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\CopyRight{2008 by Fn1. Last\_name1, Fn2. Last\_name2, Fn3.
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%\CopyRight{2008 by T. Bouhadada, M.-T. Laskri}


\Headings{Fn1. Last\_name1, Fn2. Last\_name2, Fn3. Last\_name3}
         {Short title of the paper \ldots}
%\Headings{T. Bouhadada, M.-T. Laskri}

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\title{The title of the paper\thanks{This work was supported by
YYYYYYY project Ref. Nr. XXXXXXXXX}}

\author{First\_name1 Last\_name1, First\_name2 Last\_name2, \and First\_name3 Last\_name3}
%\author{Tahar Bouhadada, Mohamed-Tayeb Laskri}

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\begin{document}
\maketitle

\begin{abstract}
This paper describes the architecture of an Interactive Learning
Environment (ILE) on internet using companions, one of which is a
human and geographically distant from the learning site. The
achieved system rests on a three-tier customer/server architecture
(customer, web server, data and applications server) where human
and software actors can communicate via the internet and use the
DTL learning strategy. It contains five main actors: a tutor actor
in charge to guide the learner; a system actor whose role is to
manage and to control the accesses to the system; a teacher actor
in charge of the management and the updating of the different
bases; a learner actor who represents the main actor of the system
for whom is dedicated the teaching. Also, a learning companion
actor whose role can be sometimes as an assistant, and other times
as a troublemaker.

\textbf{Keywords:} Interactive learning environment, LCS, DTL
strategy, companion, distance learning, troublemaker.
\end{abstract}

\section{Introduction}

The distant teaching pedagogy differs from the teaching in a
classroom. Indeed, the absence of the teacher influences the
incentive and the concentration of the learner, what encourages
the isolation feeling and so, moves him away of the stimulating
context as in a real classroom.

In a distant learning context, the pedagogical triangle [1],[2]
must take into account two elements that, in this case, take a
particular importance: the group and the mediation context (Figure
1).

\begin{figure}
\begin{center}
\includegraphics{fig1.eps}
%%% or with the help of command
%%% \epsfig{file=fig_1.eps,height=27.1mm,width=64.9mm}
\caption{The pedagogical triangle} \label{figure01}
\end{center}
\end{figure}

The group is an instituted set of learners and teachers in
interaction, sharing some common objectives. The introduction of
the group element puts in evidence the social character of the
knowledge construction [3]. Indeed, the group constitutes a
psychological support factor [4]. The mediation context
constitutes the material or a virtual environment in which occurs
the interactions.

In the present work, we describe an interactive learning
environment (ILE) in a distant-teaching context with learning
companions and using Internet as the environment of communication
and interaction. The achieved system is a software framework
dedicated to the learning of the relational databases whose
customer/server architecture is based on multi-agents approach.
For the communication between the learners, we used tools, more
powerful, as the electronic mailing, the forums, that have already
been integrated in many distant-training frameworks as support for
collective learning activities [5],[6].

....

\section{Definition of the diagnosis}
Diagnosis is defined as a process of exact failure cause
localization. Once the failure is detected, it is the
responsibility of the maintenance engineer to recognize the
effects, to analyze information, to interpret the various error
messages and indications, and to leave with the true diagnosis of
the situation in term of components having caused the failure and
the reason of their failures. When the diagnosis is completed, the
replacement or the repair of the component at the origin of the
failure is the following defect correction stage. The companies
are thus confronted with this double economic challenge:
\begin{itemize}
\item to increase the productivity by increasing availability of
their equipments production;

\item to reduce the maintenance costs.
\end{itemize}

...

\noindent \textbf{Corollary 2.4} - \textit{For a graph}
\textbf{K}$_n$ \textit{with} $n\ge3$, \textit{we have}:

$$
\left\{
\begin{array}{l}
\overline{\chi}(\textbf{K}_n)=\frac{9k^2-7k}{3} \hspace{1.0cm} \text{if} \hspace{0.5cm} n=3k\\
\overline{\chi}(\textbf{K}_n)=\frac{9k^2+k-2}{2} \hspace{0.8cm} \text{if} \hspace{0.5cm} n=3k+1\\
\overline{\chi}(\textbf{K}_n)=\frac{9k^2+5k-2}{2} \hspace{0.65cm} \text{if} \hspace{0.5cm} n=3k+2\\
\end{array}
\right.
$$

...

