radius; the exact curves should be smooth.

\begin{figure}[p]
\centering
\includegraphics[width=0.7\textwidth]{fig2}
\caption{Scaled radii of the optimal configurations as a function of $\beta$.} \label{fig2}
\end{figure}

\begin{figure}[p]
\centering
\includegraphics[width=0.7\textwidth]{EnergyEx1}
\caption{Energy (i.e.,  $\int_{\mathbb{R}^2} u_0(x) dx$)  of the optimal configurations as a function
of $\beta$.} \label{fig2ex}
\end{figure}

In Figure~\ref{fig2ex}, we plotted the energy (i.e., the $L^1$-norm $\int_{\mathbb{R}^2} u_0(x) dx$) of the
optimal configurations as in Figure~\ref{fig2}. Of course, those can be easily calculated from the
radii in Figure~\ref{fig2}.

For illustration purposes, in Figure~\ref{fig3}  we present some optimal initial shapes for some representative values of
$\beta$ in each of the intervals, for $\beta = 1,3,10,18$ (from left to right, from top to bottom).

\begin{figure}[p]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{optbeta1}
{$\beta = 1$}

\includegraphics[width=\textwidth]{optbeta10}
{$\beta = 10$}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{optbeta3}
{$\beta = 3$}

\includegraphics[width=\textwidth]{optbeta18}
{$\beta = 18$}
\end{minipage}
\caption{Optimal shapes of initial bleach for various values of $\beta$.}\label{fig3}
\end{figure}

\subsection{Problem 2}
The results from Problem 1 can also be used for  Problem 2.  Here we have the
additional restriction on the energy. The energy of some optimal configurations without constraints can
be read off from Figure~\ref{fig2ex}.

In order to find  an optimal shape, we again restrict
ourselves to $N \leq 4$. We calculate the value of the kernel for different configurations,
i.e., values of $N$, and $r_i$ and additional calculate their energy (the area of the initial bleach shape in this case).
For each of the four cases of $N$ we pick the configuration which is
optimal for each energy.  Note that this comes with a slight
difficulty because due to the discretization of the radii, the
energy values are discrete and in order to compare configurations,
we have to interpolate these values to a common energy grid. Note
that for the case $N = 1,$ when fixing the energy, the disk
is uniquely determined by the energy and there is no need to
optimize in this case because no degree of freedom is left.

\begin{figure}[p]\label{fig4}
\centering
\includegraphics[width=0.7\textwidth]{Energyfig}
\caption{Color-coded illustration of optimal configuration with $N \leq 4$ for different values of $\beta$ and
with fixed values of the energy (Problem 2). The colors indicates which configuration was found optimal.}
\end{figure}

In Figure~\ref{fig4}, we
display again a color-coded plot indicating the optimal configurations (up to $N\leq 4$).
For each value of the energy and each value of $\beta,$ the color at
the corresponding pixel indicates which of the configurations are
found optimal. We observe that in the majority of cases the double
annulus is optimal, but this is only because we restricted ourselves
to $N \leq 4$. It is to be expected that initial shapes with more
annuli are optimal in general and the problem does not reveal an
obvious pattern. Furthermore, we observe that the plot has  quite
irregular regions with not always clearly defined boundaries. This
can be an artefact of the discretization and we consider the plot
only as rough indication of what is going on.

Only in a tiny region around $\beta \sim 0$ energy $\sim 2,$ we have
a situation where the disk is optimal, but it is uncertain
if this is a valid result or an artefact from our discretization and
the interpolation of the energy values.  As a summary, it is quite
likely that simple-shaped objects (i.e., those with small  $N$) are
rarely (or even never) optimal for Problem~2.

Although, we have a put some model assumptions like symmetric
observation domain and simple diffusion with constant parameter,
we think that our results are relevant and---with modification---
valid also in more general cases. In particular, when the observational
domain is not a disk but an square, we expect the optimal bleach shape for Problem~1
being a rounded square or several nested rounded squares, where the connectivity
again increases with $\beta$. For non-constant diffusion, the sensitivity
is represented by a matrix and the optimal experimental design
leads to the problem of maximizing some matrix norm of the inverse of the
Fisher information matrix. Then, the optimal bleach shape for Problem~1 might depend on
several additional design setups like the basis representation of the
diffusion coefficient. Still it is not unlikely, that even in this case
a similar pattern of optimal shapes with increasing connectivity
emerges.

\section{Conclusion}
Our study started with the question: How does the bleach shape (and
topology) influence the accuracy of resulting parameter estimates?
Then the problem of the optimal initial shape for the identification
of a constant diffusion parameter was formulated.
 As
optimality criterion we choose to maximize a sensitivity measure in
order to have the expected error minimal; cf. \eqref{sensi}.
We studied
two problems, the first
 with the fixed bleach depth
 and the
second with an fixed energy as an additional restriction.
We found out the analytical expressions for the sensitivity measure
$S_{int}$, cf. Proposition 3.2, allowing to compare different initial
bleach shapes.
Our numerical calculations revealed rather surprising results.
For  the first problem without restriction on energy, a clear
pattern is revealed. For small values of the scaled inverse
diffusion coefficient, the disk is the optimal shape and
for higher values, shapes with more and more components (i.e.
annuli-type shapes) become optimal. In particular, the disk
is not always the best shape. For practically relevant values of the
parameters,  sometimes an annulus can be better leading  to a
significant improvement in the confidence intervals.
For the later problem with restriction on energy, is seems to have
in most of the cases or even always only  highly
oscillating solutions.
However, our ongoing research is directed to
this problem.

