text:
\section{Introduction}

In the turbid rivers of Africa and South America, some species of
fish generate a stable, high frequency, weak electric field
(0.1-10 kHz, $\leq 100$ mV/cm)  which is not enough for defense
purpose. In 1958, Lissmann and Machin \cite{lissmann1958mechanism}
discovered that the emitted electrical signal is in fact used for
active electro-sensing. The weakly electric fish have thousands of
receptor organs at the surface of their skin. A nearby target with
different admittivity than the surrounding water is detected by
measurements of the electric organ discharge modulations at the
receptor organs \cite{bastian, moller1995}. Targets with large
permittivity cause appreciable phase shifts, which can be measured
by receptors called T-type units \cite{book_nelson}. It is an
important input for the fish, and thus it will be the central
point in this paper for shape classification.

Active electro-sensing has driven an increasing number of
experimental, behavioral, biological, and computational studies
since Lissmann and Machin's work
\cite{assad1997electric,babineau2006modeling, budelli2000electric,
chen2005modeling,maciver2001computational,maciver2001prey,rasnow1989simulation,von1993electric}.
Behavioral experiments have shown that weakly electric fish are
able to locate a target \cite{von1993electric} and discriminate
between targets with different shapes \cite{von2007distance}
or/and electric parameters (conductivity and permittivity)
\cite{von1999active}. The growing interest in electro-sensing
could be explained not only by the curiosity of discovering a
``sixth sense'', the electric perception, that is not accessible
by our own senses, but also by potential bio-inspired applications
in underwater robotics. It is challenging to equip robots with
electric perception and provide them, by mimicking weakly electric
fish, with imaging and classification capabilities in dark or
turbid environments \cite{mciver3, boyer, boyer3, boyer2, mciver2,
mciver, von1999active}.

Mathematically speaking, the problem is to locate the target and
identify its shape and material parameters given the current
distribution over the skin.  Due to the fundamental ill-posedness
character of this imaging problem, it is very intriguing to see
how much information weakly electric fish are  able to recover.
The electric field due to the target is a complicated highly
nonlinear function of its shape, admittivity, and distance from
the fish. Thus, understanding analytically this electric sense is
likely to give us insights in this regard \cite{
assad1997electric,babineau2006modeling, boyer4,
budelli2000electric, maciver2001computational,boyer2,
von2007distance}. While locating targets from the electric field
perturbations induced on the skin of the fish is now understood
(see \cite{electroloc, boyer3} and references therein),
identifying and classifying their shapes is considered to be one
of the most challenging problems in electro-sensing. Although the
neuroethology of these fish has significantly been advanced last
years (see \cite{neuro} and references therein), the neural
mechanisms encoding the shape of a target is far beyond the scope
of our study. Rather, this work focuses on the physical
feasibility of such a process, which was not explained until now.

In \cite{electroloc}, a rigorous model for the electro-location of
a target around the fish was derived. Using the fact that the
electric current produced by the electric organ is time-harmonic
with a known fundamental frequency, a space-frequency location
search algorithm was introduced. Its robustness with respect to
measurement noise and its sensitivity with respect to the number
of frequencies, the number of sensors, and the distance to the
target were illustrated. In the case of disk- and ellipse-shaped
targets,  the conductivity, the permittivity, and the size of the
targets can be reconstructed separately from multifrequency
measurements. Such measurements have been used successfully in
trans-admittance scanners of breast tumors \cite{seo1,seo2,
scholz2002towards}.

In this paper, we tackle the challenging problem of shape
recognition and classification. In order to explain how the shape
information is encoded in measured data, we first derive a
multipolar expansion for the perturbations of the electric field
induced by a nearby target in terms of the characteristic size of
the target. Our asymptotic expansion generalizes Rasnow's formula
\cite{rasnowformula} in two directions: (i) it is a higher-order
approximation of the effect of a nearby target and it is valid for
an arbitrary shape and admittivity contrast and (ii) it takes also
into account the fish's body. As it has been first proved in
\cite{electroloc}, by postprocessing the measured data using layer
potentials associated only to the fish's body, one can reduce the
multipolar formula to the one in free space, {\it i.e.}, without
the fish. Then we show how to identify and classify a target,
knowing by advance that the latter belongs to a dictionary of
pre-computed shapes. The shapes considered in this paper have been
experimentally tested and results reported in \cite{gerhard}. This
idea comes naturally in mind when modeling behavioral experiments
such as in \cite{von1999active,von2007distance,von1993electric}.
The pre-computed shapes would then be a model for the fish's
memory (trained to recognize specific shapes), and the experience
of recognition presented here would simulate the discrimination
exercises that are then imposed to them. We develop two algorithms
for shape classification: one based on shape descriptors while the
second is based on spectral induced polarizations. We first
extract, from the data, generalized (or high-order) polarization
tensors of the target (GPTs) \cite{dico}. These tensors, first
introduced in \cite{ammarisima02},  are intrinsic geometric
quantities and constitute the right class of features to represent
the target shapes \cite{AGKLY11, ammarikang2007GPT}. The shape
features are encoded in the high-order polarization tensors. The
extraction of the GPTs can be achieved by a least-squares method.
The noise level in the reconstructed generalized polarization
tensors depends on the angle of view. Larger is the angle of view,
more stable is the reconstruction. $l_1$-regularization techniques
could be used.  Then we compute from the extracted features
invariants under rigid motions and scaling. Comparing these
invariants with those in a dictionary of pre-computed shapes, we
successfully classify the target. Since the measurements are taken
at multiple frequencies, we make use of the spectral content of
the generalized polarization tensor in order to dramatically
improve the stability with respect to measurement noise of the
physics-based classification procedure. In fact, we show
numerically that the first-order polarization tensor at multiple
frequencies is enough for the purpose of classification.

\section{Feature extraction from induced current measurements}

\subsection{Electro-sensing model}
Let us recall the nondimensionalized model of electro-sensing: the
body of the fish is $\Omega$ (of size of order $1$), an open
bounded set in $\mathbb{R}^d, d=2,3$, of class
$\mathcal{C}^{1,\alpha}$, $0<\alpha<1$, with outward normal unit
vector $\nu$. The electric organ is a dipole $f$ placed at
$z_{0}\in\Omega$ or a sum of point sources inside $\Omega$
satisfying the neutrality condition. We refer to \cite{electroloc}
where the equations are nondimensionalized and the different
scales are identified. The fish's skin is very thin and highly
resistive. Its effective thickness, that is, the skin thickness
times the contrast between the water and the skin conductivities,
is denoted by $\xi$ and is of order $10^{-1}$
\cite{assad1997electric}. We assume that the conductivity of the
background medium is $1$ and that its permittivity is vanishing.
Consider a target $D=z+\delta B$, where $\delta \ll 1$ is the
characteristic size of $D$, $z$ is its location, and $B$ a smooth
bounded domain containing the origin. We assume that $D$ is of
complex admittivity $k=\sigma+i\varepsilon\omega$, with $\omega$
being the operating frequency in the range $[1,10]$ and $\sigma$
and $\varepsilon$ being respectively the conductivity and the
 permittivity of the target. It has been also shown in~\cite{electroloc} that, in the
presence of $D$,
 the electric
field $u$ generated by the fish is the solution of the following
system:
\begin{equation}
\left\{ \begin{alignedat}{2}\Delta u & =f & \,\, \mbox{in } \Omega,\\
\nabla\cdot(1+(k-1)\chi_{D})\nabla u & =0 & \,\,   \mbox{in } \mathbb{R}^{d}\setminus\overline{\Omega},\\
\left.\frac{\partial u}{\partial\nu}\right|_{-} & =0 & \,\, \mbox{on } \partial\Omega,\\
{} u |_+ - u|_- & =\xi\left.\frac{\partial u}{\partial\nu}\right|_{+} & \,\, \mbox{on } \partial\Omega\\
\left|u(x)\right| & =O(\vert x\vert^{-d+1}), &
\,\,\left|x\right|\rightarrow\infty.
\end{alignedat}
\right.\label{eq:system-u}
\end{equation}
Here, $\chi_{D}$ is the characteristic function of $D$.
Fig.~\ref{figmodel} shows
 isopotentials with and without a target with zero permittivity but different conductivity from the surrounding
medium. Note that if the target's admittivity depends on the
frequency ({\it i.e.}, if the permittivity is nonzero), then a
phase shift in the electrical potential is induced.

\begin{figure}[!h]
\centering%
\begin{tabular}{cc}
\includegraphics[width=4cm]{fish_background.pdf} & \includegraphics[width=4cm]{fish.pdf}\tabularnewline
(a) & (b)\tabularnewline
\end{tabular}
\caption{\label{figmodel}Isopotentials without (a) and with (b) a
target with $\sigma=5$ and $\varepsilon=0$.}
\end{figure}

In a previous study~\cite{dico}, we have extracted the GPTs of a
target by multistatic measurements using arrays of sources and
receptors. These GPTs were then arranged and compared to a
dictionary of already known shapes. This study aims at adapting
this method to the electro-sensing problem.

\subsection{Asymptotic formalism} \label{sec:reconstruction-GPT}

The first step is to compute the GPTs from the measurements. In
this regard, the next result will be useful. Except when
mentioned, we will fix in this section the frequency~$\omega$,
leading to a fixed complex admittivity~$k$. We will need the
following notation. For a multi-index $\alpha \in \mathbb{N}^d$,
let $x^\alpha = x_1^{\alpha_1} \ldots x_d^{\alpha_d},
\partial^\alpha = \partial_1^{\alpha_1} \ldots
\partial_d^{\alpha_d},$ with $\partial_j = \partial /
\partial x_j$. Let $G(x)$ be the Green function of the Laplacian
in $\mathbb{R}^d$ which satisfies $\Delta G =\delta$ (where
$\delta$ is the Dirac function at the origin) and is given by
$$
G(x) = \left\{
\begin{array}{ll}
\displaystyle \frac{1}{2\pi} \ln |x|, \quad & d=2 , \\
\noalign{\medskip} \displaystyle -\frac{1}{4\pi} \frac{1}{|x|}, \quad & d=3.
\end{array}
\right.
$$
We denote the single and double layer potentials of a function
$\phi \in L^2(\partial \Omega)$ as $\mathcal{S}_\Omega[\phi]$ and
$\mathcal{D}_\Omega[\phi]$, where
\begin{align} \label{defs}
\mathcal{S}_\Omega[\phi](x) &:= \int_{\partial \Omega} G(x-y) \phi
(y) \, d \sigma(y), \quad x \in \mathbb{R}^d,
\end{align}
and
\begin{equation} \label{defd}
\mathcal{D}_\Omega[\phi](x) := \int_{\partial \Omega}
\frac{\partial G}{\partial  {\nu(y)}} (x-y) \phi (y) \, d
\sigma(y), \quad x \in \mathbb{R}^d \setminus
\partial \Omega.
\end{equation}
 We also define the  boundary integral operator $\mathcal{K}^*_\Omega$ on
$L^2(\partial \Omega)$ by
\begin{equation} \label{defk}
\mathcal{K}_\Omega^*[\phi](x) := \int_{\partial \Omega} \frac
{\partial G}{\partial {\nu(x)}}(x-y) \phi (y)\,d\sigma(y), \quad
\phi \in L^2(\partial \Omega).
 \end{equation}
The operator $\mathcal{K}^*_\Omega$ is called the
Neumann-Poincar\'e operator. We assume that the target is away
from the fish, {\it i.e.}, the distance between the fish and the
target is much larger than the target's characteristic  size but
smaller than the range of the electrolocation which does not
exceed two fish's body lengths. The following theorem holds.
\begin{thm}
\label{thm:asymptotic-formula}Let us define the function $H:\mathbb{R}^{d}\rightarrow\mathbb{C}$
by
\begin{equation}
H(x)=p(x)+\mathcal{S}_{\Omega}\left[\left.\frac{\partial
u}{\partial\nu}\right|_{+}\right]-\xi\mathcal{D}_{\Omega}\left[\left.\frac{\partial
u}{\partial\nu}\right|_{+}\right],\label{eq:H-def}
\end{equation}
where $p$ is the field created by the dipole $f$, {\it i.e.},
$\Delta p=f$ in $\mathbb{R}^{d}$. Then, for every integer
$K\geq1$, the following expansion holds
\begin{equation} \begin{array}{l}
\displaystyle
u(x)=H(x)+\delta^{d-2}\sum_{\left|\alpha\right|=1}^{K}\sum_{\left|\beta\right|=1}^{K-\left|\alpha\right|+1}
\frac{(-1)^{\vert\beta\vert}\delta^{\left|\alpha\right|+\left|\beta\right|}}{\alpha!\beta!}
\partial^{\alpha}H(z) \\ \noalign{\medskip} \qquad \qquad  \displaystyle \times M_{\alpha\beta}(\lambda,B)\partial^{\beta}G(x-z)+O(\delta^{d+K}),
\end{array}\label{eq:asymptotic formula}
\end{equation}
uniformly for $x \in \partial \Omega$, where $z$ is the location
of the target $D$ and
$$M_{\alpha\beta}(\lambda,B):= \int_{\partial B} (\lambda
I-\mathcal{K}_{B}^{*})^{-1}[\frac{\partial y^\alpha}{\partial
\nu}] y^\beta \, d\sigma(y)$$ is the generalized polarization
tensor (of order $(\alpha,\beta)$) associated with the domain $B$
and the contrast $\lambda=(k+1)/2(k-1)$; see
\cite{ammarikang2007GPT}. Here, $\mathcal{K}_{B}^{*}$ is the
Neumann-Poincar\'e operator associated with $B$ and $I$ is the
identity operator.
\end{thm}
Let us make a few remarks. First, the definition of the GPTs still
holds for complex-valued $\lambda$. However, some properties are
lost by this change; thus one has to study them more carefully in
this situation. Second, the function $H$, which is computed from
the boundary measurements,  still depends on $\delta$ but this is
not important for our present study. Indeed, formula
(\ref{eq:asymptotic formula}) could have been derived with $U$,
the background solution in the absence of the target, instead of
$H$ and $G_{R}$ - the Green function associated to Robin
conditions on $\partial\Omega$ - instead of $G$, but it is much
easier to compute $\partial^{\alpha}H(z)$ and
$\partial^{\beta}G(x-z)$ once $z$ is known. This leads us to the
third remark: the location $z$ is supposed to be known from the
algorithm developed in~\cite{electroloc}. Electrolocation
algorithms are either based on a space-frequency approach in the
case of multifrequency measurements or on the fish's movement if
only one frequency is used \cite{electroloc, boyer3}. Finally, it
is worth mentioning that the knowledge of
$M_{\alpha\beta}(\lambda,B)$ for all $\alpha, \beta\in
\mathbb{N}^d$ determines uniquely $B$ and $\lambda$ \cite{mms,
LNM1846}. Moreover, the following scaling relation holds:
$$
M_{\alpha\beta}(\lambda, \delta B) = \delta^{d-2 + |\alpha| +
|\beta|} M_{\alpha\beta}(\lambda,B).
$$