At each iteration, the algorithm computes the overall minimum
distance between clusters and merges those being at that distance
from each other. Thus, for any $i < j$, we have

\begin{equation}
\label{eq1} d(C_{i} ,C_{j} ) = d_{\min}  \Rightarrow C_{i} : =
C_{i} \cup C_{j} ,C_{j} : = \phi
\end{equation}

\noindent where

\begin{equation}
\label{eq2} d_{\min}  = {\mathop {\min} \limits_{i \ne j \in
[1,n]}} d(C_{i} ,C_{j} ),
\end{equation}

...

\section{Dimension of d-Convex Simple Planar\\ Graphs}
\subsection{Example of subsection A}
Next, the feature extraction procedure described in section 2 is
performed on the five images. For each image we consider square
blocks of size $a = 8$ and compute their standard deviation. The
resulted dispersion-based graphical feature set, $\{V(I_{1}
),...,V(I_{5} )\}$ is represented in Fig. 2. Each image feature
vector $V(I_{i} )$ constitutes a [31x41] matrix. It is displayed
as a $3D$ surface plot in Fig. 2, being indicated by its number,
$i$.

\begin{figure}[htbp]
\centerline{\includegraphics[width=5.47in,height=3.07in]{fig2.eps}}
%%% or with the help of command
%%% \epsfig{file=fig_1.eps,height=27.1mm,width=64.9mm}
\caption{The feature vector set} \label{fig2}
\end{figure}

The automatic classification provided in section 3 is applied to
this feature set. The Euclidean distance is used in the
classification process, because the image feature vectors have
identical sizes. In the next table there are registered the
computed distance values between all the pairs of feature vectors.

\begin{table}[htbp]
\caption{Distances between image feature vectors} \vspace{3mm}
\begin{tabular}
{|p{34pt}|p{44pt}|p{44pt}|p{44pt}|p{44pt}|p{44pt}|} \hline &
$V(I_{1} )$& $V(I_{2} )$& $V(I_{3} )$& $V(I_{4} )$&
$V(I_{5} )$ \\
\hline $V(I_{1} )$& 0& 571.3183& 293.0381& 675.6527&
319.3169 \\
\hline $V(I_{2} )$& 571.3183& 0& 599.5098& 359.3718&
618.9163 \\
\hline $V(I_{3} )$& 293.0381& 599.5098& 0& 686.5573&
361.6215 \\
\hline $V(I_{4} )$& 675.6527& 359.3718& 686.5573& 0&
712.8829 \\
\hline $V(I_{5} )$& 319.3169& 618.9163& 361.6215& 712.8829&
0 \\
\hline
\end{tabular}
\label{tab1}
\end{table}

...

\subsection{Example of subsection B}

In case of d-convex graph here we have always equality
$|D|=n+m+2$. Let us denote by $\Gamma(x)$ the neighborhood of
vertex $x$, i.e. $\Gamma(x) = \{ y\in X | x\sim y \} $.

\begin{definition}
\bf{[5].} \it{A vertex $y$ is called copy for vertex $x\: (x \neq
y)$, in graph $G=(X;\: U)$ if $\Gamma(x) = \Gamma(y)$.}
\end{definition}

Let $T$ be a tree with at least 3 vertexes and $T_0$ a sub-graph
of $T$, that consists of all vertexes and edges of $T$, without
those suspended. So, for each unsuspended vertex $x$ from $T$, we
have a uniquely correspondent vertex $\bar x$ from $T_0$, and for
each vertex of $T_0$ we have a uniquely correspondent vertex from
$T$. Let $L(T,\: T_0)$ be a graph obtained from $T$, $T_0$ and by
adding the following edges: every vertex $\bar x$ of $T_0$ will be
adjacent with all vertexes from $\Gamma(x)$ from $T$, where $x$
and $\bar x$ are correspondent vertexes. It is easy to see that in
graph $L(T,\: T_0)$ every vertex of degree at least 3 has a unique
copy and there are no suspended vertexes.

The next theorem is true:

\begin{theorem}
\bf{[6].} \it{If $T$ is a tree with at least 3 vertexes, then
graph $G = L(T,\: T_0)$ is d-convex simple and planar.}
\end{theorem}

...