\subsection*{Acknowledgement}
  This work was supported  by
the Ministry of Education, Youth and Sports of the Czech Republic --
projects CENAKVA (No. CZ.1.05/2.1.00/01.0024), the Cenakva Centre
Development CZ.1.05/2.1.00/19.0380, and
 by the OeAD (Austrian agency for
international mobility and cooperation in education, science and
research) within the programme "Aktion Oesterreich-Tschechien
(AOeCZ-Universitaetslehrerstipendien)".

redpajama_set_name: RedPajamaArXiv


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text:
import { mount } from '@vue/test-utils'
import Vue from 'vue'
import Layout from './index'
import Header from './header'
import Aside from './aside'
import Main from './main'
import Footer from './footer'

const components = {
  'c-layout': Layout,
  'c-header': Header,
  'c-main': Main,
  'c-aside': Aside,
  'c-footer': Footer
}

describe('[Layout] basics layout', () => {
  it('should throw with no content', () => {
    const testApp = {
      components,
      render(h) {
        return <c-layout />
      }
    }
    // eslint-disable-next-line
    const spy = jest.spyOn(global.console, 'error').mockImplementation(() => {})
    try {
      mount(testApp)
    } catch (err) {
      expect(err.message).toBeTruthy()
    } finally {
      spy.mockRestore()
    }
  })

  it('should stack vertically without aside', () => {
    const testApp = {
      components,
      render(h) {
        return (
          <c-layout>
            <c-header>header</c-header>
            <c-main>main</c-main>
            <c-footer>footer</c-footer>
          </c-layout>
        )
      }
    }
    const wrapper = mount(testApp)
    const layout = wrapper.find('.c-layout')
    const header = wrapper.find('.c-layout__header')
    const main = wrapper.find('.c-layout__main')
    const footer = wrapper.find('.c-layout__footer')
    expect(layout).toBeTruthy()
    expect(layout.classes('c-layout--has-aside')).toBe(false)
    expect(header).toBeTruthy()
    expect(footer).toBeTruthy()
    expect(main).toBeTruthy()
  })
})

describe('[Aside] sidebar', () => {
  it('should display horizontally with sidebar', async () => {
    const testApp = {
      components,
      render(h) {
        return (
          <c-layout>
            <c-aside>aside</c-aside>
            <c-main>main</c-main>
          </c-layout>
        )
      }
    }
    const wrapper = mount(testApp)
    const layout = wrapper.find('.c-layout')
    expect(layout).toBeTruthy()
    expect(layout.classes('c-layout--has-aside')).toBe(true)
  })

  it('should be collapsed by prop', async () => {
    const testApp = {
      components,
      data() {
        return { collapsed: false }
      },
      render(h) {
        return (
          <c-layout>
            <c-aside collapsed={this.collapsed}>aside</c-aside>
            <c-main>main</c-main>
          </c-layout>
        )
      }
    }
    const wrapper = mount(testApp)
    const aside = wrapper.find('.c-layout__aside')
    expect(aside.classes('c-layout__aside--collapsed')).toBe(false)

    wrapper.setData({ collapsed: true })
    await Vue.nextTick()
    expect(aside.classes('c-layout__aside--collapsed')).toBe(true)
  })

  it('should be toggle collapse by click button', async () => {
    const testApp = {
      components,
      data() {
        return { collapsed: false }
      },
      render(h) {
        return (
          <c-layout>
            <c-aside vModel={this.collapsed} collapsible>
              aside
            </c-aside>
            <c-main>main</c-main>
          </c-layout>
        )
      }
    }
    const wrapper = mount(testApp)
    const aside = wrapper.find('.c-layout__aside')
    expect(aside.classes('c-layout__aside--collapsed')).toBe(false)

    const trigger = wrapper.find('.c-layout__collapse-trigger')
    trigger.trigger('click')
    await Vue.nextTick()
    expect(aside.classes('c-layout__aside--collapsed')).toBe(true)
  })

  it('should make main scrolling when header and aside fixed', async () => {
    const testApp = {
      components,
      data() {
        return { showHeader: true }
      },
      render(h) {
        const header = this.showHeader ? (
          <c-header fixed>header</c-header>
        ) : null
        return (
          <c-layout>
            {header}
            <c-layout>
              <c-aside fixed>aside</c-aside>
              <c-main>main</c-main>
            </c-layout>
          </c-layout>
        )
      }
    }
    const wrapper = mount(testApp)
    await Vue.nextTick()
    const layout = wrapper.find('.c-layout')
    expect(layout.classes('c-layout--scroll-main')).toBe(true)

    wrapper.setData({ showHeader: false })
    await Vue.nextTick()
    expect(layout.classes('c-layout__aside--collapsed')).toBe(false)
  })
})

redpajama_set_name: RedPajamaGithub


text:
package org.xbmc.kore.ui.sections.video;