We will follow the proof of \cite[Theorem 4.8]{LNM1846}. In a first
step, let us show the following formula.
\begin{lem}
For $x\in\mathbb{R}^{d}$,
\begin{equation}
u(x)=H(x)+\mathcal{S}_{D}(\lambda
I-\mathcal{K}_{D}^{*})^{-1}\left[\left.\frac{\partial
H}{\partial\nu}\right|_{\partial D}\right](x),\label{eq:u-H}
\end{equation}
where $\nu$ is the outward normal unit vector at $\partial D$ and
$\mathcal{S}_{D}$ and $\mathcal{K}_{D}^{*}$ are respectively
defined by (\ref{defs}) and (\ref{defk}) with $\Omega$ replaced
with $D$.\end{lem}
\begin{proof}
In \cite[Section 4.1.2]{electroloc}, it is shown that
\[
u(x)=p(x)+\mathcal{S}_{\Omega}[\psi](x)+\mathcal{D}_{\Omega}[\phi](x)+\mathcal{S}_{D}[\phi](x),
\]
where the functions $\psi$, $\phi\in L^{2}(\partial\Omega)$ and
$\phi\in L^{2}(\partial D)$ verify the following system
\[
\left\{ \begin{alignedat}{2}\phi & =-\xi\psi & \,\, \mbox{on } \partial\Omega,\\
\left(\frac{I}{2}-\mathcal{K}_{\Omega}^{*}+\xi\frac{\partial\mathcal{D}_{\Omega}}{\partial\nu}\right)[\psi]
-\frac{\partial}{\partial\nu}(\mathcal{S}_{D}[\phi]) & =\frac{\partial p}{\partial\nu} & \,\, \mbox{on } \partial\Omega,\\
-\frac{\partial}{\partial\nu}(\mathcal{S}_{\Omega}[\psi])+\xi\frac{\partial}{\partial\nu}(\mathcal{D}_{\Omega}[\psi])
+(\lambda I-\mathcal{K}_{D}^{*}) & [\phi]=\frac{\partial
p}{\partial\nu} & \,\mbox{on } \partial D.
\end{alignedat}
\right.
\]
The third line gives us
\[
\phi=(\lambda
I-\mathcal{K}_{D}^{*})^{-1}\left[\left.\frac{\partial
H}{\partial\nu}\right|_{\partial D}\right],
\]
and the jump formulas for the single and double layer
potentials~\cite[Theorem 2.17]{ammarikang2007GPT} give us
\[
\psi= \frac{\partial u}{\partial\nu} \bigg|_+ \quad \mbox{ and
}\phi=- u|_+ + u|_-,
\]
so that, from the boundary conditions of the
system~(\ref{eq:system-u}), we obtain
$p+\mathcal{S}_{\Omega}[\psi]+\mathcal{D}_{\Omega}[\phi]=H$ and
the lemma is proved.
\end{proof}

We can now prove Theorem~\ref{thm:asymptotic-formula}, using the
arguments in \cite[pp. 72-73]{LNM1846}. Starting with formula
(\ref{eq:u-H}), the proof relies on a Taylor expansion of $H$ and
the Green function involved in the single layer potential. Indeed,
denoting
\[
H_{K}(x)=\sum_{\left|\alpha\right|=0}^{K}\frac{1}{\alpha!}\partial^{\alpha}H(z)(x-z)^{\alpha},
\]
a Taylor expansion gives us
\[
\left\Vert \frac{\partial H}{\partial\nu}-
\frac{\partial H_{K}}{\partial\nu}\right\Vert _{L^{2}(\partial D)}\leq C\delta^{K}\left|\partial D\right|^{1/2},
\]
and from \cite[Formula (4.10)]{LNM1846}, we have for any $h\in
L^2(\partial D)$ such that $\int_{\partial D}h=0$:
$$
\forall x\in\partial\Omega,\,\,\left|\mathcal{S}_{D}(\lambda
I-\mathcal{K}_{D}^{*})^{-1}[h](x)\right|\leq C\delta
\left|\partial D\right|^{1/2} \left\Vert h\right\Vert
_{L^{2}(\partial D)}.$$ Hence, using the fact that $\left|\partial
D\right| = \delta^{d-1} \left|\partial B\right|$, we obtain
\[\begin{array}{l}
\displaystyle \left\Vert \mathcal{S}_{D}(\lambda
I-\mathcal{K}_{D}^{*})^{-1}\left[\frac{\partial
H}{\partial\nu}-\frac{\partial
H_{K}}{\partial\nu}\right]\right\Vert
_{L^{\infty}(\partial\Omega)} \\
\noalign{\medskip} \qquad \leq \displaystyle C\delta\left|\partial D\right|^{1/2}\left\Vert
\frac{\partial H}{\partial\nu}-\frac{\partial
H_{K}}{\partial\nu}\right\Vert _{L^{2}(\partial D)} \leq
C\delta^{d+K}.
\end{array}
\]
Plugging this inequality into (\ref{eq:u-H}) enables us to write,
for $x\in\partial\Omega$,
\[
u(x)=H(x)+\mathcal{S}_{D}(\lambda
I-\mathcal{K}_{D}^{*})^{-1}\left[\frac{\partial
H_{K}}{\partial\nu}\right](x)+O(\delta^{d+K}).
\]
By a change of variables $y'=(y-z)/\delta$,  denoting
$\phi_{\alpha}(y')= (\lambda
I-\mathcal{K}_{B}^{*})^{-1}[\nu\cdot\nabla w^{\alpha}](y')$ for
$y'\in\partial B$ (where $\nu$ is here the outward normal unit
vector to $\partial B$), we have (see for example the arguments
in~\cite[Section 3]{AGKLY11})
\[\begin{array}{l}
\displaystyle
u(x)-H(x)=\sum_{\left|\alpha\right|=0}^{K}\frac{1}{\alpha!}\partial^{\alpha}H(z)
\delta^{\left|\alpha\right|+d-2} \\ \noalign{\medskip} \qquad \qquad \displaystyle \times
\int_{\partial B}G(x-z-\delta
y')\phi_{\alpha}(y')d\sigma(y)+O(\delta^{d+K}).
\end{array} \] We can now conclude by injecting a Taylor expansion of
the Green function
\[
G(x-z-\delta y)=\sum_{\left|\beta\right|=0}^{\infty}\frac{(-\delta)^{\left|\beta\right|}}{\beta!}\partial^{\beta}G(x-z)y^{\beta},
\]
in the integrand, giving
\[\begin{array}{l}
\displaystyle
u(x)-H(x)=\delta^{d-2}\sum_{\left|\alpha\right|=0}^{K}\sum_{\left|\beta\right|=0}^{K-\left|
\alpha\right|+1}\frac{(-1)^{\vert\beta\vert}\delta^{\left|\alpha\right|+\left|\beta\right|}}{\alpha!\beta!}
\partial^{\alpha}H(z)\\ \noalign{\medskip} \qquad \qquad \displaystyle \times \partial^{\beta}G(x-z)\int_{\partial
B}y^{\beta}\phi_{\alpha}(y)d\sigma(y)+O(\delta^{d+K}).
\end{array} \] The last term is $M_{\alpha\beta}(\lambda,B)$ by
definition~\cite{ammarikang2007GPT}; it then suffices to show that
the terms with $\left|\alpha\right|=0$ or $\left|\beta\right|=0$
vanish, which is the case because $\int_{\partial
B}\phi_{\alpha}=0$ and $\phi_{\alpha}=0$ if
$\left|\alpha\right|=0$. Thus,
Theorem~\ref{thm:asymptotic-formula} is proved. \qed

\subsection{Data acquisition and reduction}

Let us suppose that the fish is moving, and let us take a sample
of $S\in\mathbb{N}^{*}$ different positions
$\left(\Omega_{s}\right)_{1\leq s\leq S}$. This gives us $2S$
different functions $\left(u_{s}\right)_{1\leq s\leq S}$ and
$\left(H_{s}\right)_{1\leq s\leq S}$, leading us to the following
data matrix
\begin{equation}
\mathbb{Q}:=\left(Q_{sr}\right)_{1\leq s\leq S,1\leq r\leq R}:=
\left(u_{s}(x_{r}^{(s)})-H_{s}(x_{r}^{(s)})\right)_{1\leq s\leq S,1\leq r\leq R},\label{eq:def-Q}
\end{equation}
where $(x_{r}^{(s)}\in\partial\Omega_{s})_{1\leq r\leq R}$ are the
receptors of the fish being in the $s^{\mbox{th}}$ position. The
choices of indices are motivated by the fact that the different
positions play the role of sources.

