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\begin{document}
\selectlanguage{english}
\title{ \textbf{ Un titlu remarcabil de unic}}
\author{ Frentescu Stefan}
\maketitle

\abstract
\noindent \blindtext
%}
\section*{Section 1}
\lipsum
\section*{Section 2}
\lipsum
\section*{Section 3}
\lipsum
\newpage
\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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\chead{LATEST TRENDS on COMPUTERS (Volume I)}
\cfoot{170}
\rfoot{ISBN: 978-960-474-201-1}
\lfoot{ISNN: 1792-4251}
\hspace{-5mm}tion, signature recognition, keystroke, teeth image recognition , 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.

\subsection*{3.1 Fuzzy Model Construction}

Fuzzy sets were introduced by Zadeh (1965) in order to represent and manipulate date that was not precise, but rather fuzzy. Similarly with the crisp case, a fuzzy subset. A of a set X is defined as a collection of ordered pairs with the first element from X and the second element from the interval [0,1] ; the set X is referred to as the universe of discourse for the fuzzy subset A.
\newline \newline \textbf{Definition 1.} \textit{If X is a nonempty set then a fuzzy set A in X is defined by its membership function} $\mu_A : X \rightarrow [0,1]$ ,\textit{ where} $\mu_A(x)$ \textit{represents the membership degree of the element} $x$ \textit{in the fuzzy set A ; then A is represented as A} $=$ \{ $(x,\mu_A(x)) / x\in X$ \}.
\newline

Because the majority of practical applications work with trapezoidal or triangular distributions and these representations are still a subject of various recent papers ( [20], [21] for instance) we will work with membership functions represented by trapezoidal fuzzy numbers. Such a number $N = (\underline{m},\overline{m},\alpha,\beta)$ is defined as

\[\mu_N(x) = \left\{
  \begin{array}{lr}
    %x^2 & : x < 0\\
    %x^3 & : x \ge 0
        0 \hspace{2mm} for \hspace{2mm} x < \underline{m} - \alpha \\
        \frac{x-\underline{m} + \alpha}{\alpha} for x \in [\underline{m} - \alpha , \underline{m}] \\
        1 \hspace{2mm} for\hspace{2mm} x \in [\underline{m} , \overline{m}] \\
        \frac{\overline{m} + \beta - x}{\beta} \hspace{2mm} for \hspace{2mm} x \in [\overline{m},\overline m + \beta]\\
        0 \hspace{2mm} for \hspace{2mm} x > \overline m + \beta
  \end{array}
\right.
\]
We are interested to compute the gravity center of a trapezoidal fuzzy numer. \hspace{2mm} For a nonself-intersecting polygon defined by n vertices $(x_0,y0) , (x_1,y_1), ... , (x_{n-1},y_{n-1}) $ the gravity center $ G = ( G^x, G^y)$ is given by [22]
$$G^x = \frac{1}{6A} \sum_{i=0}^{n-1} (x_i + x_{i+1})(x_i y_{i + 1 } - x_{i+1} y_i) $$


$$G^y = \frac{1}{6A} \sum_{i=0}^{n-1} (y_i + y_{i+1})(x_i y_{i + 1 } - x_{i+1} y_i) $$
where A is the polygon's area
$$ A = \frac{1}{2} \sum_{i=0}^{n-1} (x_i y_{i+1} - x_{i+1} y_i). $$
For the trapezium $(\underline m , \overline m , \alpha , \beta)$ the previous formulas give
$$ G^x = \frac{(\overline m + \beta)^2 - (\underline m - \alpha)^2 + 2(\overline m ^2 - \underline m ^2)}{6(\overline m - \underline m ) + 3(\alpha + \beta)}+  $$
$$ + \frac{\underline m \alpha + \overline m \beta}{6(\overline m - \underline m ) + 3(\alpha + B)} $$
$$ G^y = \frac{3(\overline m - \underline m )+ \alpha + \beta}{6(\overline m - \underline m )+ 3(\alpha + \beta)} $$
Given a lot of fuzzy sets \{$F_1,F_2,...,F_n\}$ with trapezoidal membership functions, we define the composition of two fuzzy sets $F_i$ and $F_j$ as
\begin{itemize}
\item compute the gravity centers $G_i$\hspace{1mm}$=(G_{i}^x, G_{i}^y)$ and \hspace{-2mm}$G_j \hspace{2mm}\mathrm{=} \hspace{2mm} ((G_{j}^x, G_{j}^y))$ respectively, corresponding \hspace{-0.1mm}of trapezoidal numbers $(\underline m _i,\overline m _i, \alpha_i, \beta_i)$ and $(\underline m_j,\overline m _j, \alpha_j,\beta_j)$ respectively