import android.database.Cursor;
import android.net.Uri;
import android.os.Bundle;
import android.provider.BaseColumns;
import android.support.v4.app.LoaderManager;
import android.support.v4.content.CursorLoader;
import android.support.v4.content.Loader;

import org.xbmc.kore.R;
import org.xbmc.kore.Settings;
import org.xbmc.kore.jsonrpc.event.MediaSyncEvent;
import org.xbmc.kore.provider.MediaContract;
import org.xbmc.kore.service.library.LibrarySyncService;
import org.xbmc.kore.ui.AbstractAdditionalInfoFragment;
import org.xbmc.kore.ui.AbstractInfoFragment;
import org.xbmc.kore.ui.generic.RefreshItem;
import org.xbmc.kore.ui.widgets.fabspeeddial.FABSpeedDial;
import org.xbmc.kore.utils.LogUtils;

/**
 * Presents a TV Show overview
 */
public class TVShowInfoFragment extends AbstractInfoFragment
        implements LoaderManager.LoaderCallbacks<Cursor> {
    private static final String TAG = LogUtils.makeLogTag(TVShowInfoFragment.class);

    // Loader IDs
    private static final int LOADER_TVSHOW = 0;

    // Controls whether a automatic sync refresh has been issued for this show
    private static boolean hasIssuedOutdatedRefresh = false;

    @Override
    protected RefreshItem createRefreshItem() {
        RefreshItem refreshItem = new RefreshItem(getActivity(),
                                                  LibrarySyncService.SYNC_SINGLE_TVSHOW);
        refreshItem.setSyncItem(LibrarySyncService.SYNC_TVSHOWID, getDataHolder().getId());
        refreshItem.setListener(new RefreshItem.RefreshItemListener() {
            @Override
            public void onSyncProcessEnded(MediaSyncEvent event) {
                if (event.status == MediaSyncEvent.STATUS_SUCCESS) {
                    getLoaderManager().restartLoader(LOADER_TVSHOW, null,
                                                     TVShowInfoFragment.this);
                    refreshAdditionInfoFragment();
                }
            }
        });

        return refreshItem;
    }

    @Override
    protected boolean setupMediaActionBar() {
        return false;
    }

    @Override
    protected boolean setupFAB(FABSpeedDial FAB) {
        return false;
    }

    @Override
    protected AbstractAdditionalInfoFragment getAdditionalInfoFragment() {
        TVShowProgressFragment tvShowProgressFragment = new TVShowProgressFragment();
        tvShowProgressFragment.setArgs(getDataHolder().getId(), getDataHolder().getTitle());
        return tvShowProgressFragment;
    }

    @Override
    public void onActivityCreated (Bundle savedInstanceState) {
        super.onActivityCreated(savedInstanceState);

        hasIssuedOutdatedRefresh = false;

        getLoaderManager().initLoader(LOADER_TVSHOW, null, this);
    }

    /**
     * Loader callbacks
     */
    /** {@inheritDoc} */
    @Override
    public Loader<Cursor> onCreateLoader(int i, Bundle bundle) {
        Uri uri = MediaContract.TVShows.buildTVShowUri(getHostInfo().getId(), getDataHolder().getId());
        return new CursorLoader(getActivity(), uri,
                                TVShowDetailsQuery.PROJECTION, null, null, null);

    }

    /** {@inheritDoc} */
    @Override
    public void onLoadFinished(Loader<Cursor> cursorLoader, Cursor cursor) {
        if (cursor != null) {
            switch (cursorLoader.getId()) {
                case LOADER_TVSHOW:
                    cursor.moveToFirst();

                    DataHolder dataHolder = getDataHolder();

                    dataHolder.setFanArtUrl(cursor.getString(TVShowDetailsQuery.FANART));

                    dataHolder.setPosterUrl(cursor.getString(TVShowDetailsQuery.THUMBNAIL));
                    dataHolder.setRating(cursor.getDouble(TVShowDetailsQuery.RATING));
                    dataHolder.setMaxRating(10);

                    String premiered = cursor.getString(TVShowDetailsQuery.PREMIERED);
                    String studio = cursor.getString(TVShowDetailsQuery.STUDIO);

                    dataHolder.setDetails(String.format(getString(R.string.premiered), premiered) + "  |  " + studio +
                                          "\n" +
                                          cursor.getString(TVShowDetailsQuery.GENRES));

                    dataHolder.setTitle(cursor.getString(TVShowDetailsQuery.TITLE));

                    int numEpisodes = cursor.getInt(TVShowDetailsQuery.EPISODE),
                            watchedEpisodes = cursor.getInt(TVShowDetailsQuery.WATCHEDEPISODES);

                    dataHolder.setUndertitle(String.format(getString(R.string.num_episodes),
                                                           numEpisodes, numEpisodes - watchedEpisodes));

                    dataHolder.setDescription(cursor.getString(TVShowDetailsQuery.PLOT));

                    updateView(dataHolder);
                    checkOutdatedTVShowDetails(cursor);
                    break;
            }
        }
    }

    /** {@inheritDoc} */
    @Override
    public void onLoaderReset(Loader<Cursor> cursorLoader) {
        // Release loader's data
    }