The goal of this subsection is to simplify this data set in order
to extract the GPTs (precisely, the contracted GPTs as it will be
seen below). Indeed, from~(\ref{eq:asymptotic formula}), one has
\begin{equation}\begin{array}{l}
\displaystyle
Q_{sr}=\delta^{d-2}\sum_{\left|\alpha\right|=1}^{K}\sum_{\left|\beta\right|=1}^{K-\left|\alpha\right|+1}(-1)^{\vert\beta
\vert}\frac{\delta^{\left|\alpha\right|+\left|\beta\right|}}{\alpha!\beta!}\partial^{\alpha}H_{s}(z)
\\ \noalign{\medskip} \qquad \qquad \displaystyle \times  M_{\alpha\beta}(\lambda,B)\partial^{\beta}G(x^{(s)}_{r}-z)+O(\delta^{d+K}).\label{eq:Qsr}
\end{array} \end{equation} As in~\cite{dico}, we will express this formula
in terms of contracted GPTs (CGPTs), when the dimension of the
space is $d=2$. Let us first recall the definitions of these
contracted GPTs. For a target $B$ with contrast ratio $\lambda$,
knowing the GPTs $M_{\alpha\beta}(\lambda,B)$ for all indices
$\alpha$ and $\beta$ such that $\left|\alpha\right|=m$ and
$\left|\beta\right|=n$ leads us to construct the following
combinations, called CGPTs,
\[
\begin{array}{l} \displaystyle M_{mn}^{cc}
=\sum_{|\alpha|=m} \sum_{|\beta|=n }
a_{\alpha}^{m}a_{\beta}^{n}M_{\alpha\beta},\\
\noalign{\medskip}
\displaystyle M_{mn}^{cs}  =\sum_{|\alpha|=m} \sum_{|\beta|=n} a_{\alpha}^{m}b_{\beta}^{n}M_{\alpha\beta},\\
\noalign{\medskip} \displaystyle M_{mn}^{sc}
=\sum_{\left|\alpha\right|=m}\sum_{\left|\beta\right|=n}b_{\alpha}^{m}a_{\beta}^{n}M_{\alpha\beta},\\
\noalign{\medskip} \displaystyle M_{mn}^{ss}
=\sum_{\left|\alpha\right|=m}\sum_{\left|\beta\right|=n}b_{\alpha}^{m}b_{\beta}^{n}M_{\alpha\beta},
\end{array}
\]
where the real numbers $a_{\alpha}^{m}$ and $b_{\beta}^{m}$ are
defined by the following relation
\[
(x_{1}+ix_{2})^{m}=\sum_{\left|\alpha\right|=m}a_{\alpha}^{m}x^{\alpha}+i\sum_{\left|\beta\right|=m}b_{\beta}^{m}x^{\beta}.
\]
In the polar coordinates, $x=r_{x}e^{i\theta_{x}}$, these
coefficients also verify the following characterization
\[
\sum_{\left|\alpha\right|=m}a_{\alpha}^{m}x^{\alpha}=r_{x}^{m}\cos
m\theta_{x} \mbox{ and
}\sum_{\left|\beta\right|=m}b_{\beta}^{m}x^{\beta}=r_{x}^{m}\sin
m\theta_{x}.
\]
This enables us to show~\cite[Appendix A.2]{dico} that
\begin{equation}
\frac{(-1)^{|\alpha|}}{\alpha!}\partial^{\alpha}G(x)=\frac{-1}{2\pi\left|\alpha\right|}\left(a_{\alpha}^{\left|\alpha\right|}\frac{\cos\left|\alpha\right|\theta_{x}}{r_{x}^{\left|\alpha\right|}}+b_{\alpha}^{\left|\alpha\right|}\frac{\sin\left|\alpha\right|\theta_{x}}{r_{x}^{\left|\alpha\right|}}\right).\label{eq:coeff-G}
\end{equation}
From the definition of $H$, and with the help of the previous
formula, one can prove the following lemma.
\begin{lem}
Let the source $f$ be a dipole of moment $\mathbf{p}_{s}$ placed
at $z_{s}$:
\begin{equation} \label{formps}
p_{s}(x)=\mathbf{p}_{s}\cdot\nabla G(x-z_{s}),
\end{equation}
Then, for any $\alpha\in\mathbb{N}^{2}$, there exist two real
numbers $A_{\left|\alpha\right|,s,z}$ and
$B_{\left|\alpha\right|,s,z}$ such that
\[
\frac{1}{\alpha!}\partial^{\alpha}H_{s}(z)=A_{\left|\alpha\right|,s,z}a_{\alpha}^{\left|\alpha\right|}
+B_{\left|\alpha\right|,s,z}b_{\alpha}^{\left|\alpha\right|}.
\]
Moreover, $A_{\left|\alpha\right|,s,z}$ and
$B_{\left|\alpha\right|,s,z}$ can be expressed in the following
way
\[
\begin{array}{lll}A_{m,s,z} & =&\displaystyle \frac{(-1)^{m}}{2\pi}\mathbf{p}_{s}\cdot\left(\begin{array}{c}
\phi_{m+1}(z-z_{s})\\
\psi_{m+1}(z-z_{s})
\end{array}\right)\\ \noalign{\medskip} &&\displaystyle -\frac{1}{2\pi m}\int_{\partial\Omega}\left.\frac{\partial u_{s}}{\partial\nu}
\right|_{+}(y)\phi_{m}(y-z)d\sigma(y),\\
\noalign{\medskip} &&\displaystyle -\frac{\xi}{2\pi }\int_{\partial\Omega}
\left(\begin{array}{c}
\phi_{m+1}(y-z)\\
\psi_{m+1}(y-z)
\end{array}\right)\cdot \nu_y \left.\frac{\partial u_{s}}{\partial\nu}
\right|_{+}(y)\, d\sigma(y),
\\
\noalign{\medskip} B_{m,s,z} &=&\displaystyle
\frac{(-1)^{m}}{2\pi}\mathbf{p}_{s}\cdot\left(\begin{array}{c}
\psi_{m+1}(z-z_{s})\\
-\phi_{m+1}(z-z_{s})
\end{array}\right)\\ \noalign{\medskip} && \displaystyle -\frac{1}{2\pi m}
\int_{\partial\Omega}\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}(y)\psi_{m}(y-z)d\sigma(y)\\
\noalign{\medskip} &&\displaystyle -\frac{\xi}{2\pi }\int_{\partial\Omega}
\left(\begin{array}{c}
\psi_{m+1}(y-z)\\
-\phi_{m+1}(y-z)
\end{array}\right)\cdot \nu_y \left.\frac{\partial u_{s}}{\partial\nu}
\right|_{+}(y)\, d\sigma(y),
\end{array}
\]
where the functions $\phi_{m}$ and $\psi_{m}$ are defined for
$x\in\mathbb{R}^{2}$, $x=(r_{x},\theta_{x})$ in the polar
coordinates, by
\[
\begin{alignedat}{1}\phi_{m}(x) & =\frac{\cos
m\theta_{x}}{r_{x}^{m}},\quad \psi_{m}(x) & =\frac{\sin
m\theta_{x}}{r_{x}^{m}}.
\end{alignedat}
\]
\end{lem}
\begin{proof}
Let us fix $\alpha\in\mathbb{N}^{2}$ and define $m=\vert\alpha\vert$.
Let us recall the definition of $H$, given in (\ref{eq:H-def})
\[
H_{s}(x)=p_{s}(x)+\mathcal{S}_{\Omega_{s}}\left[\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}\right]-\xi\mathcal{D}_{\Omega_{s}}\left[\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}\right],
\]
where $\Delta p_{s}=f_{s}$ in $\mathbb{R}^{2}$. From
(\ref{formps}) it follows that
\[
\partial^{\alpha}p_{s}(x)=\mathbf{p}_{s}\cdot\nabla\partial^{\alpha}G(x-z_{s}).
\]
Hence, (\ref{eq:coeff-G}) yields
\[\begin{array}{l}
\displaystyle \frac{(-1)^{|\alpha|}}{\alpha!}\partial^{\alpha}p_{s}(z) = \displaystyle
a_{\alpha}^{m}\left[\frac{-1}{2\pi
m}\mathbf{p}_{s}\cdot\nabla\phi_{m}(z-z_{s})\right]
\\ \qquad \qquad \displaystyle +b_{\alpha}^{m}\left[\frac{-1}{2\pi
m}\mathbf{p}_{s}\cdot\nabla\psi_{m}(z-z_{s})\right]. \end{array}
\] Moreover, we have
\[
\begin{alignedat}{1}\nabla\phi_{m} & =-m\left(\begin{array}{c}
\phi_{m+1}\\
\psi_{m+1}
\end{array}\right),\quad
\nabla\psi_{m} & =-m\left(\begin{array}{c}
\psi_{m+1}\\
- \phi_{m+1}
\end{array}\right).
\end{alignedat}
\]
In the same manner, from
\[
\mathcal{S}_{\Omega_{s}}\left[\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}\right](x)=\int_{\partial\Omega_{s}}\left.\frac{\partial
u_{s}(y)}{\partial\nu}\right|_{+} G(y-x)d\sigma(y),
\]
we can deduce
\[
\begin{array}{l}
\displaystyle \frac{1}{\alpha!}
\partial^{\alpha}\mathcal{S}_{\Omega_{s}}\left[\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}\right](z)  \\ \noalign{\medskip} \displaystyle =
\frac{1}{\alpha!} \int_{\partial\Omega_{s}}\left.\frac{\partial
u_{s}}{\partial\nu}\right|_{+}(y)(-1)^{|\alpha|}\partial^{\alpha}G(y-z)d\sigma(y)\\
 \noalign{\medskip} \displaystyle =a_{\alpha}^{m}\left(\int_{\partial\Omega_{s}}\frac{-1}{2\pi m}\left.
 \frac{\partial u_{s}}{\partial\nu}\right|_{+}(y)\frac{\cos
 m\theta_{y-z}}{r_{y-z}^{m}}d\sigma(y)\right)\\
\noalign{\medskip} \displaystyle
+b_{\alpha}^{m}\left(\int_{\partial\Omega_{s}}\frac{-1}{2\pi
m}\left.\frac{\partial u_{s}}{\partial\nu}\right|_{+}(y)
 \frac{\sin m\theta_{y-z}}{r_{y-z}^{m}}d\sigma(y)\right).
\end{array}
\]
Combining those two equations leads us to the desired result.
\end{proof}
From (\ref{eq:Qsr}), the data matrix is then expressed as follows
\begin{equation}
\begin{alignedat}{1}Q_{rs} & =\sum_{\left|\alpha\right|+\left|\beta\right|=1}^{K+1}
\left(A_{\left|\alpha\right|,s,z}a_{\alpha}^{\left|\alpha\right|}
+B_{\left|\alpha\right|,s,z}b_{\alpha}^{\left|\alpha\right|}\right)\\ &\times M_{\alpha\beta}(\lambda, \delta B)
\frac{-a_{\beta}^{\left|\beta\right|}\cos\left|\beta\right|\theta_{x_{r}^{(s)}-z}
-b_{\beta}^{\left|\beta\right|}\sin\left|\beta\right|\theta_{x_{r}^{(s)}-z}}{2\pi
\left|\beta\right|r_{x_{r}^{(s)}-z}^{\left|\beta\right|}}\\ & +O(\delta^{K+2})\\
 & =\sum_{m+n=1}^{K+1}\underbrace{\left(\begin{array}{cc}
A_{m,s,z} & B_{m,s,z}\end{array}\right)}_{\mathbf{S}_{sm}}\underbrace{\left(\begin{array}{cc}
M_{mn}^{cc} & M_{mn}^{cs}\\
M_{mn}^{sc} & M_{mn}^{ss}
\end{array}\right)}_{\mathbf{M}_{mn}}\\ & \times \underbrace{\left(\begin{array}{c}
\cos n\theta_{x_{r}^{(s)}-z}\\
\sin n\theta_{x_{r}^{(s)}-z}
\end{array}\right)\frac{-1}{2\pi nr_{x_{r}^{(s)}-z}^{n}}}_{\mathbf{G^{(s)}}_{nr}}
+\underbrace{O(\delta^{K+2})}_{{E}_{rs}}.
\end{alignedat}
\label{eq:Qrs-developped}
\end{equation}
Thus, defining the following matrices
\[
\mathbb{M}=\left(\begin{array}{cccc}
\mathbf{M}_{11} & \mathbf{M}_{12} & \ldots & \mathbf{M}_{1K}\\
\mathbf{M}_{21} &  & \iddots & 0\\
\vdots & \iddots & \iddots & \vdots\\
\mathbf{M}_{K1} & 0 & \ldots & 0
\end{array}\right),\,\,\mathbb{E}=\left({E}_{rs}\right)_{1\leq r\leq R,1\leq s\leq S},
\]
the problem is to recover the matrix $\mathbb{M}$ knowing the matrix
\[
\mathbb{Q}=\mathcal{L}(\mathbb{M})+\mathbb{E},
\]
where $\mathcal{L}$ is the linear operator defined by
(\ref{eq:Qrs-developped}).

