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  1. \RequirePackage{fix-cm}
  2. \documentclass[10pt,a4paper]{article}
  3.  
  4. %\usepackage{titlesec}
  5. %\renewcommand{\thesubsubsection}{(\roman{subsubsection})}
  6. %\usepackage{textgreek}
  7. \usepackage[english,romanian]{babel}
  8. \usepackage{newunicodechar}
  9. %\usepackage[romanian]{babel}
  10. \usepackage{combelow}
  11. \usepackage{memhfixc}
  12. \usepackage{fancyhdr}  
  13. \usepackage[utf8]{inputenc}
  14. \usepackage{libertine}
  15. \usepackage{blindtext}
  16. \usepackage{geometry}
  17. \usepackage{lipsum}
  18. \usepackage{multicol}
  19. \usepackage [reqno]{amsmath}
  20. \usepackage {bm}
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  22. \usepackage{graphicx}
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  25. \usepackage{siunitx}
  26. \usepackage{upgreek}
  27. \usepackage{float}
  28. \usepackage{tempora}  % Times for numbers in math mode
  29. \usepackage{amsmath}
  30. \usepackage{pifont}
  31. \usepackage{mathtools}
  32. \usepackage{makecell,adjmulticol}
  33. \usepackage{multirow}
  34. %\usepackage{newtxmath}
  35. \graphicspath{ {C:/Users/Frentescu Stefan/Desktop} }
  36. \makeatletter
  37. \newcommand{\leqnomode}{\tagsleft@true}
  38. \newcommand{\reqnomode}{\tagsleft@false}
  39. \newcommand\tab[1][5mm]{\hspace*{#1}}
  40. \makeatother
  41.  
  42. \geometry{
  43. a4paper,
  44. total={170mm,257mm},
  45. left=20mm,
  46. top=20mm,
  47. }
  48. \setlength{\columnsep}{1cm}
  49. \lhead{\headerL}
  50. \chead{\headerC}
  51. \rhead{\headerR}
  52. \def\headerL{}
  53. \def\headerC{}
  54. \def\headerR{}
  55. \newcommand{\thispageheader}[2][R]{\expandafter\def\csname header#1\endcsname{#2}}
  56.  
  57. \begin{document}
  58. \selectlanguage{english}   
  59. \title{ \textbf{ Un titlu remarcabil de unic}}
  60. \author{ Frentescu Stefan}
  61. \maketitle
  62.  
  63. \abstract
  64. \noindent \blindtext
  65. %}
  66. \section*{Section 1}
  67. \lipsum
  68. \section*{Section 2}
  69. \lipsum
  70. \section*{Section 3}
  71. \lipsum
  72. \newpage
  73.  
  74. \begin{multicols}{2}
  75.    
  76. \section*{DOUBLE EXPONENTIAL DISTRIBUTIONS}
  77.  
  78. Let be $\mathrm{a_t}$ be a serios of independent identically double-exponentially (Laplace) distributed random variables, i.e. with probability density function(\textbf{PDF}) given by $$\displaystyle{ f(a) = \frac{\lambda}{2}e^{-\lambda |a|} , \lambda > 0 \forall a}$$
  80. Let be the observed stationary time series $ \mathrm{\{X_t\}} $ be generated by the \textbf{ARMA} scheme
  81. $$\Phi ( B ) X_t = \Theta ( B ) a_t $$ where $$ \Theta ( B ) = (1 - \phi_{_1} B - ... - \phi_q B^q ) $$
  82. $$ \Phi ( B )  = ( 1 - \phi_{_1} B - \phi_{_2} B^2 - ... - \phi_p B^p ) $$ and B is the backward shift operator so that $$\mathrm{ B^k X_t = X_{t-k}.} $$
  84. Since the series $\mathrm{\{X_t\}} $ is assumed to be stationary, all the roots of lie outside the unit circle, and we can write the moving average
  85. $$ X_{_t} = \Phi^{-1} ( B )  \Theta ( B ) a_t = \Psi (B ) \sum\nolimits_{j = 0}^{m} \mathit{\Psi_j a_{t-j}} $$
  86. where the coefficients \boldsymbol{$ \Psi ( B )  = 1 + \psi_1 B + ...    $} can be found by equating coefficients in
  87. $$ \Phi( B )  \Psi ( B ) = \Theta ( B ). $$
  88. Let
  89. \begin{equation} Z_{_n} = \sum_{j = 0}^n \mathit{\Psi_{_j} a_{_{t-j}}} \end{equation}
  90. and assume $\scriptstyle{\Psi_y \neq \Psi_j} $ for i $\neq$ j.