    /**
     * Checks wether we should refresh the TV Show details with the info on XBMC
     * The details will be updated if the last update is older than what is configured in the
     * settings
     *
     * @param cursor Cursor with the data
     */
    private void checkOutdatedTVShowDetails(Cursor cursor) {
        if (hasIssuedOutdatedRefresh)
            return;

        cursor.moveToFirst();
        long lastUpdated = cursor.getLong(TVShowDetailsQuery.UPDATED);

        if (System.currentTimeMillis() > lastUpdated + Settings.DB_UPDATE_INTERVAL) {
            // Trigger a silent refresh
            hasIssuedOutdatedRefresh = true;
            getRefreshItem().startSync(true);
        }
    }

    /**
     * TV Show details query parameters.
     */
    private interface TVShowDetailsQuery {
        String[] PROJECTION = {
                BaseColumns._ID,
                MediaContract.TVShows.TITLE,
                MediaContract.TVShows.THUMBNAIL,
                MediaContract.TVShows.FANART,
                MediaContract.TVShows.PREMIERED,
                MediaContract.TVShows.STUDIO,
                MediaContract.TVShows.EPISODE,
                MediaContract.TVShows.WATCHEDEPISODES,
                MediaContract.TVShows.RATING,
                MediaContract.TVShows.PLOT,
                MediaContract.TVShows.PLAYCOUNT,
                MediaContract.TVShows.IMDBNUMBER,
                MediaContract.TVShows.GENRES,
                MediaContract.SyncColumns.UPDATED,
                };

        int ID = 0;
        int TITLE = 1;
        int THUMBNAIL = 2;
        int FANART = 3;
        int PREMIERED = 4;
        int STUDIO = 5;
        int EPISODE = 6;
        int WATCHEDEPISODES = 7;
        int RATING = 8;
        int PLOT = 9;
        int PLAYCOUNT = 10;
        int IMDBNUMBER = 11;
        int GENRES = 12;
        int UPDATED = 13;
    }
}


The air is getting cooler (sort of) and the days are getting shorter. Fall is in the air and that means Halloween is just around the corner. Get pumped up for the best holiday out there by checking out this Wicked Halloween Playlist! Rock out to great hits like "Thriller," "Bark at the Moon" and, of course, "Monster Mash."


text:
From Friendship to Lifelong Love: Scott and Lauren's Journey
January 19, 2023 January 19, 2023 | Chapter Web Service | Advocacy
Finding love is something many of us hope for, including people with disabilities. Lauren and Scott met while they were participating in programming at Favarh (The Arc of Farmington Valley). At first, they forged a strong friendship, but as the years went on, their relationship eventually blossomed into love. Now married and living together independently, Scott and Lauren spoke with us about their journey and what it means to make their own choices in love and life.
LAUREN: At the age of 21, I was looking for a job and found Favarh. I was nervous and didn't know who was going to be friends with me, but I met Scott on my first day. It was amazing because that night, I went home and I'm like "Wow, I think I made an actual friend." When I met him, he stuck by me all the time. He worked with me for two years at Michaels, he and I had a very good friendship. Everyone kind of knew that he and I would end up together. I was coming out of a relationship, and I tried some dating sites. I thought it would be really nice to see what it was like.
SCOTT: I actually talked her out of it.
LAUREN: He told me I shouldn't be on apps, and said "Let me be the one to be with you."
SCOTT: I didn't want anything to happen to her.
LAUREN: I was nervous because I didn't know what to expect out of him. I had been verbally abused before. But I knew at some point, there's got to be someone out there who can put me first and for me to rely on.
How did you start dating?
LAUREN: No one asked the other person out, we just did it together. We just wanted to go out and see what it's like.
When Scott and I became boyfriend and girlfriend, my grandfather passed away. It was really weird timing. For me to have Scott was really special. He was the first one I messaged after I heard the news because I knew I needed my friend to understand what was happening. He came to my grandfather's memorial service and it was so nice.
My mom and dad were very supportive of us going out and took us places (and still do.) It kind of amazes me now to look back and say wow, this really has happened. He's been by my side throughout the years. It's really hard to be away from him.
We got engaged in 2019. He needed my parents' permission to propose to me. Scott and my dad went out to dinner together, and then Scott and my mom separately.
Why was getting married important to you and what does it mean to you? What was the process like?
LAUREN: Just the fact that we're together. Knowing he's there for me is a great feeling.
SCOTT: Relying on each other more. Knowing that we're husband and wife.
LAUREN: Planning our wedding was a long process. It was the hottest day of the year that we got married. He wanted to wear his bathing suit under his wedding suit! It was really hard trying to figure out everything. I have a brother who lives in Denver, and he was a part of the wedding. We had two best men and I had junior bridesmaid and a maid of honor. It was really fun having my hair and makeup done. Everyone in the wedding sat with us at our table. All I wanted was a small wedding, but we ended up inviting a lot of people!
Why is being independent important to you?
LAUREN: It's important to be in control of what you do.
SCOTT: Our parents don't come over as much as they used to. We've had less staffing, they've backed off of helping us. They used to observe and supervise us cooking dinner, they used to check on us a lot more. Now we do our own laundry, plan our days, go to work, and come home. It is awesome to be able to stay up as late as we want. And now [living together], we don't have to message each other – we don't have to use our phones as much because we're right here.
What do you see for your future?
LAUREN: Having kids—someday being parents. I can see us having independent jobs.
SCOTT: Growing old together. I can't imagine being with anyone else.
The post From Friendship to Lifelong Love: Scott and Lauren's Journey appeared first on The Arc.
Get Ready for Medicaid Renewals in 2023
Statement on the Killing of Tyre Nichols
A Recommitment to Care with the Introduction of the Better Care Better Jobs Act