Therefore,  the CGPTs of the target $D$ can be reconstructed as
the least-squares solution of the above linear system, {\it i.e.},
\begin{equation}
\widehat{\mathbb{M}} = \underset{\mathbb{M} \perp \mathrm{ker }
(\mathcal{L})}{\mbox{argmin}} \| \mathbb{Q} -
\mathcal{L}(\mathbb{M}) \|^2_{F}, \label{eq:lsqr}
\end{equation}
where $\mathrm{ker } (\mathcal{L})$ denotes the kernel of
$\mathcal{L}$ and $\|\cdot\|_F$ denotes the Frobenius norm of
matrices \cite{dico}.

Let us remark that, in the case of multifrequency measurements
$(\omega_{1},\ldots,\omega_{F})$, we can reconstruct
$\left(\widehat{\mathbb{M}}^{(f)}\right)_{1\leq f\leq F}$ from
$\left(\mathbb{Q}^{(f)}\right)_{1\leq f\leq F}$ analogously.

It is worth mentioning that the location of the target detected by
the fish may be different from the true one. Moreover, the target
may be rotated and hence, the reconstructed CGPTs correspond to a
translated, scaled, and rotated target $B$. In order to recognize
the shape $B$, it is therefore fundamental for the recognition
procedure to have size invariance, rotational invariance, and
translational invariance. This could be related to the behavioral
experiments that have shown that weakly electric fish categorize
targets according to their shapes but not according to  sizes,
locations, or orientations \cite{gerhard}.

\section{Recognition and classification}

\label{sub:identification-algo}

Depending on whether we consider multifrequency measurements or
not, we will not identify the CGPTs in the same way.

\subsection{Fixed frequency setting: shape descriptor based classification}

In~\cite{dico}, an algorithm based on shape descriptors was
developed for the recognition of a target in a more classical
electrical sensing setup, ({\it i.e.}, multiple sources/receptors
placed on the surface of a disk). In this paper, we apply this
algorithm  in the context of electro-sensing.

We recall here the concept and properties of shape descriptors in
two dimensions \cite{dico}. For a double index $mn$, with
$m,n=1,2,\ldots$, we introduce the following complex combinations,
$\mathbf{N}^{(1)} = (\mathcal{N}^{(1)}_{mn})_{m,n}, \mathbf{N}^{(2)} =
(\mathcal{N}^{(2)}_{mn})_{m,n}$,  of CGPTs:
\begin{equation}
\begin{aligned}
\mathcal{N}^{(1)}_{mn}(\lambda, D) &= (M^{cc}_{mn} - M^{ss}_{mn})
+
i(M^{cs}_{mn} + M^{sc}_{mn}),\\
\mathcal{N}^{(2)}_{mn}(\lambda, D) &= (M^{cc}_{mn} + M^{ss}_{mn})
+ i(M^{cs}_{mn} - M^{sc}_{mn}).
\end{aligned}
\label{eq:Mccdef}
\end{equation}

Let
$$u=\frac{\mathcal{N}^{(2)}_{12}(D)}{2\mathcal{N}^{(2)}_{11}(D)},
\quad T_{-u}D = \{x-u, x\in D\}.$$ We define  the following
quantities which are translation invariant:
\begin{align}
  \label{eq:shape_descrp_trans}
  {\mathbf \Tau^{(1)}}(D) &= \mathbf{N}^{(1)}(T_{-u}D) = \mathbf{C}^{-u}\mathbf{N}^{(1)}(D)(\mathbf{C}^{-u})^t, \\
  \noalign{\medskip}
  {\mathbf \Tau^{(2)}}(D) &= \mathbf{N}^{(2)}(T_{-u}D) = \overline{\mathbf{C}^{-u}}\mathbf{N}^{(2)}(D)(\mathbf{C}^{-u})^t,
\end{align}
with $t$ being the transpose and the matrix $\mathbf{C}^{-u}$
being a lower triangular matrix with the $m,n$-th entry given by
  $$
  \mathbf{C}^{-u}_{mn}= \binom{m}{n}  (-u)^{m-n}.
  $$
From ${\mathbf \Tau^{(1)}}(D) =(\Tau^{(1)}_{mm}(D))_{m,n}$ and ${\mathbf
\Tau^{(2)}}(D) =(\Tau^{(2)}_{mm}(D))_{m,n}$, we define, for any indices
$m,n$, the scaling invariant quantities:
\begin{align}
  \label{eq:shape_descrp_scl}
  \Scl^{(j)}_{mn}(D) =
  \frac{\Tau^{(j)}_{mn}(D)}{\left(\Tau^{(2)}_{mm}(D)\Tau^{(2)}_{nn}(D)\right)^{1/2}},
  \quad j=1,2.

\end{align}
Finally, we introduce the CGPT-based shape descriptors ${\mathbf
\Dcrp^{(1)}} = ( \Dcrp^{(1)}_{mn})_{m,n}$ and ${\mathbf \Dcrp^{(2)}} =
(\Dcrp^{(2)}_{mn})_{m,n}$:
\begin{align}
  \label{eq:shape_descrp}
  \Dcrp^{(1)}_{mn}(D) = |\Scl^{(1)}_{mn}(D)|, \qquad \ \Dcrp^{(2)}_{mn}(D) =
  |\Scl^{(2)}_{mn}(D)|,
\end{align}
where $|\cdot|$ denotes the modulus of a complex number.
Constructed in this way, ${\mathbf \Dcrp^{(1)}}$ and ${\mathbf \Dcrp^{(2)}}$
are invariant under translation, rotation, and scaling. Only shape
descriptors of order $2$, {\it i.e.}, for $m,n=1,2$ will be used
in the sequel. Shape descriptors in three dimensions were derived
in \cite{invariants3D}.

\subsection{Multifrequency setting: Spectral induced polarization based classification}

When multiple frequencies are involved, we can use the shape
descriptors $\Dcrp^{(1)}_{mn}(D)$ and $\Dcrp^{(2)}_{mn}(D)$ at frequencies
$\omega_1, \ldots, \omega_F$ to enhance the stability of the
classification. However, as it will be shown later, this does not
yield a very stable classification procedure.

Here we rather focus on the first-order polarization tensor (PT),
that is,  the  $2\times2$ complex matrix $\mathcal{M}^{(f)}(D)$
associated with the target $D$ and frequency $f$:
$$
\mathcal{M}^{(f)}(D):= \int_{\partial D} \bigg(\frac{\sigma + 1 +
i \omega_f \varepsilon}{2(\sigma -1 + i \omega_f \varepsilon)}
I-\mathcal{K}_{D}^{*} \bigg)^{-1}[\nu] y \, d\sigma(y),
$$
for $f=1,\ldots, F$. We will show that they are sufficient to
identify  efficiently the targets. Note that it is not possible to
compute the shape descriptors $\Dcrp^{(1)}_{mn}(D)$ and
$\Dcrp^{(2)}_{mn}(D)$ based only on first-order PT, because they
require at least second-order polarization tensors. This limits
the use of shape descriptors in the limited-view case where the
reconstruction of higher-order GPTs is not accurate
\cite{ammari2012tracking}.

Here  we use the spectral content of the first-order PTs for
recognition. We have the following
properties~\cite{ammarikang2007GPT}.
\begin{prop}
For any scaling $\delta>0$, rotation angle $\theta\in\mathbb{R}$
and translation vector $z\in\mathbb{R}^{2}$, let us denote
\[
D= z+\delta R_{\theta} B:=\left\{ x=z+\delta R_{\theta}u,\,\, u\in
B\right\} ,
\]
where
\[
R_{\theta}:=\left(\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{array}\right),
\]
is the rotation matrix of angle $\theta$. Then,
\begin{equation} \label{eqrota}
\mathcal{M}^{(f)}(D)=\delta^{2}R_{\theta}\mathcal{M}^{(f)}(B)R_{\theta}^{T}.
\end{equation}
Hence, if we denote by $\tau_{1}^{(f)}(D)$ and $\tau_{2}^{(f)}(D)$
the singular values of $\mathcal{M}^{(f)}(D)$, we obtain
\[
\forall
j\in\{1,2\},\,\,\tau_{j}^{(f)}(D)=\delta^{2}\tau_{j}^{(f)}(B).
\]

\end{prop}
This gives an idea for two algorithms:
\begin{enumerate}
\item The first one, matching the singular values of all the
first-order PT $\left(\mathcal{M}^{(f)}\right)_{1\leq f\leq F}$,
would be dependent of the characteristic scale $\delta$ of the
targets in the dictionary; \item The second one, independent of
the scale of the target, would match the following quantities
\begin{equation}
\mu_{j}^{(f)}=\frac{\tau_{j}^{(f)}}{\tau_{j}^{(F)}},\label{eq:ratio_sv}
\end{equation}
for $j=1,2$ and $f=1,\ldots,F-1$.
\end{enumerate}
Some comments are in order. First, the reason why we consider the
first one, even if it is scale-dependent, is because it is far
more stable. Also, in some biological experiments, two targets of
different scales are considered as
different~\cite{von2007distance}. A question raised was then: how
is it possible to discriminate between a nearby small target and
an extended one situated far away? With the second algorithm, we
have an answer. The last remark concerns
equation~(\ref{eq:ratio_sv}). We could have also considered other
scale-dependent ratios, such as
\[
\frac{\tau_{j}^{(f)}}{\tau_{j}^{(1)}}\mbox{ or
}\frac{\tau_{j}^{(f)}}{\sum_{f'=1}^{F}\tau_{j}^{(f')}},
\]
but since $\tau_{j}^{(F)}$ happens to be the largest one (the
frequencies are sorted in increasing order), it is more stable to
consider~(\ref{eq:ratio_sv}). It is worth mentioning that if there
exists an integer $p>2$ such that $R_{2\pi/p} D = D$, then
$\mathcal{M}^{(f)}(D)$ is proportional to identity.

\subsection{Background field elimination}

We can also improve stability of reconstruction by eliminating the
background field. Let us denote by $U(x)$ the background electric
field ({\it i.e.}, the solution of (\ref{eq:system-u}) with
$k=1$). In~\cite[Proposition 2]{electroloc}, we have proved the
following formula:
\begin{equation}
\mathcal{P}_\Omega \left(  \left.\frac{\partial u_f}{\partial
\nu}\right|_+ - \left.\frac{\partial U}{\partial \nu}\right|_+
\right) \approx \nabla U(z)^T \mathcal{M}^{(f)}(D) \nabla_z \left(
\frac{\partial G}{\partial \nu_x}\right),
\label{eq:approx-dipol-U}
\end{equation}
where $u_f$ is $u$ associated with the $f^{\textrm{th}}$
frequency, $\mathcal{M}^{(f)}(D)$ is the first-order polarization
tensor at the  $f^{\textrm{th}}$ frequency, and
$\mathcal{P}_\Omega$ is the (real-valued) postprocessing operator
given by
$$
\mathcal{P}_\Omega:= \frac{1}{2} I - \mathcal{K}_\Omega^* - \xi
\frac{\partial \mathcal{D}_\Omega}{\partial \nu},
$$
with $\mathcal{D}_\Omega$ and $\mathcal{K}_\Omega^*$  being
defined by (\ref{defd}) and (\ref{defk}); see \cite{electroloc}.
Hence, if the emitted signal $U$ is real-valued, then taking the
imaginary part leads us to
\begin{equation}
\mathcal{P}_\Omega \left[ \Im m \, \left( \left.\frac{\partial
u_f}{\partial \nu}\right|_+  \right) \right] \approx \nabla U(z)^T
\Im m \,  \mathcal{M}^{(f)}(D)  \nabla_z \left( \frac{\partial
G}{\partial \nu_x}\right). \label{eq:approx-dipol-imag}
\end{equation}
Note that in biological sciences, the restriction on $U$ to be
real is justified since the permittivities of water and the fish
are negligible \cite{electroloc}. Now, from
(\ref{eq:approx-dipol-imag}), we can extract $\Im m\,
\mathcal{M}^{(f)}(D)$ by solving a least-squares problem similar
to (\ref{eq:lsqr}). Then, we have the singular values of the
imaginary part of $\mathcal{M}^{(f)}(D)$, which would be
sufficient for shape recognition and classification. The goal of
this procedure is to get rid off the computation of $
\partial U/\partial \nu$ in (\ref{eq:approx-dipol-U}), which is
supposed to be performed numerically in real-world applications,
thus subject to errors. Note that the postprocessing operator
${\mathcal P}_\Omega$ makes the data independent of the shape of
the fish's body.