  92. The \textbf{PDF} of $ \mathrm{Z_n} $ is given by
  93. \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{where $\mathrm{\alpha_{j}^{(n)} }$ are functions of $\mathrm{\scriptstyle{\{\Psi_i\}}}$
  94. the given by
  95. } $$ \alpha_{_j}^{(n)} = \prod_{i=0,i \neq j} (1 - \left|\mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| )^{-1}, \mathit{j} = 1,2,...,n $$
  97. Now
  98. \begin{equation} \displaystyle\ f(x) = \lim_{_{n \rightarrow \infty}} f_n(x)  $$
  99. and obtain the following expression for the marginal \textbf{PDF} of $\mathrm{X_t}$
  100. $$ 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| )   \
  101. \end{equation}
  102. and $$ \alpha_{_j} = \prod_{_{i=0,i \neq j}}^{\infty} {(1 - \left| \mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| ) }^{-1} $$
  104. 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 between two independent $\chi^2$ variables. \textit{Preda} [8] generalized the above fact to mixed double-exponentially.
  106. \section*{\center{1. STATISTICAL MODEL}}
  107. \subsection*{\center{Autoregressive model}}
  109. Let be the time series Z and Y be represented by autoregressive models of order p
  110. $$ Z_{_t} = \mu + \sum\nolimits_{i=1}^p \phi_{_i}(Z_{_{t-i}} -\mu) + a_{_t} $$
  111. and
  112. $$ Y_{_t} = \xi + \sum\nolimits_{i=1}^p \mathit{\Theta}_{_i}(Y_{_{t-i}} -\xi) + b_{_t} $$
  113. 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 processes with $\mathrm{E(a_t)=E(b_t)=0,}$ and Cov($\mathrm{a_t , b_{t+r}=0,}r \neq 0$) and Var($\mathrm{a_t}$)$=\mathrm{ \sigma^2_{z}}$ and Var($\mathrm{b_t}) = \mathrm{\sigma^2_{y}}.$
  115. 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($\mathrm{a_t,b_{t+r}}$) $=\mathrm{\rho \sigma_z \sigma_y \forall r.}$
  117. 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.
  119. We get
  120. $$ Z_{_{n+L}}(L) = \mu + \sum\nolimits_{i=1}^p \phi_{_i}(Z_{_n}(L-i)-\mu) $$
  122. 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
  123. $$e_n(L) = \sum\nolimits_{i=0}^{L-1} R_{_i}a_{n+L-i}$$
  124. where
  125. $$ \mathrm{R_{_0} = 1, R_{_1} = \phi_{_1} , ... , R_{_j} }= \sum\nolimits_{i=1}^p \phi_{_i} R_{_{j-i}} $$
  126. %// footer pt urmatoarea pagina sa nu uiti
  127. \pagenumbering{gobble}
  128.    
  129.     \pagebreak
  130.    
  131. \pagestyle{fancy}
  132. \renewcommand{\headrulewidth}{0pt} % no line in header area
  133. \fancyhead{}
  134. \chead{LATEST TRENDS on COMPUTERS (Volume I)}
  135. \cfoot{170}
  136. \rfoot{ISBN: 978-960-474-201-1}
  137. \lfoot{ISNN: 1792-4251}
  138. \hspace{-5mm}tion, signature recognition, keystroke, teeth image recognition , ADN, etc.
  139.  
  140. 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.
  141.  
  142. \section*{3.1 Fuzzy Model Construction}
  143.  
  144. 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
  145. \newline
  146. \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} $=$ \{ $(x,\mu_A(x)) / x\in X$ \}.
  147. \newline
  148.  
  149. 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
  150.  
  151. \[\mu_N(x) = \left\{
  152.  \begin{array}{lr}
  153.     %x^2 & : x < 0\\
  154.     %x^3 & : x \ge 0
  155.         0\,\,for\,\,x < \underline{m} - \alpha \\
  156.         \frac{x-\underline{m} + \alpha}{\alpha}\,\,for\,\, x\,\, \in [\underline{m} - \alpha , \underline{m}] \\
  157.         1\,\,for\,\,x\,\, \in [\underline{m} , \overline{m}] \\
  158.         \frac{\overline{m} + \beta - x}{\beta}\,\,for \,\, x\,\, \in [\overline{m},\overline m + \beta]\\
  159.         0\,\,for\,\,x\,\, > \overline m + \beta
  160.  \end{array}
  161. \right.