redpajama_set_name: RedPajamaCommonCrawl


Over the past 10 years, TCAH's Green Teens' program has trained and inspired hundreds of young leaders. Interns that participate in our program are not only taught urban farming, they also learn leadership skills and team building. Our Green Teens often receive mentoring and support to become certified to prepare tax returns to work as paid assistants in the free tax prep service we offer. Green Teens may also work as a sous chef in one of our many nutrition education courses or cooking demonstrations. Lastly, we encourage all Green Teens to attend food justice workshops or events to bring their personal passion for farming, food justice, and improved nutrition back to their own communities. The skills learned at our comprehensive Urban Agro-Ecological Centers are invaluable and prepare them for the next phase in their lives and careers. This page highlights the accomplishments of our interns past and present.
Graduated from Hunter College 2016, majored in Sociology.
Graduated from Laguardia Community College in 2017, majored in Mas Communication.
Now works for Heritage Human Services.
Graduated from Hunter College in 2017, majored in Accounting.
Currently enrolled in Medgar Evers College, intended degree Occupational Therapy.
Now works for Phoenix Rehabilitation Center.
Currently enrolled in Medgar Evers College, Early Childhood Ed.
Now works as a Teacher's Assistant at a private school.
© Copyright The Campaign Against Hunger. All Rights Reserved.


text:
How I Met Your Mother S6E9 season 6 episode 9 - Glitter
Added by himym.org
Watch online series How I Met Your Mother season 6 episode 9 S6E9 - Glitter 2010 with English subtitles
While Barney finds a new video from Robins Canadian teen star past, Lily fears that a baby could come between her friendship with Robin. Meanwhile, Teds obnoxious old friend, "Punchy," comes from Cleveland for a visit.
Director : Pamela Fryman
IMDB rating : 8.6
Released : 15 Nov 2010
Info IMDB click here
Stars : Josh Radnor, Jason Segel, Cobie Smulders, Neil Patrick Harris
Writer : Carter Bays (created by), Craig Thomas (created by), Kourtney Kang, Matt Kuhn (executive story editor)
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Watch online series How I Met Your Mother season 6 episode 9 S6E9 - Glitter 2010 with English subtitles While Barney finds a new video from Robins Can...