Because of the following relation which follows from
(\ref{eqrota})
\begin{equation*}
\mathcal{M}^{(f)}(D) = O(\delta^2 \mathcal{M}^{(f)}(B)),
\end{equation*}
taking the imaginary part would lead us to only compute  $\nabla
U(z)$ in (\ref{eq:approx-dipol-imag}) and hence, the error made
here would be modulated by a factor of order $\delta^2$.

\section{Numerical illustrations}

In this section, we illustrate the performance of the algorithms
developed in the previous section. We use the CGPTs obtained in
order to classify the targets. We present an example with fixed
frequency, and another with multifrequency measurements. As it
will be seen, the latter does not lead us to a significantly more
stable classification in the presence of noise or for limited
aperture. The errors in the reconstruction of the high-order
polarization tensors due to measurement noise or the limited-view
aspect deteriorate the stability of the proposed algorithm.
However, when, at multiple frequencies, only the first-order
polarization tensor is used, we arrive at a very robust and
efficient classification procedure.

For the sake of clarity, and due to the large numbers of computations
performed, the results are presented in the appendix.

\subsection{Setup and methods}
We describe the dictionary as well as the measurement systems. We
consider two different shapes for the fish: ellipses and twisted
ellipses. Note that this variety of shapes exists in nature. On
the one hand, twisted ellipses would represent electric eels
(\emph{Electrophorus electricus}), whereas on the other hand
straight ellipses would look like \emph{Apteronotids}
\cite{moller1995}. This simplified representation shows that the
principle of our algorithms can be generalized to any kind of
fish's shape (hence modeling, for example, electro-sensing for
\emph{Mormyrids} as well). It also enhances the fact that, for
bio-inspired engineering applications, the shape of the robot is
not determining. Moreover, as we will see later, our simplified
representation is a good model to tackle aperture issues.

\subsubsection{Dictionary}

The dictionary $\mathcal{D}$ that we consider is composed by 8
different targets: a disk, an ellipse, the letter 'A', the letter
'E', a rectangle, a square, a triangle, and an ellipse with
different electrical parameters (see Fig.~\ref{fig:dico}). Indeed,
all the other targets have conductivity $\sigma=2$ and
permittivity $\varepsilon=1$ whereas the last one has conductivity
$\sigma=5$ and permittivity $\varepsilon=2$. Except when
mentioned, the characteristic size of the target will not matter,
and will be fixed to be of order~$1$.

\begin{figure}[!h]
\centering\includegraphics[width=9cm]{dico.pdf}

\caption{\label{fig:dico}The 8 elements of the dictionary. The
dotted lines indicate a target with different electrical
parameters.}

\end{figure}

\subsubsection{Measurements}

In each numerical experiment, one target of the dictionary is
placed at the origin, while the fish swims around it. As it has
been mentioned, we consider two different shapes for the fish's
body: ellipses and twisted ellipses. The measured data is built
taking $20$ positions of the fish around the target (see
Fig.~\ref{fig:Setup}).
\begin{figure}[!h]
\centering%
\begin{tabular}{cccc}
\includegraphics[width=2cm]{banana2.pdf} & \includegraphics[width=2cm]{banana3.pdf} &\includegraphics[width=2cm]{banana4.pdf}
\\ \tabularnewline
\includegraphics[width=2cm]{ellipse2.pdf} &
\includegraphics[width=2cm]{ellipse3.pdf} & \includegraphics[width=2cm]{ellipse4.pdf}
\tabularnewline
\end{tabular}
\caption{\label{fig:Setup}Two different kinds of experiences
involve (on the top) a twisted-ellipse shape or (on the bottom)
ellipse shape. The real part of the electric field is plotted, for
$3$ (out of $20$) positions that the fish takes around the target
(placed at the origin).}
\end{figure}

In Fig. \ref{fig:aperture}, a smaller aperture than the one in
Fig. \ref{fig:Setup} is considered.

\begin{figure}
\centering
\includegraphics[width=5.2cm, height=4.1cm]{aperture.pdf}
\caption{\label{fig:aperture} A fish having the shape of a twisted
ellipse with a smaller aperture than in Fig. \ref{fig:Setup}.}
\end{figure}

The typical size of the target is $\delta=0.3$ while the fish is
turning around a disk of radius $R=1$; the twisted ellipse's
semi-axes are $a=1.8$ and $b=0.2$ while the straight ellipse's
semi axis are $a=1$ and $b=0.2$. The effective thickness  of the
skin is set at $\xi=0$. The fish has $2^{7}$ receptors uniformly
distributed on its skin, and the electric organ emits $F=10$
frequencies, equally distributed from $\omega_{0}:=1$ to
$F\omega_{0}$. Again, we refer to \cite{electroloc} for
nondimensionalization of the underlying model equations with
proper quantities and realistic values.

\subsubsection{Classification}

The recognition process is as follows. When measurements are
acquired, we perform least-square reconstruction of the (first- or
second-order) CGPT of the targets. From this CGPT, we compute
quantities of interest $q$ (\emph{i.e.} Shape Descriptors or
singular values or ratios of singular values). Then, for each
element $n$ in the dictionary $\mathcal{D}$, we compute
$\norm{q-q_n}$, where $q_n$ is the -~pre-computed~- quantity of
interest for the $n$th shape. This leads us to charts such as the
ones presented in Figs.~\ref{fig:results-SD-banana},
\ref{fig:results-SD-ellipse}, and \ref{fig:results-aperture}.

  \begin{itemize}
    \item[]{Framework for algorithms of multifrequency classification:}
    \begin{algorithmic}[1]
      \STATE Input: the quantities of interest $\left(q^{(f)}\right)_{1\leq f\leq F}$ calculated from
      the measurement of an
      unknown shape $D$;
      %
      \FOR {$B_n\in \mathcal{D}$}
      %
      \STATE $e_n\leftarrow \sum_{1\leq f \leq F} \| q(B_n)^{(f)} - q^{(f)} \|^2 $ where $\left( q(B_n)^{(f)}
      \right)_{1\leq f \leq F}$ is the same type of quantities of interest of the shape $B_n$;
      %
      \STATE $n\leftarrow n+1$;
      %
      \ENDFOR
      %
      \STATE Output: the true dictionary element $n^*\leftarrow \mbox{argmin}_n e_n$.
    \end{algorithmic}
  \end{itemize}

\subsubsection{Stability analysis}

First, let us explain what kind of noise is considered. We will
add a random matrix (with Gaussian entries) to the data matrix $
$$\mathbb{Q}$ defined in~(\ref{eq:def-Q}). More precisely, we will
consider
\[
\tilde{\mathbb{Q}}:=\mathbb{Q}+\varepsilon\mbox{\ensuremath{\mathbb{W}}},
\]
where $\mathbb{W}$ is a $S\times R$ matrix whose coefficients
follow a Gaussian distribution with mean $0$ and variance $1$. The
real number $\varepsilon$ is the strength of the noise, and will
be given in percentage of the fluctuations of $\mathbb{Q}$, ({\it
i.e.},  $\max_{s,r}Q_{sr}-\min_{s,r}Q_{sr}$). The recognition
procedure remains the same.

Stability analysis was then carried out empirically: for each
noise level, we performed $N_{\textrm{stabil}}$ independent
recognition
 process (with $N_{\textrm{stabil}}$ being precised for each experiment), and computed the ratio of good detection. The computation ends when we reach the threshold of $12.5\%$ probability of detection that corresponds to a
 uniform random choice of the object. This gives us Figs.~\ref{fig:stability-SD-mono} to
 \ref{fig:results-imag-ellipse-ratio}.

\subsection{Results and discussion}

\label{sub:results}

In this subsection, we compare the respective stability of our algorithms. Due to the large number of situations and computations, only significant figures were plotted, giving the following classification of recognition processes, according to their range of application.

\subsubsection{Fixed frequency setting: shape descriptors}

If only one frequency is accessible for the measurements, then the
only algorithm possible is classification based on shape
descriptors. Indeed, first-order PTs are not enough to
discriminate between objects. However, the use of shape
descriptors is limited to the twisted-ellipse case with nearly
full aperture (see Figs.~\ref{fig:results-SD-ellipse} and
~\ref{fig:results-aperture}, where some targets are not
recognized, such as the ellipse in
Fig.~\ref{fig:results-SD-ellipse} or the disk in
Fig.~\ref{fig:results-aperture}). Moreover, its stability with
respect to measurement noise is quite poor (see
Fig.~\ref{fig:stability-SD-mono}).

\subsubsection{Multifrequency setting: spectral content of PTs}

In the case of multifrequency measurements, shape descriptors do
not increase their stability enough compared to singular values
(see Fig.~\ref{fig:stability-SD-multi}). Hence, it is better to
use singular values of the PTs (see
Figs.~\ref{fig:results-multi-complete-banana} to
\ref{fig:results-multi-ratio-ellipse}). One can see that:
\begin{itemize}
\item considering all singular values
(Figs.~\ref{fig:results-multi-complete-banana} and
\ref{fig:results-multi-complete-ellipse}) is much more stable than
considering ratios of singular values
(Figs.~\ref{fig:results-multi-ratio-banana} and
\ref{fig:results-multi-ratio-ellipse}); \item the aperture does
not change very much the stability.
\end{itemize}
In this regard, the most stable algorithm is to recognize all
singular values when the fish is a twisted ellipse
(Fig.~\ref{fig:results-multi-complete-banana}), leading us to a
probability of detection superior to $90\%$ with noise level of
$125\%$. This is a huge gap when compared to the recognition
process with shape descriptors, allowing only a few percents of
noise. Note that the noise level is computed with respect to the
perturbation in the measurements $\mathbb{Q}$ given by
(\ref{eq:def-Q}), which is of order of the target volume, see
(\ref{eq:Qsr}). Hence, a noise level of $125\%$ remains small
compared to the actual transdermal potential $u$.

\subsubsection{Background field elimination}

We can see in Figs.~\ref{fig:results-imag-ellipse-complete} and
\ref{fig:results-imag-ellipse-ratio} that taking the imaginary
part of the measurements in order to avoid the computation of the
background field does not significantly decrease the stability of
the reconstruction based on spectral content. Since the
reconstruction of CGPTs is very fast, the background field
elimination technique would yield to real-time imaging.

\section{Concluding remarks}

In this paper, we have successfully exhibited the physical
mechanism underlying shape recognition and classification in
active electrolocation. We have shown that extracting generalized
polarization tensors from the data and comparing invariants with
those of learned elements in a dictionary yields a classification
procedure with a good performance in the full-view case and with
moderate measurement noise level. However, this shape descriptor
based classification is instable in the limited-view case and for
higher noise level. When measurements at multiple frequencies are
used, the stability of our classification approach is
significantly improved by using phase shifts and keeping only the
first-order polarization tensor. The resulting spectral induced
polarization based classification is very robust.

Our results open the door for the application of the extended
Kalman filter developed in~\cite{ammari2012tracking} to show the
feasibility of a tracking of both location and orientation of a
target from perturbations of the electric field on the skin
surface of the fish. It also remains to understand to what extent
the spectral induced polarization approach could help us retrieve
the electric parameters of the target or locate and recognize
multiple targets.

\begin{acknowledgments} This work was supported by the ERC Advanced Grant Project
MULTIMOD--267184. \end{acknowledgments}

\bibliographystyle{plain} \bibliographystyle{plain} \bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


It's The HFCS (saturating all processed food, pharma and beverages) Stupid.
Overweight carries a greatly increased risk of cirrhosis of the liver in men, reveals a new study from the Sahlgrenska Academy. "Given the increasing problem of overweight in Sweden, there is reason to fear that more people will develop cirrhosis of the liver," says Jerzy Kaczynski, docent at the Sahlgrenska Academy and doctor at Sahlgrenska University Hospital.


