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\begin{document}

\title{ \textbf{ Un titlu remarcabil de unic}}
\author{ Frentescu Stefan}
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\section*{Section 1}
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\section*{Section 2}
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\section*{Section 3}
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\begin{multicols}{2}

\hspace{-5.3mm}\textbf{DOUBLE EXPONENTIAL \\
DISTRIBUTIONS}

\hspace{5mm}Let be $\mathrm{a_t}$ be a serios of independent identically \hspace{5mm} double-exponentially \hspace{5mm} (Laplace) distributed random variables, i.e. with probability density function(\textbf{PDF}) given by $$\displaystyle{ f(a) = \frac{\lambda}{2}e^{-\lambda |a|} , \lambda > 0 \hspace{5mm} \forall a}$$

Let be the observed stationary time series $ \mathrm{\{X_t\}} $ be generated by the \textbf{ARMA} scheme
$$\Phi ( B ) X_t = \Theta ( B ) a_t $$ where $$ \Theta ( B ) = (1 - \phi_{_1} B - ... - \phi_q B^q ) $$
$$ \Phi ( B )  = ( 1 - \phi_{_1} B - \phi_{_2} B^2 - ... - \phi_p B^p ) $$ and B is the backward shift operator so that $$\mathrm{ B^k X_t = X_{t-k}.} $$

\hspace{5mm} Since the series $\mathrm{\{X_t\}} $ is assumed to be stationary, all the roots of \hspace{2mm} lie outside the unit circle, and we can write the moving average
$$ X_{_t} = \Phi^{-1} ( B )  \Theta ( B ) a_t = \Psi (B ) \sum\nolimits_{j = 0}^{m} \mathit{\Psi_j a_{t-j}} $$
where the coefficients \boldsymbol{$ \Psi ( B )  = 1 + \psi_1 B + ...    $} can be found by equating coefficients in
$$ \Phi( B )  \Psi ( B ) = \Theta ( B ). $$
Let
\begin{equation} Z_{_n} = \sum_{j = 0}^n \mathit{\Psi_{_j} a_{_{t-j}}} \end{equation}
and assume $\scriptstyle{\Psi_y \neq \Psi_j} $ for i $\neq$ j.

\hspace{5mm} The \textbf{PDF} of $ \mathrm{Z_n} $ is given by
\large{$$ \displaystyle{ f_n (z) = \frac{\lambda}{2} \sum\nolimits_{j=0}^{n} \alpha_{_j}^{ ( n )} \left| \mathit{\Psi}_{_j} \right|^{-1}  \mathrm{exp} ( - \lambda \left| \frac{z}{\mathit{\Psi}_{_j}} \right|)}$$} \normalsize{\hspace{-1.5mm}where $\mathrm{\alpha_{j}^{(n)} }$ are functions of \hspace{1mm} $\mathrm{\scriptstyle{\{\Psi_i\}}}$
the given by
} $$ \alpha_{_j}^{(n)} = \prod_{i=0,i \neq j} (1 - \left|\mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| )^{-1}, \hspace{2mm} \mathit{j} = 1,2,...,n $$

\hspace{5mm}Now
\begin{equation} \displaystyle\ f(x) = \lim_{_{n \rightarrow \infty}} f_n(x)  $$
and obtain the following expression for the marginal \textbf{PDF} of $\mathrm{X_t}$
$$ f(x) = \frac{\lambda}{2} \sum\nolimits_{j=0}^{\infty} \alpha_{_j} \left| \mathit{\Psi_{_j}} \right|^{-1 } \mathrm{exp} (-\lambda \left| \mathit{\frac{x}{\Psi_{_j}}} \right| )   \
\end{equation}
and $$ \alpha_{_j} = \prod_{_{i=0,i \neq j}}^{\infty} {(1 - \left| \mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| ) }^{-1} $$

\hspace{5mm} These results follows as special case of \textit{Box's} [3], where he derives the distribution of any linear combination of independent $ \chi^2 $ variables with even degree of freedom by nothing that each $\mathrm{a_i}$ may be written as a constant times as difference \hspace{3mm} between \hspace{3mm} two independent \hspace{3mm} $\chi^2$ variables. \textit{Preda} [8] generalized the above fact to mixed double-exponentially.

\subsection*{\center{1. STATISTICAL MODEL}}
\subsubsection*{\center{Autoregressive model}}

\hspace{16.4mm}Let be the time series Z and Y be represented by autoregressive models of order p
$$ Z_{_t} = \mu + \sum\nolimits_{i=1}^p \phi_{_i}(Z_{_{t-i}} -\mu) + a_{_t} $$
and
$$ Y_{_t} = \xi + \sum\nolimits_{i=1}^p \mathit{\Theta}_{_i}(Y_{_{t-i}} -\xi) + b_{_t} $$
where $\phi_{_i}$ and $\mathit{\Theta_{_i}}$ (i=1,...p) are the autoregressive parameters and $\mu$ and $\xi$ are the mean of the series Z and Y, respectively. $\mathrm{\{a_{t}  \} and \{ b_t \}}$ are white-noise \hspace{3mm} processes \hspace{2mm} with $\mathrm{E(a_t)=E(b_t)=0,}$ \hspace{2mm} and Cov($\mathrm{a_t , b_{t+r}=0,}r \neq 0$) \hspace{2mm} and Var($\mathrm{a_t}$)$=\mathrm{\hspace{2mm} \sigma^2_{z}}$ \hspace{2mm} and Var($\mathrm{b_t}) = \mathrm{\sigma^2_{y}}.$

\hspace{5mm}Since the assumption of independence is very limited in practive, we assume that the joint distribution of $\mathrm{a_t}$ and $\mathrm{b_t}$ is bivariate , so that Cov($\mathrm{a_t,b_{t+r}}$) $=\mathrm{\rho \sigma_z \sigma_y \forall r.}$

\hspace{5mm} Let denote by $\mathrm{Z_n(L) = E(Z_{n+L} |z^{(n)})}$ the expecte value of Z at time n+L and $\mathrm{z^{(n)} = z_1,...,z_n}$ the set of observations form Z. Similarly denote by $\mathrm{Y_m(K) = E(Z_{m+K}|y^{(m)})}$ the expected value of Y at time m+K and the set of observations form Y.

\hspace{10mm} We get
$$ Z_{_{n+L}}(L) = \mu + \sum\nolimits_{i=1}^p \phi_{_i}(Z_{_n}(L-i)-\mu) $$

\hspace{5mm} The difference $\mathrm{Z_{n+L}-Z_n(L)}$ between actual and expected value Z at time n+L will be denoted by $\mathrm{e_n(L)}$.We can show that
$$e_n(L) = \sum\nolimits_{i=0}^{L-1} R_{_i}a_{n+L-i}$$
where
$$ \mathrm{R_{_0} = 1, R_{_1} = \phi_{_1} , ... , R_{_j} }= \sum\nolimits_{i=1}^p \phi_{_i} R_{_{j-i}} $$
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tion, signature recognition, keystroke, teeth image recoginition , ADN, etc.\\
The protocol can be used not only for two persons who wants to communicate but also for a group communication. The second situation is much more complex due to the number of the authentications that must be made. We suppose that all the users that have been authenticated (even if they are two or more) have acces to the messages transmitted on the network. An authenticated user can send messages, read messages even if they are not addresed to him, and modify messages that he did not send. All the users that have been authenticated have the same common key so they all can see if one message have been modified and what are the modifications.
\section{Fuzzy Model Construction}
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