\item compute $x_{ij} = \frac{G_{i}^x + G_{j}^x}{2}$

\item $ k \in {1,2,...,n}$ is given by $$  \mu_F_k ( x_{ij}) = max_{l=1,n} \mu_F_l (x_{ij}) $$

\item define $ F_i \otimes F_j = F_k.$

\end{itemize}

Let A and B be fuzzy sets in the universe $ U\subseteq R,$ represented by trapezoidal numbers $(\underline m _A , \overline m_A, \alpha _A, \beta_A ) $ and $(\underline m_B, \overline m_B , \alpha _B , \beta_B)$ respectively, with the gravity centers $G_A$ and $G_B$ respectively. We define the distance between A and B as follows:
\setcounter{equation}{0}
\begin{equation}
d(A,B) = \frac{d_E(G_A,G_B)}{|U|}
\end{equation}
where $d_E$ is the Euclidean distance and $|U|$ is the length of $U$; it is obviously that $d(A,B) \in [0,1]$.
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\footnotesize{2} \vspace{-5mm}\center{ \footnotesize{N.S PAPAGEORGIOU, V. RĂDULESCU, AND D.REPOVS}}
\end{flushleft}

Let $u \in {K_\hat\sigma}_{_\lambda}$. Then

\leqnomode
\begin{align}
 \left\langle   A_p (u),h \right\rangle  + \left\langle A(u),h \right\rangle = \int_\Omega \lambda \hat g (z,u) hdz \mathrm{\,\, for\,\, all } \,\, h \in W_{0}^{1,p} (\Omega).%\tag{e}
\end{align}

In (2) we choose $h = (u - u_{\lambda}^*)^+ \in W_{0}^{1,p} (\Omega). $ Then
$$ \left\langle A_p(u),(u-u_\lambda^*)^+ \right\rangle + \left\langle A(u), (u-u_\lambda^*)^+ \right\rangle$$
\leqnomode

\begin {align*}
\longrightarrow\tab =\tab& \lambda \int_\lambda f(z,u_\lambda^*)(u-u_\lambda^*)^+ dz \mathrm{(see (1))} \tag{2'}& \\
\longleftrightarrow \tab=\tab &  \left\langle A_p(u_\lambda^*),(u-u_\lambda^*)^+ \right\rangle + \left\langle A(u), (u-u_\lambda^*)^+\right\rangle \mathrm{(since}\,\, u_\lambda^* \in S_+ \mathrm{)},& \\
\Rightarrow \hspace{4.5mm} & \left\langle  A_p(u) - A_p(u_\lambda^*),(u-u_\lambda^*)^+ \right\rangle + \left\langle A(u) - A(u_\lambda^*),(u-u_\lambda^*)^+ \right\rangle = 0, & \\
\Rightarrow \hspace{4.5mm} & ||D(u-u_\lambda^*)^+ ||_{2}^2 = 0, & \\
\Rightarrow \hspace{4.5mm}& u \leq u_\lambda^* &
\end{align*}

Similarly, if in (2) we choose $h = (v_\lambda^* -u)^+ \in W_0^{1,p} (\Omega),$ then we obtain
%\leqnomode
\begin{align}\label{$\A_\alpha$}
\tag{$A_\alpha$}
v_\lambda^* \leq u %\tag{A_\alpha}
\end{align}
So, we have proved that ~\eqref{$A_\alpha$}
$$ u \in [v_\lambda^*, u_\lambda^*] $$

Moreover, as before the nonlinear regularity theory ((see the proof of Proposition ??), implies that $u \in C_0^1(\overline \Omega)$
. Therefore we conclude that
$$ K_{\overline{\sigma}}_\lambda  \subseteq [v_\lambda^*,u_\lambda^*] \cap C_0^1 (\overline\Omega) $$

In a similar fashion we show that
\begin{align} \label{$P_\lambda$}
\tag{$P_\lambda$}
K{_\hat{\sigma}_\lambda^+} \subseteq [0,u_\lambda^*] \cap C_+ \,\,\mathrm{and}\,\, K{_\hat{\sigma}_\lambda^-} \subseteq [v_\lambda^*,0] \cap (-C_+)
\end{align}
The extremality of $u_\lambda^*$ and $v_\lambda^*$ ,implies that: ~\eqref{$A_\alpha$} and ~\eqref{$P_\lambda$}
$$ K{_\hat{\sigma}_\lambda^+} = \{0,u_\lambda^*\} \,\,\mathrm{and}\,\, K{_\hat{\sigma}_\lambda^-} = \{ v_\lambda^*,0\}. $$

This proves Claim 1.