Weight Loss Changed My Outlook on Life!
That's what happened for one member who started her weight-loss journey in her 50s. See how she looks today (and learn how she dropped 63 pounds) in this video.
I Lost 150 Pounds and Ran a 15K Race!
Inspirational Video: You Can Do It!
Today, I'm Nearly Half My Size!
Small Changes Helped Me Lose 90 Pounds!
Is High-Fructose Corn Syrup Hazardous to Your Health?
Those 95 Pounds Will Never Weigh Me Down Again!


Have you found an online vendor selling Ron Botran Anejo 12?
Nothing really special about this fellow. Had it not been for the good company and great mood when tasting it, would probably have forgotten all about it already.
To try just once, just for the shake of it, ask the bartender to perform some extra orange peel treatment on the glass before serving so that you might get close to a "decent" outcome.
Pourer said age statement is solera (sigh), so there's only trace amounts of 12 year-old juice in here. But hey, it's fine; tastes how you'd expect. It also appears to be priced appropriately, and is not as dosed as many others from Central America.
Skip it for sipping, but highly recommended as a mixer.
For the price this is a good choice for adding to your favorite cocktail. Bit of a burn and harshness going down to deter from drinking neat.
Good classic smell. Nice taste but a bit spicy at the finish.
Unlike Botran Reserva its way too alcoholic with plain body. Better go for older ones which are awesome.
Wasn't impressed with this bottle.Tasting it neat was not great, but it improved with a bit of water. It's quite sharp, but you are not rewarded with a nice after taste. It makes for a nice sharp Cuba Libre though, so I'll use it for mixing. I paid €19 for this bottle and at that price I guess you can't expect a real sipper, so you get what you pay for.
It truly is a surprise how close the 15 and the 12 year Botran are flavor profilewise. Neither is really a neat sipper since the alcohol smell and taste is rather strong and the burn at the end is not a nice surprise. Mix it with Coke and you will have a glimmer of hope. The 12 year Anejo is just marginally worse than the much more expensive Reserva 15 year old Botran.
ES UN RON AÑEJO COMERCIAL, BUENA OPCIÓN PARA COCTELERÍA.
Heavy and chocolate taste. Try it with a chocolate together.
Firstly: it's cheap. Also people should note, the Botran 12 is, like most Botrans and most Zacapas, a solera-style rum, so it's not actually 12 years old. When I was in Guatemala I drank this far more than the often praised but wildy overrated Zacapa 23 (which to me is an overly sweet cloying licqueur, not a proper rum). This is a lot dryer, more honest. You do get some alcohol, and a somewhat unrefined sensation, but at half the price of the Zacapa 23 this was always the better option for me.
Had higher expectations from a 12y old. Don't like it with coke too much, like it better on the rocks. Strong alcohol flavour, lacking character. Don't like the aftertaste a lot, but not too bad considering the price.
A good one for a reasonnable price. Tasty and perfumed, this is the kind of rum you'll enjoy after dinner along with a piece of dark chocolate.. Not revolutionnary but a good choice that every amateur should possess.
Un ron de sabor suave y dulce, en México la botella es más bonita q la original con tapón de corcho.
Según dicen es un Zacapa económico. Ya que también son de la misma región.


Keystone LG Welded Hardware Cloth ¼-in. x ¼-in. x 36-in. - 10-ft.
Home » Welded Hardware Cloth » Keystone LG Welded Hardware Cloth ¼-in. x ¼-in. x 36-in. - 10-ft.
Keystone LG Keystone LG Welded Hardware Cloth ¼-in. x ¼-in. x 36-in. - 10-ft.


text:
Is It Time to Rebrand?
An overview of the reasons for rebranding and the steps needed to successfully give your brand image a complete makeover.
BY LEE NELSON
Rebranding has become the hopeful catalyst for many organizations—including many real estate companies—to grow, become more modern and resonate with consumers. Some have been successful, and some, not so much.
"Your brand is the heart, soul and promise of what you are to your customers," said Jim Heininger, rebranding strategist at his company Rebranding Experts in Chicago. "But you need to revisit and redefine what your promise is to your customers. Once you answer that, it will indicate if you need to rename your organization, have a new visual identity or design a new logo for it."
But many organizations that rebrand put a lot of attention on the logo, he adds. "But that's not the most important thing to customers."
When Heininger sits down with his clients, he tells them the first step is to answer fundamental questions. What has changed in the marketplace for you do to a rebranding? Why is your brand still not relative for return or new customers? Where is the industry going? Can a new technology application help better service your customers? How do you represent that in your brand? What about your current brand is limiting your business growth?
In a survey about rebranding success by The Dixon|James Co., Heininger's public relations' firm, it showed 81 percent of organizations which rebranded introduced an entirely new logo, 69 percent introduced an entirely new name and 75 percent staged an external launch event and public announcement.
Most Recent Rebranding in Real Estate
In March, Coldwell Banker Real Estate LLC announced a new vision when it kicked off Project North Star. Included in that was a new logo with a star in the top right corner, symbolizing the company's consistent, reliable presence guiding everyone home. The company also announced its new mission statement: "We empower our people to leave their mark on the world of real estate."
This was the first time in nearly 40 years that it changed its logo and adopted a fresh mission. The brokerage's box logo was limiting in this digital media era. In research, the mission statement had an 80 percent approval rating, and consumers said the new logo represented an innovative brand 80 percent more often than the exiting logo.
"I love it. I think it still contains all the Coldwell Banker heritage with a new twist," says Rosemary Buerger, broker associate at Coldwell Banker Advantage in Cameron.
She read about the changes via social media. She will have to order new business cards, but the cost will be minimal such as $60-$150 for the changes at most.
"As an agent, it gives me the opportunity to sing the praises of my company further by showcasing the rebrand," she adds. "It's really about staying present in business and showcasing what we do, from the logo to sort of boots on the ground working with clients.
Overall, the response has been positive toward the rebranding in her circle.
"It will be rolled out over the next year, so the hard launch date is not here yet. It's about staying in front of the changing footprint of real estate and keeping current," Buerger explains. "A logo is one part of who I am as an agent, so it's part of my marketing tools. I love that my brand is forward thinking and embraces change as it's needed."
Kent Hanley, CEO of Coldwell Banker Howard Perry and Walston, headquartered in Raleigh, has 15 offices in central North Carolina.
"We had a feeling there was going to be a big announcement this year, and that was it," Hanley says. He learned about the new logo and branding at the opening event of the Coldwell Banker GenBlue Conference in Las Vegas in March.
"There was a lot of excitement around it, and I was very thrilled about the new logo. We are looking at the logo as a new, crisp representation of the CB brand," he adds. "The logo will scale better now in our marketing. Coldwell Banker is giving us two years to implement these changes, which will be good."
He doesn't know yet how much all of this change will cost his company with letterheads, yard signs and more. "Currently, we are using and replenishing the materials that are vital for our day-to-day needs, but are not going out and ordering thousands of business cards or yard signs. We will wait for January 2020 to start the rebrand," Hanley states.
What it takes for companies to rebrand
Back in 1978, when RE/MAX came up with its balloon logo, there was no digital or social media advertising. So, the company decided in the fall of 2015 to bring the idea of changing it to the CEO.
"When we were changing the logo, it was very important to test it in different mediums so it shined through, and it had to adapt to whatever size and font that went into the creative mediums," says Abby Lee, senior vice president of marketing and communications for RE/MAX.
The company did a lot of testing for more than two years with more than 20,000 consumers surveyed in Canada and the United States. "We weren't in a desperate situation. We just wanted to update the logo but not go too far," she says. "We realized it didn't have to be a complete rebrand or reposition ourselves, but just a refresh."
So, they decided on removing the stripes on the balloon, sticking with the original colors and changing the lettering slightly. There's still the balloon with subtle changes.
When Hadi Atri was apprised of RE/MAX going through a rebranding, he questioned why they were doing this—and he was on their advisory board.
"I knew quite a while back. It was not an easy decision. You are used to seeing the balloon in one way. I questioned why we are doing this," says Atri, president and CEO of RE/MAX Executive in Charlotte.
With more than 350 agents and 11 locations, Atri continues to get new signage. It has been a costly change.
It will cost $15,000 or more to just replace the lighted sign in his main office in Charlotte. When he opened that office in 2012, the sign was $12,000.
"We have four new offices since the rebranding started. We'll eventually get them all redone. It will be expensive," he says.
It has taken about 1 1/2 years to get through all the former agents signs, and since the rebranding, he has added 22 new agents. "Since I was involved with the rebranding from the beginning, I told them that it won't happen overnight. A yard sign costs $500. The agents won't just throw them away," he adds. "The change isn't that extreme that you can really differentiate it right away. That's a good thing."
He says he is very conservative and when it comes to branding, he likes it clean cut with not too many sharp colors or extreme changes.
If you have a successful brand, you have to be convinced the change is for the good, he says.
"New generations and clients have changed. The new logo is much more clean and better for production on paper. My marketing department and the people in that division say it does come out better," Atri says.
Reasons to Rebrand
Heininger says there are eight top reasons a company should rebrand:
1. To more accurately reflect its expanded scope of services or product line.
2. The mission no longer accurately represents the work or service lines.
3. Current brand elements are no longer relevant and limiting business success.
4. A merger or acquisition.
5. Transformative changes to the business model, product line or services.
6. Realign with changes in the marketplace or external influences.
7. The business has lost its meaning and its key differentiators to rally around.
8. Negative reputation makes it difficult to operate successfully.
Heininger has observed that a lot of real estate companies have been named after original families that founded them.
"Those names might not be relevant in the community. The founding members may not have any involvement anymore, and the brand does not explain what the business is all about," he adds.
Recommendations for a great rollout
Heininger encourages a real estate company to do a one-day flip the switch announcement. Everything should flip over, from the signage to social media to all the materials updated and used from that day on.
"Everything old is discarded so it is not confusing."
He suggests building internal excitement the week leading up to it. Every place your brand is portrayed needs to change, from advertising to car logos to jackets. You can even have an event for your customers and leaders in your community, and explain why you are doing the change.
He has seen failures with execution when the rebranding wasn't strong enough to achieve a positive outcome or the new brand was not well thought through, didn't make sense or wasn't relevant to their customers who couldn't support it.
"If rebranding is done correctly, thoughtfully and all proper planning is done, and you get employees on board and effectively tell your customers, it can translate into increased sales," he states.
Lee Nelson is a freelance journalist from the Chicago area. She has written for Yahoo! Homes, TravelNursing.org, MyMortgageInsider.com, and REALTOR® Magazine. She also writes a bi-monthly blog on Unigo.com.

redpajama_set_name: RedPajamaCommonCrawl


text:
Les Girondins Etoro
Is Etoro Anonymous 2021 Today – Best Trader Resources Online
Looking to get Is Etoro Anonymous?…eToro is a multi-asset and foreign exchange trading company that specializes in providing foreign exchange and financial trading services around the world. It has branches in the United Kingdom, Canada, Australia, Cyprus, Israel and the United States. In 2021, it had valued its value at more than $800 million. The company is listed on the Boston Stock Exchange and is one of the largest futures and options trading companies in the world.
Is Etoro Anonymous
Although there are other trading platforms out there, none has the staying power of eToro. This is because it offers a lot more than competing exchanges. In addition to having a low commission, it has several unique features that set it apart from other platforms. For example, it lets traders trade in three major trading currencies, as well as ether (the most famous form of currency today), and four major commodities. Traders will also be able to place a limit order, which is similar to a traditional stock trade, allowing them to limit the risks they are taking on a particular trade.Is Etoro Anonymous
Rakuten Vs Etoro 2021 Today – Best Trader Resources Online
Long Position On Etoro 2022
Can I Send From Binance To Etoro 2022
Will Tron Add To Etoro In The Near Future 2022
Bitpay Vs Etoro Wallet 2022
Gemini Fees Vs Etoro Fees 2022
Why Is Etoro Blowing Up, Pump And Dump? 2022
© 2022 Les Girondins Etoro • Built with GeneratePress