Let $\V$ be a finite generating set of the monoid of nonnegative
integer solutions
$(\psi_0,\psi_1,\ldots,\psi_m,\eta_1,\ldots,\eta_m,\eta_0)$ of:
\begin{equation}\label{miller4.1}
\psi_0 \mdeg(g) + \sum_{\nu=1}^m \psi_{\nu} \mdeg(f_{\nu}) =
\sum_{\nu=1}^m \eta_{\nu} \mdeg(f_{\nu}) + \eta_0 \mdeg(h).
\end{equation}

...

\begin{algorithm}\label{millerAlg2ref} \hspace{1cm} \\
\framebox[1.1\width]{
\begin{minipage}[c]{.8\textwidth}
Input: $g,h \in A$, a finite SAGBI basis $F$ for $A$\\
Output: A syzygy family $\mathrm{SyzFam}(g,h)$ for $g$ and $h$\\
Initialisation: $\mathrm{SyzFam}(g,h):=\emptyset$,
$\mathcal{PV}:=\emptyset$\\
Compute a generating set $\V$ for the solutions of
system~(\ref{miller4.1}).\\
$\mathcal{PV}:=\{\vec{v} \in \V : c_0 = d_0 = 1\}$\\
For Each $\vec{v} \in \mathcal{PV}$:\\
\makebox{$\hspace{1.5em}$} $s_{\vec{v}}:=\lc(H^{\vec{v}^r})\cdot
G^{\vec{v}^l} -
\lc(G^{\vec{v}^l})\cdot H^{\vec{v}^r}$\\
$ \mathrm{SyzFam}(g,h):=\bigcup_{\vec{v}\in \mathcal{PV}}
\{s_{\vec{v}}\} $
\end{minipage}
}
\end{algorithm}

An implementation of this algorithm is included in the author's
Maple package for SAGBI and SAGBI-Gr\"obner computations,
see~\cite{myMaplePackage}. For calculating the Hilbert bases the
Maple package uses Dmitrii~V. Pasechnik's implementation of the
algorithm described in~\cite{PasechnikHB}.

As an application of Algorithm~\ref{millerAlg2ref} we consider
example 4.7 and 5.2 in~\cite{millerSG}.
\begin{example}
Let $A = \mathbb{Q}[x^2,xy] \subseteq \mathbb{Q}[x,y]$ and use the
degree lexicographical order with $x > y$. The set $F=\{x^2,xy\}$
is a SAGBI basis for $A$. Let $g = x^3 y + x^2$ and $h = x^4 + x^2
y^2$ in $A$. A Hilbert basis for the set of solutions of the
equation~(\ref{miller4.1}) is:
\[ \begin{array}{ccc}
\vec{v_1} = (0,0,1,0,1,0), & \vec{v_2} = (0,1,0,1,0,0), &
\vec{v_3} =
(0,2,0,0,0,1), \\
\vec{v_4} = (1,0,0,1,1,0), & \vec{v_5} = (1,1,0,0,1,1), &
\vec{v_6} = (2,0,0,0,2,1).
\end{array}\]
Thus $\mathcal{PV} =\{\vec{v_5}\}$, so by
Algorithm~\ref{millerAlg2ref} a syzygy family for $(g,h)$ is
$\{G^{(1,1,0)} - H^{(0,1,1)}\} =\{ -x^3 y^3 + x^4\}$.

In the original version of this example (example~4.7 in
\cite{millerSG}) the syzygy family was $\{-x^5 y^3 + x^6, -x^3 y^3
+ x^4 \}$ instead. It should however be noted (as proved in
example~5.2, \cite{millerSG}) that the extra syzygy polynomial
$-x^5 y^3 + x^6$ SI-reduces to zero over $\{g,h\}$. Thus this
extra polynomial does not affect the final result of
SAGBI-Gr\"obner basis computations.
\end{example}



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\bibitem{SwedId}
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\bibitem{myMaplePackage}
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\end{thebibliography}


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\begin{parbox}{118mm}{\footnotesize
First\_name1 Last\_name1, First\_name2 Last\_name2,   \hfill
Received October 17, 2007

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Institution

Address

Phone:

E--mail:

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Institution

Address

Phone:

E--mail:

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First\_name3 Last\_name3

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}

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\end{center}

\end{document}