  162. \]
  163. We are interested to compute the gravity center of a trapezoidal fuzzy number. For a non-self-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]
  164. $$G^x = \frac{1}{6A} \sum_{i=0}^{n-1} (x_i + x_{i+1})(x_i y_{i + 1 } - x_{i+1} y_i) $$
  165. $$G^y = \frac{1}{6A} \sum_{i=0}^{n-1} (y_i + y_{i+1})(x_i y_{i + 1 } - x_{i+1} y_i) $$
  166. where A is the polygon's area
  167. $$ A = \frac{1}{2} \sum_{i=0}^{n-1} (x_i y_{i+1} - x_{i+1} y_i). $$
  168. For the trapezium $(\underline m , \overline m , \alpha , \beta)$ the previous formulas give
  169. $$ G^x = \frac{(\overline m + \beta)^2 - (\underline m - \alpha)^2 + 2(\overline m ^2 - \underline m ^2)}{6(\overline m - \underline m ) + 3(\alpha + \beta)}+  $$
  170. $$ + \frac{\underline m \alpha + \overline m \beta}{6(\overline m - \underline m ) + 3(\alpha + B)} $$
  171. $$ G^y = \frac{3(\overline m - \underline m )+ \alpha + \beta}{6(\overline m - \underline m )+ 3(\alpha + \beta)} $$
  172. %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
  173. \begin{itemize}
  174. \item compute the gravity centers $G_i$ $=(G_{i}^x, G_{i}^y)$ and $G_j \mathrm{=} ((G_{j}^x, G_{j}^y))$ respectively, corresponding 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
  175.  
  176. \item compute $x_{ij} = \frac{G_{i}^x + G_{j}^x}{2}$
  177.  
  178. \item $ k \in \{1,2,...,n\}$ is given by $$\mu_{F_k} ( x_{ij}) = max_{l=1,n} \mu_{F_l} (x_{ij}) $$
  180. \item define $ F_i \otimes F_j = F_k.$
  182. \end{itemize}
  184. 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:
  185. \setcounter{equation}{0}
  186. \begin{equation}
  187. d(A,B) = \frac{d_E(G_A,G_B)}{|U|}
  188. \end{equation}
  189. where $d_E$ is the Euclidean distance and $|U|$ is the length of $U$; it is obviously that $d(A,B) \in [0,1]$.
  190. \pagebreak
  191. \end{multicols}
  192.  
  193. \newpage
  194. \pagestyle{fancy}
  195. \renewcommand{\headrulewidth}{0pt} % no line in header area
  196. \setlength{\headheight}{50pt}
  197. \fancyhead{}
  198. \chead{N.S PAPAGEORGIOU, V. RĂDULESCU, AND D.REPOVS}
  199. \lhead{2}
  200. \cfoot{}
  201. \rfoot{}
  202. \lfoot{}
  203. Let $u \in K_{\hat{\sigma}_{\lambda}}$. Then
  204.  
  205. \leqnomode
  206. \begin{align}
  207. \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}
  208. \end{align}
  209.  
  210. In (2) we choose $h = (u - u_{\lambda}^*)^+ \in W_{0}^{1,p} (\Omega). $ Then
  211. $$ \left\langle A_p(u),(u-u_\lambda^*)^+ \right\rangle + \left\langle A(u), (u-u_\lambda^*)^+ \right\rangle$$
  212. \leqnomode
  214. \begin {align*}
  215. \longrightarrow\tab =\tab& \lambda \int_\lambda f(z,u_\lambda^*)(u-u_\lambda^*)^+ dz \mathrm{(see (1))} \tag{2'}& \\
  216. \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{)},& \\
  217. \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, & \\
  218. \Rightarrow \hspace{4.5mm} & ||D(u-u_\lambda^*)^+ ||_{2}^2 = 0, & \\
  219. \Rightarrow \hspace{4.5mm}& u \leq u_\lambda^* &
  220. \end{align*}
  223. Similarly, if in (2) we choose $h = (v_\lambda^* -u)^+ \in W_0^{1,p} (\Omega),$ then we obtain
  224. %\leqnomode
  225. \begin{align}
  226. \tag{$A_\alpha$}
  227. v_\lambda^* \leq u
  228. \label{A de alpha}
  229. \end{align}
  230. So, we have proved that ~\eqref{A de alpha}
  231. $$ u \in [v_\lambda^*, u_\lambda^*] $$
  233. Moreover, as before the nonlinear regularity theory ((see the proof of Proposition ??), implies that $u \in C_0^1(\overline \Omega)$
  234. . Therefore we conclude that
  235. $$ K_{\overline{\sigma}_{\lambda}} \subseteq [v_{\lambda}^{*},u_{\lambda}^{*}] \cap C_0^1 (\overline{\Omega}) $$
  237. In a similar fashion we show that
  238. \begin{align}
  239. \tag{$P_\lambda$}
  240. K_{\hat{\sigma}_{\lambda}^+} \subseteq [0,u_{\lambda}^*] \cap C_+ \,\,\mathrm{and}\,\, K_{\hat{\sigma}_{\lambda}^{-}} \subseteq [v_{\lambda}^{*},0] \cap (-C_+)  \label{P de lambda}
  241. \end{align}
  242. The extremality of $u_\lambda^*$ and $v_\lambda^*$ ,implies that: ~\eqref{A de alpha} and ~\eqref{P de lambda}
  243. $$ K_{\hat{\sigma}_\lambda^+} = \{0,u_\lambda^*\} \,\,\mathrm{and}\,\, K_{\hat{\sigma}_\lambda^-} = \{ v_\lambda^*,0\}. $$