On account of Claim 1, we see that we may assume that
\begin{align}
K{_\hat{\sigma}_\lambda}\mathrm{ \,is \,\,infinite}
\end{align}

Otherwise evidently we already have an infinity of smooth nodal solutions and so we are done.\\ \newline
\large{\textbf{Claim 2.}} \normalsize{$u_\lambda^* \in D_+ \,\,and\,\, v_\lambda^* \in \,\,-D_+ \,\,are \,\,local \,\,minimizers \,\,of \,\,the \,\,functional\,\, \hat{\sigma}_\lambda$}.

From(1) it is clear that $\hat{\sigma}_\lambda^+ $ is coercive. Also, $\hat{\sigma}_\lambda^+$  is sequentially weakly lower semicontinuous. So, we can find $\hat{u}_\lambda^* \in W_0^{1,p}(\Omega) $ such that
\begin{align}
\label{asta}
\hat{\sigma}_\lambda^+(\hat{u}_\lambda^*) = inf \left[\hat{\sigma}_\lambda^+(u) : u \in W_0^{1,p} (\Omega) \right].
\end{align}

Let $\hat{u_1}(p)$ be the positive principal eigenfunction of$(- \Delta_p , \W_0^{1,p}(\Omega))$.\tab We know that $\hat{u}_1(p) \in D_+$ (see Motreanu-Motreanu-Papageorgiou [16]). Recall that $u_\lambda^* \in D_+.$ So, invoking Lemma 3.6 of Filippakis-Papageorgiou [9], we can find $\tau$ $>$ 0 such that
\begin{align}
\tau \hat{u}_1(p) = \left[\frac{1}{2}u_\lambda^* , u_\lambda^* \right]
\end{align}

Evidently $\displaystyle{\frac{1}{2} \leq \tau \leq 2.}$ Hypothesis $H_1(ii)$ implies that there exists $\epsilon >0$ such that
\begin{align}
F(z,x) \geq \xi |x|^\beta \mathrm{\,\,for\,\, almost\,\, all\,\,} z \in \Omega , \mathrm{\,\,all\,\,} 0 \leq x \leq \eta
\end{align}
\pagebreak
\newpage
\begin{center}
NODAL SOLUTIONS FOR THE ROBIN P-LACIAN PLUS AND INDEFINITE POTENTIAL AND A GENERAL REACTION
\end{center}

\newline
(here $\eta \in (0,\delta_1)$ is as in (\ref{fail})). We have
$$ \hat\sigma_\lambda^+(\tau\hat u_1(p)) \leq \frac{\tau^p}{p}\hat\lambda_1(p) + \frac{\tau^2}{2}||D\hat u_1(p)||_2^2 - \lambda\xi\tau^\beta ||\hat u_1(p)||_\beta^\beta$$
with $\hat\lambda_1(p) > 0$ being the principal eigenvalue of $(-\Delta_p,W_0^{1,p}(\Omega))$.

It follow that
$$ \hat\sigma _\lambda^+(\tau\hat u_1(p)) < 0 $$
\begin{center}
if and only if
\end{center}
\begin{align}
\tag{$7_\mathrm{f}$}
\end{align}
\vspace{-10mm}
\begin{align*}
\frac{\frac{\tau^{p-2}}{p}\hat\lambda_1(p) + \frac{1}{2} ||D\hat u_1(p)||_2^2}{\xi \tau^{\beta-2}||\hat u_1(p)||_\beta^\beta} < \lambda
\end{align*}