Intel and its partners will kick of CeBIT 2007 with the launch of Intel's Bearlake family of chipsets for desktop PCs. The new series introduces support for DDR3 memory, front side bus speeds up to 1333MHz and are designed to support Intel's upcoming 45nm-process CPUs, the Core 2 Duo (Wolfdale) and Core 2 Quad (Yorkfield).
The main chipset on show is known as the P35 and is to be positioned as the successor to Intel's current 965 Express series for mainstream desktops, according to sources. Motherboards based on the P35 will be able to support FSB speeds up to 1333MHz (up from the 1066MHz of the 965) which will be required for several of the CPUs Intel is planning to launch in the second half of the year. The P35 also introduces support for DDR3 memory which is currently specified to run at up to 1066MHz, versus DDR2's 800MHz, and so offers greater memory bandwidth. DDR3 also operates at just 1.5V compared to DRR2's 1.8V.
Supplies of DDR3 are, however, currently low and prices relatively high, which does not match the mainstream position of the P35. For this reason Intel has also designed the P35 with optional legacy support for DDR2 which motherboard makers can choose to implement is several ways: cutting-edge boards that only support DDR3 but which increase system costs because new, expensive RAM needs to be bought; legacy boards that support DDR2, but not DDR3, allowing users to take advantage of the abundance of DDR2 now, but may be limited in the future when DDR3 becomes more common; and so-called "combo" or "hybrid" boards which support both DDR3 and DDR2, but not at the same time, meaning that consumers can save some money by using DDR2 now and upgrade to DDR3 later, but the system is limited to only two DIMMs at a time compared to the four that are available in the other implementations.
Several Taiwan-based motherboard makers will be showing P35-based boards at CeBIT. Micro-Star International (MSI) will display a total of six boards, one pure DDR3, two hybrid-boards, which will target high- and mid-range markets, two pure DDR2 boards, again for the high- and mid-range segments, and finally an mATX-board. ECS will be showing a hybrid-board which will be released as the P35T-A. Also Gigabyte United, which has recently gained a reputation for having some of the most overclockable motherboards in the 965 range will be hoping to do it again with its P35 series.
P35-based motherboards are expected to be available in April or May.
Today, in Part 2, the other chipsets in the Bearlake family will be covered, including a high-end version which introduces support for PCI Express 2.0 and integrated graphics processor (IGP) designs.
While the P35 can be found openly shown at booths around CeBIT, the G33 ? a chip with similar features but which adds integrated graphics ? can only be seen behind closed doors. As with the P35 the G33 supports DDR3 and/or DDR2 memory, front side bus speeds up to 1333MHz, and is designed to support Intel's upcoming Wolfdale and Yorkfield CPUs. The G33's IGP supports DirectX 9 and is scheduled to launch in the second quarter, according to sources.
Intel will expand the Bearlake family in the third quarter starting at the high-end with the X38. The successor to the current 975X Express chipset, the X38 also supports FSB speeds of 1333MHz and upcoming Wolfdale and Yorkfield CPUs, but unlike the P35 and G33 will not include legacy support for DDR2. X38-based motherboards will only support DDR3, but in the high-end market this makes sense. The X38 will also feature support for PCI Express (PCIe) 2.0 which offers double the bit-rate of current PCIe for higher bandwidth ? up to 5GT/s. X38-boards will support two x16 PCIe slots, but it is unknown whether Crossfire and/or SLI will be possible due to licensing issues.
Also in the third quarter, Intel is scheduled to release the G35, a chipset similar to the G33 but which features a DirectX 10 based IGP. Finally, two chipsets aimed at business platforms, the Q35 and Q33, and a value segment chip ? which will not sport DDR3 or the quad-core Yorkfield CPUs ? will be also be introduced under the name G31.


text:
Q: How do I create an inverted index containing information on the number of occurrences of each word in each document I have a dataset containing 250 data points of text information. Now I need to create an inverted index containing information on the occurrences of each word within each key. I have around 250 keys and a list of the preprocessed collection of words.
I have tried playing around with Whoosh libraries but with little success. The documentation seems a little vague on how it works. I receive Unknown Field Error.
from whoosh.fields import Schema, TEXT, KEYWORD, ID, STORED
from whoosh.analysis import StemmingAnalyzer
schema = Schema(Title = ID(stored = True), Content = `enter code here`KEYWORD(stored = True))
import os, os.path
from whoosh import index
if not os.path.exists('inv_indexdir'):
    os.mkdir('inv_indexdir')
ix = index.create_in('inv_indexdir', schema)
import whoosh.index as index
ix = index.open_dir('inv_indexdir')
writer = ix.writer()
writer.add_document(title = u'Title', content = u'Content', path = u'json_data.json')

redpajama_set_name: RedPajamaStackExchange


text:
Professional impersonator, Johnny Vallis, is playing the part of Buddy Holly in the Buddy, Roy and Elvis show at the renovated and rebranded Elements Casino Victoria. (Lindsey Horsting/News Gazette staff)
West Shore resident back on Legends of Rock'n'Roll tour
Johnny Vallis was captivated by Buddy Holly's musical innovation
Lindsey Horsting
May. 8, 2018 5:30 a.m.
Rock'n'roll and the stage is something Johnny Vallis has always had an affinity for.
He had his first big gig was when he was 11 years old as an Elvis impersonator. Back then he was blond, tall and thin, didn't look too much like Elvis, but dug the music. Another Elvis impersonator suggested Vallis impersonate Buddy Holly because of his uncanny resemblance.
Vallis did some research on Holly and found his music to be so creative, and that there was overlap from Holly to bands like The Beatles and The Rolling Stones, who idolized Holly, he said.
Vallis and his friend started to try out some Holly tunes, Vallis singing and on guitar, his friend on the drums. They were playing at a diner in Victoria when Vallis was discovered by a musician who was a guest at the restaurant who then got in touch with the promoter of Legends of Rock'n'Roll and they brought Vallis over to Vancouver for a photo shoot.
Vallis first joined the Legends when he was 16 years old and has now rejoined them for a tour.
Over the years, Vallis has traveled the world touring with the with Legends – his favourite an Australia 66-date tour – Puerto Rico, and has played for the King and Queen of Malaysia. The highlight of his career was working on tour with Veronica Bennett of the Ronettes.
"She was great to work with, a really nice lady," he said. "She's the "bad girl" of rock'n'roll, but truthfully, she's a sweetheart."
Through touring, Vallis has met people who worked in the same studio as Holly, like Sunny West who wrote "Oh, Boy!" and "Rave On". He met Buddy Knox, who told him to take his glasses off, because Vallis looked too much like Holly, he said.
Vallis said meeting people who knew Holly helped him get to know the man he lives on stage, and having great respect for the artist helps him portray Holly honestly and authentically.
He is sensitive to Holly's story, who passed away at just 22 years old, when "he was at the height of his popularity. It was rock'n'roll's first tragedy, " Vallis said. "Holly was only around for 18 months and is equal billing to these people [Roy Orbison and Elvis]."
The most enjoyable part of his job is bringing fond memories to people who remember Holly's music and creating an escape that brings joy, Vallis said. The Legends of Rock'n'Roll tour features artists that have all passed away, so that is the illusion part of the show, he said, being able to present something to the public they couldn't otherwise have.
There is no performer in Vallis' immediate bloodline, but he thinks his grandmother would have liked to be on the stage.
As his day job, Vallis performs as the 'Man of Many Voices' where he impersonates the likes of Mick Jagger and John Lennon. Vallis said he would like to try Mick Jagger for a Legends tour one day.
The Legends of Rock'n'Roll at Elements Casino Victoria features the trifecta of Roy Orbison played by Larry Branson, Elvis played by Ben Klein and Holly played by Vallis. The concert dates are May 11 and 12. People can purchase tickets in person at the casino, by phone at 1-855-985-5000 or online bit.ly/2HX196k.
lindsey.horsting@goldstreamgazette.com
REVIEW: Building a better burger
Esquimalt hooks the region with the return of Buccaneer Days


For many people, the thought of asking that special someone in your life to marry you is enough to scare even the most hardened of souls. If you're planning on asking your partner to spend the rest of their lives with you then you've probably thought about it a lot, including just how to do it the right way.
Before you get down on one knee we thought we'd round up a selection of marriage proposal stories to get you inspired.
One fortunate family member was lucky enough to witness their mum's boyfriend get down on one knee, in full fancy dress, at a Halloween party, surrounded by ghastly ghouls and scary souls. The spur-of-the-moment proposal meant that he didn't actually have a ring, so plucked one from a Haribo packet as a sweet substitute. She didn't mind though, and still said yes!
A great family story came out of one couple's proposal, when the sure-of-himself groom-to-be booked the registry office before asking his partner the big question. Then, his proposal involved asking her if she was free on that date! Fortunately, she was, and the tale is still told by their children to this day.
One lucky commuter witnessed a proposal first-hand at St Pancras International train station recently, which all focused around one of the famous pianos. A small crowd had gathered around a gentleman playing on one of the pianos, and when he stopped a guy standing behind him proposed to his girlfriend. Only once she said yes did a group of family and friends emerge from amongst the gathered crowd to celebrate with them – there was plenty of clapping and cheering too!
As English tradition dictates, many men like to ask the permission of their potential bride's father before popping the question. This becomes far more difficult when your father-in-law doesn't speak English. Wanting to propose to his German partner, one romantic hopeful had to enlist the help of his brother-in-law to help translate his intentions. Lots of confused looks followed, even when he flashed the ring – later, it was clarified that it's not actually common to have an engagement ring in Germany, hence why this didn't help the situation. Fortunately the translation was all cleared up before his bride-to-be returned, leaving him to get down on one knee in the wine hills surrounding her home.
Do you have an interesting or romantic proposal story? We'd love to hear about it! Comment below or tweet us at @SpicersofHythe. And be sure to take a look at the range of Valentines Day hampers for an extra special treat for your partner.


Subject: A PhD for Technical Writing?; was, "starting on page 2?"
Phil, I totally misread your original message. I thought the gist of your message was to suggest that the likes of Christ Morton shouldn't presume to judge the knowledge of his betters, "his betters" in this case being the greenhorn PhD who was set to oversee his group.
Scientific disciplines and possibly engineering excepted, I tend to think a PhD is a disadvantage when it comes working in the trenches. As we all know, "PhD" stands for "Doctor of Philosophy." Philosophy is a predominantly theoretical endeavor that doesn't necessarily translate into practical ability. And as far as technical writing is concerned, I think a PhD is counterproductive, on several levels, for anyone not interested in a career in academia. Having a PhD in English or technical writing proves that one has mastered at least one facet of English or technical communications theory. It in no way proves that one can work competently as a technical writer or manage a technical writing group.
Fred, with all due respect, you seem to be contradicting yourself.
On the one hand you agree with my point that a style guide is specific to a context, and on the other you suggest that Chicago is the "most broadly applicable".
In any given context, a stye guide is either applicable or not. How large the domain of applicability is outside of that context is irrelevant as to whether it should be used or not in any given situation.
> > APA is for academic writing, Chicago is for 'trade' or 'professional' writing.
> I have to disagree with both parts of this statement.
> The APA (American Psychological Association) style guide is only appropriate for a small segment of academic writing, namely psychology and the social sciences. It is basically useless and inappropriate for other types of academic writing.
> And while the Chicago Manual of Style is used in "'trade' or 'professional' writing", it's foundation is *academic* writing, since it is the house style guide of the University of Chicago Press, the largest university press in the US.
> The point is that there are many different types and subtypes of writing, not just a broad division between "academic" and "trade or professional". And because of this, there is a large number of style guides specific to many of these different types and subtypes. There are at least a half-dozen well known and widely used style guides for academic writing in specific fields (ACS for chemistry, ASA for social sciences, CSE for life sciences, MHRA and MLA for the arts and humanities, IEEE for computer science, etc.). The most comprehensive and therefore most broadly applicable academic style guide is Chicago, largely because the press that compiles it publishes such a broad range of academic and scholarly books and journals. But even Chicago is only partly appicable to most types of technical writing.
Next by Thread: Re: A PhD for Technical Writing?; was, "starting on page 2?"