Detroit Tigers and Red Wings owner Michael Ilitch recently passed at age 87. He may be gone physically from us, but his legacy and works will keep him in sports lore.
In 2011 ESPN the Magazine released its annual "Best in Sports" rankings and when it came to pro franchise owners, Ilitch was ranked #1. The rankings were reportedly based on honesty, commitment to the community and loyalty to core players.
As I think of great owners in sports, two men quickly come to mind. When I was snotty-noised journalist trying to come up in L.A., surprisingly to me Al Davis (Oakland & Los Angeles Raiders) and Dr. Jerry Buss (Los Angeles Lakers) became my personal advocates and helped me gain traction in the industry, by ensuring I got inclusion, as I was transitioning from engineering to a journalism career.
Any talk of great owners in sports should yield George Halas, Chicago Bears (1920-1983); Ted Turner, Atlanta Braves (1976-2007); George Steinbrenner, New York Yankees (1973-2010); Walter O'Malley, Brooklyn/L.A. Dodgers (1944-1979); Robert Kraft, New England Patriots (1994-present); Pittsburgh Steelers owners Art Rooney and son (1933-present), Dan, who took over in 2003 and was the linchpin for the "Rooney Rule, and finally my guys Buss and Davis.
All these men were winners and understood continuity, their athletes and each sieged the moment in front them.
To Ilitch's credit he tried to siege the moment in front of him during his ownership, not sparing anything, including money, to put both his Tigers and Red Wings in positions to win.
But what puts him up in the top rung of owners is his commitment to the community and the actions he took to make Detroit a better city.
A few years back I wrote a column beseeching, cajoling Detroit's movers and shakers to seek to make the Motor City a national sports entertainment district.
In my 29-years as a journalist, one of the most amazing transformations I have seen is the cities of San Antonio and Indianapolis. Both were little towns with nothing going on in their downtown, each had minimal restaurants, hotels and entertainments venues.
However both San Antonio and Indianapolis built basketball and football venues and all the hotels and entertainment facilities soon followed.
I think Ilitch saw what I saw in those two smaller cities that were both seeking to define themselves. Each city recognized and acknowledged the walkable sports entertainment future direction and took massive steps to regenerate themselves in that genre.
Building basketball, football and baseball facilities in downtown have worked for both San Antonio and Indianapolis and each city have seen their downtown explode.
Unfortunately Ilitch did not live to see his Little Caesars Arena open. But he can take refuge in the fact that unlike San Antonio and Indianapolis, Motown has all four professional sports franchises, and, they are in walking distance from each other.
Ilitch in his plans for the Detroit District has committed to not only a new basketball and hockey arena, but the redevelopment plans include a posh hotel, medical center, retail shops and residential housing.
Indeed Ilitch believed that a sports and walkable entertainment district would collectively help Detroit's renovation and place the city in rotation for hosting the mega-sporting events like the Final Four, Super Bowl, NBA All-Star Weekend and many other sports and entertainment events.
Maybe, just maybe the addition of Little Caesars Arena could be the linchpin that will even further thrust Detroit into the sports entertainment rotation of major sporting events.
The Motor City is the only cold weather city to host two Super Bowls, and, it has also recently been host to the Final Four and Major League Baseball All-Star Game. With the new arena the NBA and NHL All-Star Games are sure to make their way here.
Ilitch's commitment to Detroit is noteworthy. Not only has it been sports, but he showed where his heart is attached when he and his wife, Marian, took a chance on the historic but neglected Fox Theatre when they purchased it in 1987. They restored the 5,000-seat theatre built in 1928 to its original splendor.
It did not stop there as the Ilitches commissioned an extraordinary renovation of the adjacent 10-story Fox office building in 1989, relocating its suburban offices staff, and established a world headquarters for their Pizza Company and Olympia Entertainment, Inc. in the transformed office building.
Since then, the theatre district has seen a rebirth marked by the opening of other restored theatres and new restaurants, and the building of two side-by-side stadiums for the Detroit Tigers and Lions.
In all the Ilitches own Little Caesar Pizza, Olympia Entertainment, Olympia Development, Champion Foods, Blue Line Foodservice Distribution, Little Caesars Pizza Kit Fundraising Program and the Detroit Red Wings, Detroit Tigers, as well as, the Motor City Casino.
Mike Ilitch may have transitioned to the afterlife, but his life's works here in Detroit will live on and on and on and on.
ANN ARBOR – As has been the case all season with the Michigan Wolverines, they put up a gallant fight but came up short.
Again, before 113,511 in the storied Michigan "Big House" Stadium, the Wolverines put up a fight until the last few seconds of the game, before succumbing to the No. 3 ranked and still undefeated Ohio State Buckeyes, 42-41.
Talking to many of my friends in the press box just before the game commenced, most were speculating that Ohio State, a 16-point favorite, would blow Michigan out.
And who could protest, squabble or argue with that assumption, based on the way the Wolverines have played the past seven weeks.
The 2013 season started with so much hope with quarterback Devin Gardner starting his first full season at the helm of this Michigan team. Living up to those expectations, the Wolverines started out 5-0.
However, after a four overtime loss in a game they should have won, UM has descended downward. After the Penn State heart breaker, over the next seven games Michigan forgot how to block for the run, protect for their quarterback (Gardner), and, defend against the run defensively.
That Penn State loss has been followed with a beat-down by Michigan State, two close very winnable games against Nebraska and Iowa, and now this.
Coach Hoke, Gardner and the entire Michigan team all said that this was enough, and, collectively they went after a win on Saturday. After trailing 35-21 in the fourth quarter, the Wolverines showed some fighting spirit and grit. Coming back time after time, eventually tying the game at 35-35 with a little more than five minutes left in the contest, only to see OSU fly down the field to take a 42-35 lead with 2:20 left in the game.
For Gardner and his teammates this was an opportunity to arrest all the football demons that have been causing them grief. With purpose and determination the Wolverines sat sail on an 11-play, 84-yard drive that gave them an opportunity to tie the game.
Indeed in was the best thing, the only problem was the play UM called in their most crucial possession of the season.
So Michigan went for the 2-point conversion and Ohio State (12-0, 8-0) remains undefeated. With the win the Buckeyes achieved their 24th consecutive victory and keep their national championship hopes alive.
Gardner put in as good an effort as any UM quarterback versus OSU. He completed 32 of 45 passes for 451 yards and four TDs, connecting nine times for 175 yards and a score to Jeremy Gallon, and he ran for a 1-yard touchdown that gave Michigan the first lead in the shootout that went to halftime tied at 21.
The near Michigan upset almost gave the Wolverines (7-5, 3-5) their biggest upset in one of America's greatest rivalries since legendary coach Bo Schembechler's first team at Michigan beat what Woody Hayes noted was his best Buckeyes squad ever.
The loss was yet another in Gardner's first tenure at the helm of the Wolverines. He has showed that he is a skilled athlete and field leader, but the problems on the offensive line and in the running game knocked him off the pedestal as a premiere quarterback in the country.
Proposed Ilitch Downtown Detroit arena could be linchpin for area.
Ford Field hosted Super Bowl and Comerica hosted MLB All-Star Weekend.