Note that
$$ \frac{\frac{\tau^{p-2}}{p}\hat\lambda_1(p) + \frac{1}{2} ||D\hat u_1(p)||_2^2}{\xi \tau^{\beta-2}||\hat u_1(p)||_\beta^\beta} \leq \frac{\frac{2^{\beta+p-4}}{p}\hat\lambda_1(p)+2^{\beta - 3 }||D\hat u_1(p)||^2_2}{\xi||\hat u_1(p)||_\beta^\beta} $$
(recall that $\displaystyle{\frac{1}{2} \leq \tau \leq 2}$)

So, if we let $\lambda^0 = \frac{\frac{2^{\beta+p-4}}{p}\hat\lambda_1(p)+2^{\beta - 3 }||D\hat u_1(p)||^2_2}{\xi||\hat u_1(p)||_\beta^\beta}$ and define
$$ \lambda_*^0 = \mathrm{max}\{\lambda^0,\lambda_*\}, $$
then from (7) we infer that
\begin{align*}
\tab &\hat\sigma_\lambda^+(\tau \hat u_1(p)) < 0 \mathrm{\,\,for\,\,all\,\,}\lambda > \lambda_*^0, & \\
\Rightarrow \hspace{4.5mm}& \hat\sigma_\lambda^+(\hat u_\lambda^*) <0 = \hat\sigma_\lambda^+ (0) \mathrm{\,\,for\,\,all\,\,}\lambda > \lambda_*^0 (see~\eqref{asta}) & \\
\Rightarrow \hspace{4.5mm} & \hat u_\lambda^* \neq 0 \mathrm{\,\,and\,\,} \hat u_\lambda^* \in K_{\hat\sigma_\lambda^*} \mathrm{\,\,for\,\,all\,\,} \lambda > \lambda_*^0 (see~\eqref{asta}) & \\
\hspace{19.5mm}\Rightarrow\hspace{4.5mm} & \hat u_\lambda^* = u_\lambda^* \mathrm{\,\,for\,\,all\,\,} \lambda > \lambda_*^0 (\mathrm{\,\,see\,\, Claim\,\, 1}).
%From (3) and Claim 2, we see that we can find $\p\in (0,1)$ small such that
\end{align*}

From (1) it is clear that $\hat\sigma_\lambda^+|_C_+.$Since $u_\lambda^* \in D_+$, from (4) it follows that
\begin {align*}
 & u_\lambda^*\mathrm{\,\,is\,\,a\,\,local\,\,} C^1(\overline\Omega) - \mathrm{\,\,minimizer\,\,of\,\,} \hat\sigma_\lambda, & \\
\hspace{-150.5mm}\Rightarrow \hspace{4.5mm} & u_\lambda^* \mathrm{\,\,is\,\, a\,\, local\,\,} W_0^{1,p}(\Omega) - \mathrm{\,\,minimizer\,\,of\,\,} \hat\sigma_\lambda (\mathrm{see\,\,Proposition\,\,\ref{fail}}).
\end{align*}

Similarly for $v_\lambda^* \in -D_+$ using this time the functional $\hat\sigma_\lambda^-$

This proves Claim 2.

Without any loss of generality, we may assume that
\begin{align}
\tag{7'}
\hat\sigma_\lambda(v_\lambda^*) \leq \hat\sigma(u_\lambda^*).
\end{align}
\setcounter{equation}{7}

From (3) and Claim 2, we see that we can find $p\in (0,1)$ small such that
\begin{align}\label{asta2}
\hat\sigma_\lambda(v_\lambda^*)\leq\hat\sigma_\lambda(u_\lambda^*) < \mathrm{\,\,inf\,\,} [\hat\sigma_\lambda(u): ||u-u_\lambda^*|| = \rho] = \hat m_\lambda, ||v_\lambda^*-u_\lambda^*||<\rho
\end{align}
(see Aizicovici-Papageorgiou-Staicu [1], proof of Proposition 29). The functional $\hat\sigma_\lambda$ is coercive (see(1)). Hence
\begin{align}
\label{asta3}
\hat\sigma_\lambda \mathrm{\,\,satisfies \,\,the \,\,PS-condition}
\end{align}

Then ~\eqref{asta2},~\eqref{asta3} permit the use of Theorem \ref{fail} (the mountain pass theorem). So, we can find $y_\lambda\in W_0^{1,p}(\Omega)$ such that
\begin{align}
y_\lambda\in K{_\hat\sigma_\lambda} \mathrm{\,\,and\,\,}\hat m_\lambda \leq \hat\sigma_\lambda(y_\lambda)
\end{align}
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