text:
Wera Hobhouse
There is nothing more lonely or frustrating than feeling unheard. That is why I set out to build a party, and a country, where every single person has an equal opportunity to thrive, regardless of their background or where they live.
That means pulling our party firmly to the centre-left, rebuilding our local government base, securing a progressive alliance, and moving effort and resources to our regions. We cannot run as a London-centric party that lets our powerful regional voices go unheard and the Conservatives a free run.
That continues to be what I believe in more than anything.
I feel grateful for the energetic campaign that we built together, and I am proud that we asked the tough questions, dared to dream big and fought for what we believed in, the values we share and the vision we hold.
That is always the right thing to do.
But, if my supporters and I really believe in the need for a new direction, then we must accept when we are no longer the best vehicle to deliver that.
That is why I am withdrawing from the Liberal Democrat leadership election and, this afternoon, I have offered to work with Layla Moran on behalf of our party and her vision for the country.
I know this will be disappointing for those of you who joined our movement, but this leadership race is more important than any one candidate, and it is vital to the future of our country that we come together to defeat the Conservatives at the next general election.
And you might expect me to say this but a revitalised and united Liberal Democrat party is needed now, more than ever.
Our party has had knockbacks, but we always come together in the aftermath to ensure our values and beliefs endure.
And that is what we must do now: It's time to refocus, organise, and take action.
Millions of people across the country are relying on us to defeat the Conservatives' hurtful agenda, and let's be absolutely clear: The Liberal Democrats have the enthusiasm, courage and tenacity to do it.
With this in mind, I am pleased to support Layla Moran and encourage those who joined my campaign to do so too. She represents a new generation with the talent to engage and inspire. She is now the only candidate who can break with our party's damaging legacy from the coalition and adopt a centre-left position to defeat the Conservatives at the next election.
Thank you to everyone who supported our campaign, all my volunteers, staff, friends and family.
I look forward to seeing you on the campaign trail very soon!
All stories by: Wera Hobhouse


text:
Q: ValidationRules without binding I want to use the ValidationRules (and it's UI effects) on the textbox without actually binding anything to the textbox.
I want to  use the textbox for some input that doesn't bound to anything but need to validate the input after focus is lost using the ValidationRules.
Can it be done?
<TextBox.Text>
   <Binding Path="" UpdateSourceTrigger="LostFocus">
     <Binding.ValidationRules>
        <local:IntegersOnlyValidator/>
     </Binding.ValidationRules>
   </Binding>
 </TextBox.Text>

A: Your code-behind should be as independent of the GUI as possible, so I would recommend you to create a property and bind to that. When you want to pass the text to the method, just pass the value of the property.

A: You could bind to just any string, e.g. create one as the source for the binding:
xmlns:sys="clr-namespace:System;assembly=mscorlib.dll"

  <TextBox>
    <TextBox.Text>
      <Binding Path=".">
        <Binding.Source>
          <sys:String>Default Text</sys:String>
        </Binding.Source>
        <Binding.ValidationRules>
          <!-- Validation Rules -->
        </Binding.ValidationRules>
      </Binding>
    </TextBox.Text>
  </TextBox>

A: This worked for me:
<TextBox.Text>
    <Binding RelativeSource="{RelativeSource Self}" Path="Text" UpdateSourceTrigger="LostFocus">
      <Binding.ValidationRules>
        <Filters:IntegersOnlyValidator/>
      </Binding.ValidationRules>
   </Binding>
 </TextBox.Text>


This table contains information on a variety of state, provincial, and national parks from around the world. Much of the information is in free-text form within the description column, making this table an ideal platform on which to demonstrate Oracle's regular expression capabilities.
You can download a script to create the park table and populate it with data from http://oreilly.com/catalog/oracleregexpr.


Have you ever tried to discover the health benefits you can get from green tea? If not, you must read this article to learn the reasons why you must include green tea in your daily routine. Green tea is the most popular drink in China where people have been drinking it for thousands of years. However, it was not very popular in western countries but now the scientific advancement has proven its effectiveness and its popularity has been amazingly increased in western world.
Green Tea has been proven to be very effective for overall body health as it contains catechin polyphenol which is known as epigallocatechin gallate or EGCG in short.
Epigallocatechin Gallate is one of the most powerful antioxidant agents which greatly reduce the risk of various heart disease and different types of cancers.
There are lots of other benefits of EGCG such as boosting of immunity, preventing formation of blood clots, protecting against free radicals which can damage your DNA. The use of green tea and the inclusion EGCG in diet also helps in slower down the natural process of aging. Moreover, you can also boost energy levels and suppress high appetite which helps you control your diet and you don't get excessive calories.
Cancer is a chronic disease which is not easy to be treated even after years of researches, scientists have not been able to provide a permanent and solid solution to eliminate all types of cancers completely and permanently. However, the years of researches have proven that the consumption of green tea can provide amazing effects on certain types of cancers.
Different studies have shown that those who consume green tea regularly have very low risk to develop esophageal cancer. Furthermore, it was also revealed that the molecule that causes cancer in reaction of using tobacco can be suppressed with green tea. If you are addicted to tobacco or smoke, you must try to get rid of them and if you can't, you should at least increase the consumption of green tea.
If you have having unbalanced cholesterol levels, you must give a try to green tea to control cholesterol and just a cup everyday can help you keep the levels in control. The researches have proven that the consumption of green tea on regular basis can be really effective in lowering down high cholesterol and triglyceride levels.
It is often suggested that consuming at least 10 cups full of green tea every day can improve liver disorders and if there is damage to your liver because of alcohol and other toxins in your body, it can also be prevented.
Green Tea has always been very popular among Chinese to control blood sugar in body. Now there are solid evidence that this type of tea can play a very active role in preventing and controlling diabetes if adopted as a drinking habit on daily basis.
Previous ArticleDoes Alcohol Help In Weight Loss?


What do students need to know about transferring?
Many universities have agreements with community colleges that make transferring a great option for students (California's www.ASSIST.org is an example of excellent transfer options). In the absence of such an agreement, transferring can mean that courses you took at one college will not satisfy requirements at another, leading to additional time to complete your degree. Transferring from one university to another university is usually messy. Find out about transferring by searching for details on college web sites and talking to counselors at both your current and prospective colleges. Transferring can mean different things depending on the circumstances. At best, it is: 1) an excellent means of preparing for university academic demands by attending a community college; 2) a economical means of obtaining a bachelor's degree; 3) a convenient means of satifying non-major requirements; 4) a way to attend college part-time while occupied with work or other responsibilities; 5) a way to explore major and career options before investing in a four year university. At its worst, transferring is: 1) a means of correcting a poor decision; 2) a misguided attempt to do better in college by changing schools; 3) necessary in order to major in a particular academic program not offered at your current college; 4) a choice that adds time and money to the completion of your degree. Keep in mind that transferring is stressful. You will spend time researching options, meeting with counselors, completing applications, adapting to a new college; time that you would otherwise be using to focus on current college demands. As a community college transfer student you will find a lot of assistance in you college's transfer center. Visit the transfer center before your first semester be sure you are immediately on track!
for those students not ready to attend a 4 year college. It can also help a student who has decided that their college wasn't the right fit, after they got there. Perhaps a student has changed majors or decided that there is a school that better offers the academic program they want to study. It's important before transferring to know what the rules are and follow them. It's also a good idea to get an academic transfer advisor to sign off on your preliminary transcript before making a change. Nothing worse than transferring and losing a bunch of units or having to retake classes, or make up classes. Make sure you have all the pre-requisites so that you can hit the ground running when you get on your new campus! Also, for transfers, it's going to be harder to connect on campus with new students. Many of them will have been there for 1 or 2 years. You may have to work a little harder to make friends and connections on campus, go out of your way to get to know professors and instructors and to learn the ins and outs of the campus. Don't be afraid to ask questions. It will be an adjustment, just as it was for a new freshman on campus!
Many students today are presented with the trade-off of seeking admission at a first rate (expensive school) or instead opting (out of choice or coercion) attending a community college. Dreams of transferring from a second choice school to a first choice school begin to fester as soon as reality sets in. Students tell themselves, "If I buckle down in community college then I can prove that I worthy of XYZ college." But let's think about this: if a higher number of students seek entry into a college with fewer spots, then what logical explanation explains how more spots for transfers would be available for (many) of the same students who applied the year prior? Added to that, elite colleges tend to have the highest retention rate, so where how could the open spots materialize? This bring us back to basketball – or any other elite sport for which many candidate compete – with one exception: at least in sports one can hope (in a kind way) that a player could get injured. Even then, how many people would vie for that spot? So it goes with the elite colleges in the country. Harvard and Princeton announced that transfers are virtually not accepted. Stanford and Amherst decreased their transfer rate by 50%. The University of Pennsylvania has about a 1% transfer rate, Yale about 4%, and Dartmouth about 7%. Let's face it, after working so diligently to gain entrance into one of these schools, why would one leave? Hope does loom on the horizon. Some schools believe that transfer students add diversity to the student body, but the essay cannot and should not be the same as a student's first year essay – otherwise the student sends a message that he or she has not grown or learned from the year off. On a positive note, last year, Cornell, MIT, Georgetown, Notre Dame and Northeastern admitted more transfer students than expected.
Transferring is like applying to college all over again, and it can be more competitive! Transfer students face challenges: making new friends, learning about campus, and losing academic credits or adding semesters. However, if you are 100% sure you are transferring for valid reasons and know the new school is the right fit, transferring can be worth it. George Washington was a good school and I had made friends there, but it wasn't a good fit for me. I applied to transfer to Rice University after my sophomore year—it was a much better fit for me, I only lost three credit hours, and I graduated on time.
The deadlines for transferring are different, then for freshman applications, and may vary from college to college. Be aware that not all of your credit will be accepted. No grades below a C- will transfer. And often the requirements of the new schools might be very different then your original schools, so do not be too disappointed if you lose some credit. No college will accept more then 60 credits- the typical equivalent of two years work, since if they will be granting you the diploma- they require that, at the very least, 50% of your class work was completed at their campus. Usually if you do apply after two years of college, they will be only considering your college transcripts and not your high school grades or your SAT scores for admissions. So, if you did not do well in HS, but was very successful at your first college, you would now be in a better position to apply to a more selective school.
Transferring from a community college or a four-year college to another four-year institution can be exciting. Hopefully, you are transferring because the degree you want is at the new college. Keeping track of your hours and how they will transfer is, to me, one of the most important factors to consider. When students transfer to a new college the hours may 1. transfer toward the degree you are seeking or 2. transfer as elective hours or 3. won't transfer. Your goal is #1. If you know in advance that you will start at a one college and then transfer later, be sure to talk with a college advisor at the final destination college to be sure that the classes you are taking will transfer as hours toward your degree. For example, a physics course may be offered as Physics for non science majors, Physics for Natural Science majors, and Physics for Engineers. Depending on your choice of major, you want to be sure you take the physics class that will ultimately transfer toward your degree.
You are wiser and will still need to cover all the bases.
This time around make sure that you are considering all of the elements of a good fit. Each school that you apply to will need to be a good academic, social and financial fit. The biggest challenge that students face when transferring is having their hard earned credits transfer. Be certain that your credits will transfer and cross check your plan with the Dean and registrar. If you are currently receiving institutional aid you will leave that behind. Contact the financial aid office of the school that you are transferring to to review your current aid package and get a solid estimate to learn how the packages will compare.
First of all, this is a big decision and it is not one to take lightly. Let's start with why someone might think about transferring. A student might find it financially smart to spend the first two years of post-secondary education at a community college and then transfer to another four-year institution. This requires shrewd choices, hard work and guidance from advisors, but the combination can save you a bundle on the tuition bill. A student might simply be unhappy at their first university. A student's desired major might not be available at his first school. In all cases, you should research the new target school's website carefully--there should be a page outlining the requirements for a transfer application. Also, do not be shy to contact the admission officer responsible for transfers in order to ask specific questions. Be aware that all of your classes might not be accepted by the transfer school, that financial aid might be limited and that there may be limited orientation opportunities for transfer students.
When it comes to switching schools, each one is different. Some may require a certain number of credits already earned, others may be concerned about minimum gpa. Some colleges will admit you directly, while others will only offer you conditional acceptance based on your academic performance after your first year there. Some colleges have a separate office for transfer students, others include them under the admissions umbrella. It is important to check with each school and learn their transfer policy.
Go to the transfer college website and find program requirements for the school that you plan on transferring to. Then use that school requirements to take classes at the school you currently want to go to. If you have any other question feel free to email mlefort@me.com I just recently transferred to a college from an online and community college and I lost no credites because I picked really wisely.