Wake up Detroit movers and shakers! This is a crucial time that calls for bold moves and long-term vision, not only for Detroit, Michigan's largest city, but the entire state.
Now that the Ilitch organization has finally put it out on the table their vision for building a new entertainment district downtown anchored by a multipurpose arena that would be home to the Ilitch-owned Red Wings, and hopefully the Pistons, I say make it happen with the quickness.
As Detroit continues to dig itself out of the economic disaster of 2007 that sent the city, state, country and the auto industry on a precarious and uncertain future, this proposed venue would give the Motor City an enormous shot in the arm.
All of Detroit leaders need to look at transformation Indianapolis and San Antonio undergone. Two smaller cities that were both seeking to define themselves. Each city recognized and acknowledged the future and regenerated themselves as sports entertainment venues.
Indeed sports and walkable entertainment collectively is the new model to ensure a city's rotation in hosting the mega-sporting events like the Final Four, Super Bowl, NBA All-Star Weekend and many other sports and entertainment events.
Sure there are those that will Detroit's finances, the many vacant homes, and the continued Urban American homicides. Distracters will talk about the neighborhoods needing special attention and rightfully so. However, the dynamics of inner cities in America is a national problem of economics, employment, shifting population, and old infrastructure.
But one problem is no reason to hold up another potential uplift. If indeed the proposed multipurpose arena is commenced, it would not only host hockey and basketball, it would host a range of shows, concerts and other events, while the broader district would include residential housing, retail shopping, office space and more.
What more needs to be said? The City Council, Mayor's Office, and state government needs to all get on board and help turn this vision into a reality.
I have been to both San Antonio and Indianapolis and seen how the new model of building all their sports venues in a walkable proximity. In conjunction with the arenas and stadiums hotels, eateries and housing have evolved.
Take the Los Angeles Staples Center for example. I was in LA when the developers started building the arena and many said who will perform there and that it was a waste of money and resources.
Well, the Lakers and Clippers and Kings after seeing the venue quickly abandon their arenas. The Staples Center has galvanized a three block district called LA Live that has clubs, restaurants, theaters and hotels.
The LA downtown area before the Staples Center and LA Live was built was a waste land of poverty.
A number of cities have shown us how a city came use the sports entertainment model to regalvanize a downtown and city.
An Ilitch family's Olympia Development news release quoted George W. Jackson Jr., the city's top development official and president and CEO of the Detroit Economic Growth Corp., as saying the plan "makes good" business sense.
The Ilitch organization pegged the probable price tag at $650 million. Legislation introduced in Lansing would create a new "catalyst development project" that could benefit from support from the Michigan Strategic Fund and also from the use of Downtown Development Authority tax revenues that support projects in the central business district.
I too had this vision. So let's keep it moving Detroit.
Detroiter's stock rising in NFL circles.
EAST LANSING – With ESPN College GameDay crew creating the hype on Michigan State University's campus and an ESPN national television audience looking on, former Detroit Public School League (PSL) star Johnathan Hankins and his Ohio State Buckeyes outlasted MSU at Spartan Stadium, 17-16 in the Big Ten opener for both teams.
As I headed to the post-game interview area, I ran across former Detroit Southeastern High coach Donshell English. He had arguably two of the best defensive linemen in PSL history on a team that finished 11-1. Both Hankins and MSU's William Gholston made All-State and were on the Michigan Chronicle ALL-PSL team. The 6-foot-7, 278 pound Gholston left the PSL the No. 1 ranked player in the state and the 6-foot3, 322 pound Hankins was the 20th.
Now Hankins and Gholston, both juniors, are ranked as the best two defensive players in the Big Ten and both are pre-season All-America picks. Recently Big Ten Network NFL draft expert Russ Lande's weekly Big Board, flip-flopped Gholston and Hankins, making the Ohio State defensive tackle Hankins the No. 1 player in the conference, with Gholston No. 2.
English said he is not surprised that both his star players at Southeastern have continued to grow as students and athletes.
About Hankins English recalled: "Against MSU he was a force from the very first play of the game controlling the middle. You can tell the way his coach (Urban Meyer) and his teammates respond to him, he is a team leader.
With Gholston being an early commit with the Spartans, and Archie Collins, Southeastern's defensive coordinator during those seasons, a graduate assistant with Michigan State, it would seem Hankins and Gholston would have ended up together in East Lansing.
Hankins said that although the Buckeyes are on probation and will not be able to play for a Big Ten or national title, Meyer still has his team motivated and striving to be the best they can be.
When Meyer arrived this season, Hankins quickly showed him he was the type of athletic defensive tackle that could be the anchor of his defense. He is fast becoming noted around the country for his running back feet, a big smile and a nose for the ball that proves he is more than just a big body.
ANN ARBOR – Obviously this was litmus test for the University Michigan Wolverines, as they undertook their first home game of the 2102 season versus the running instead of flying Air Force Falcons.
Before 112,522 in the Big House the only thing that took flight for the Falcons was the jet flyover that took place after the nation anthem was concluded.
These groups of flyboys really are as run happy a team as there is in college football. Air Force grounded out 400 yards in its first game and pushed the Wolverines around enough to the tune of 290 rushing yards.
Still the hungry Wolverines hung on and chewed up the Falcons through the air and on the ground for a hard fought 31-25 victory. Leading the way was Wolverine quarterback Denard Robinson, who combined for 426 yards (218 rush, 208 pass), accounting for all but seven of Michigan's entire offensive output.
Helping Robinson out were junior quarterback-turned-receiver Devin Gardner and freshman tight end Devin Funchess.
Funchess had four catches for 106 yards and a noteworthy 30-yard touchdown where he out-jumped a defender. He was the first Michigan tight end to top 100 yards since Jerame Tuman had 126 against Colorado in 1997.
However, Gardner was the most comfortable target for the second straight week, catching five passes for 63 yards, including a precision route for his score. Against Alabama Gardner latched onto a 44-yard touchdown pass, showing all his potential.
Last week the Wolverines took on defending national champion and then second-ranked Alabama. The Crimson Tide put such a beating on the Wolverines that they were elevated to the top-ranked team in the country while the Wolverines sunk from No. 8 to 19.
During the fall of 2009, Gardner was rated as the No. 1 dual-threat high school quarterback in the United States by Rivals.com and the No. 5 quarterback by ESPNU. He verbally committed to the University of Michigan's 2010 recruiting class in the spring of 2009 and enrolled at the University of Michigan in January 2010.
Much to his chagrin Robinson was entrenched as the Michigan quarterback and during spring practice Gardner made the switch to receiver.
The question I asked Gardner is he now a receiver or a quarterback?
As for now Gardner is turning into the most stable receiver on the UM team. They play Massachusetts this Saturday at the Big House and it is an important test for keeping this team on track.
Gardner, 6-foot-4, is proving to be a key clog in the Wolverines passing game.


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Kona Cayenne is a warm, spicy dark reddish-orange hue that looks great with Brick and Terracotta and is a fantastic complement for Kona Blue or Kona Pickle.
Kona solids are manufactured by Robert Kaufman Fabrics.
Yardage is sold in 1/2 metre cuts. Fat quarters measure approx. 19.5" x 21". Please email info@studio39fabrics.com to inquire about availability.


text:
Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae
Maria Fernanda C. Bueno, Gabriela R. Francisco, Jessica A. O'Hara, Doroti De Oliveira Garcia, Yohei Doi
https://doi.org/10.1128/AAC.02545-15
Published - 01-2016
Methyltransferases
Klebsiella pneumoniae
beta-Lactamases
beta-lactamase CTX-2
Bueno, M. F. C., Francisco, G. R., O'Hara, J. A., De Oliveira Garcia, D., & Doi, Y. (2016). Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae. Antimicrobial agents and chemotherapy, 60(1). https://doi.org/10.1128/AAC.02545-15
Bueno, Maria Fernanda C. ; Francisco, Gabriela R. ; O'Hara, Jessica A. ; De Oliveira Garcia, Doroti ; Doi, Yohei. / Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae. In: Antimicrobial agents and chemotherapy. 2016 ; Vol. 60, No. 1.
@article{d2bdce2160d44d9eabc65d32c3a41140,
title = "Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae",
author = "Bueno, {Maria Fernanda C.} and Francisco, {Gabriela R.} and O'Hara, {Jessica A.} and {De Oliveira Garcia}, Doroti and Yohei Doi",
doi = "10.1128/AAC.02545-15",
journal = "Antimicrobial Agents and Chemotherapy",
Bueno, MFC, Francisco, GR, O'Hara, JA, De Oliveira Garcia, D & Doi, Y 2016, 'Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae', Antimicrobial agents and chemotherapy, vol. 60, no. 1. https://doi.org/10.1128/AAC.02545-15
Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae. / Bueno, Maria Fernanda C.; Francisco, Gabriela R.; O'Hara, Jessica A.; De Oliveira Garcia, Doroti; Doi, Yohei.
In: Antimicrobial agents and chemotherapy, Vol. 60, No. 1, 01.2016.
T1 - Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae
AU - Bueno, Maria Fernanda C.
AU - Francisco, Gabriela R.
AU - O'Hara, Jessica A.
AU - De Oliveira Garcia, Doroti
AU - Doi, Yohei
U2 - 10.1128/AAC.02545-15
DO - 10.1128/AAC.02545-15
JO - Antimicrobial Agents and Chemotherapy
JF - Antimicrobial Agents and Chemotherapy
Bueno MFC, Francisco GR, O'Hara JA, De Oliveira Garcia D, Doi Y. Erratum for Bueno et al., Coproduction of 16S rRNA Methyltransferase RmtD or RmtG with KPC-2 and CTX-M Group Extended-Spectrum β-Lactamases in Klebsiella pneumoniae. Antimicrobial agents and chemotherapy. 2016 Jan;60(1). https://doi.org/10.1128/AAC.02545-15
10.1128/AAC.02545-15


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If after too many New York late nights and with some help from DFA's James Murphy and Tim Goldsworthy you tried to re-create the synth pop and new wave that scored some teenager's summer of '84, the result might sound a lot like a Holy Ghost! track. In this afternoon's DJ set, though, Holy Ghost! are likely to shake off some of their own aloof nostalgia with an enervating dose of contemporary disco house. Fellow local duo Blondes and their tableful of equipment disperse a more cosmic version of disco. Theirs is a mind-clearing, palate-cleansing sound perfect for spending a Sunday afternoon forgetting the week to come. This is the first of a series of summer parties that promoters JDH and Dave P (Fixed) and Paul Raffaele (the Dog & Pony Show) are throwing in Dekalb Market, the shipping-container playground and food court open mini mall off Flatbush Avenue.