  245. This proves Claim 1.
  247. On account of Claim 1, we see that we may assume that
  248. \begin{align}
  249. K_{\hat{\sigma}_\lambda} \mathrm{ \,is \,\,infinite}
  250. \end{align}
  252. Otherwise evidently we already have an infinity of smooth nodal solutions and so we are done.\\ \newline
  253. \large{\textbf{Claim 2.}} \normalsize{$u_\lambda^* \in D_+ \,\,and\,\, v_\lambda^* \in \,\,-D_+ \,\,are \,\,local \,\,minimizers \,\,of \,\,the \,\,functional\,\, \hat{\sigma}_\lambda$}.
  255. 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
  256. \begin{align}
  257. \label{asta}
  258. \hat{\sigma}_\lambda^+(\hat{u}_\lambda^*) = inf \left[\hat{\sigma}_\lambda^+(u) : u \in W_0^{1,p} (\Omega) \right].
  259. \end{align}
  261. 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
  262. \begin{align}
  263. \tau \hat{u}_1(p) = \left[\frac{1}{2}u_\lambda^* , u_\lambda^* \right]
  264. \end{align}
  265. Evidently $\displaystyle{\frac{1}{2} \leq \tau \leq 2.}$ Hypothesis $H_1(ii)$ implies that there exists $\epsilon >0$ such that
  266. \begin{align}
  267. F(z,x) \geq \xi |x|^\beta \mathrm{\,\,for\,\, almost\,\, all\,\,} z \in \Omega , \mathrm{\,\,all\,\,} 0 \leq x \leq \eta
  268. \end{align}
  269. \pagebreak
  270. \newpage
  271. \chead{NODAL SOLUTIONS FOR THE ROBIN P-LACIAN PLUS AND INDEFINITE POTENTIAL AND A GENERAL REACTION}
  272. \rhead{}
  273. \lhead{}
  274. \newpage
  276. (here $\eta \in (0,\delta_1)$ is as in (\ref{fail})). We have
  277. $$ \hat\sigma_\lambda^+(\tau\hat u_1(p)) \leq \frac{\tau^p}{p}\hat\lambda_1(p) + \frac{\tau^2}{2}||D\hat u_1(p)||_2^2 - \lambda\xi\tau^\beta ||\hat u_1(p)||_\beta^\beta$$
  278. with $\hat\lambda_1(p) > 0$ being the principal eigenvalue of $(-\Delta_p,W_0^{1,p}(\Omega))$.
  281. It follow that
  282. $$ \hat\sigma _\lambda^+(\tau\hat u_1(p)) < 0 $$
  283. \begin{center}
  284. if and only if
  285. \end{center}
  286. \begin{align}
  287. \tag{$7_\mathrm{f}$}
  288. \end{align}
  289. \vspace{-10mm}
  290. \begin{align*}
  291. \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
  292. \end{align*}
  294. Note that
  295. $$ \frac{\frac{\tau^{p-2}}{p}\hat\lambda_1(p) + \frac{1}{2} ||D\hat u_1(p)||_2^2}{\xi \tau^{\beta-2}||\hat u_1(p)||_\beta^\beta} \leq \frac{\frac{2^{\beta+p-4}}{p}\hat\lambda_1(p)+2^{\beta - 3 }||D\hat u_1(p)||^2_2}{\xi||\hat u_1(p)||_\beta^\beta} $$
  296. (recall that $\displaystyle{\frac{1}{2} \leq \tau \leq 2}$)
  298. 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
  299. $$ \lambda_*^0 = \mathrm{max}\{\lambda^0,\lambda_*\}, $$
  300. then from (7) we infer that
  301. \begin{align*}
  302. \tab &\hat\sigma_\lambda^+(\tau \hat u_1(p)) < 0 \mathrm{\,\,for\,\,all\,\,}\lambda > \lambda_*^0, & \\
  303. \Rightarrow \hspace{4.5mm}& \hat\sigma_\lambda^+(\hat u_\lambda^*) <0 = \hat\sigma_\lambda^+ (0) \mathrm{\,\,for\,\,all\,\,}\lambda > \lambda_*^0 (see~\eqref{asta}) & \\
  304. \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}) & \\
  305. \hspace{19.5mm}\Rightarrow\hspace{4.5mm} & \hat u_\lambda^* = u_\lambda^* \mathrm{\,\,for\,\,all\,\,} \lambda > \lambda_*^0 (\mathrm{\,\,see\,\, Claim\,\, 1}).
  306. %From (3) and Claim 2, we see that we can find $\p\in (0,1)$ small such that
  307. \end{align*}
  308.  
  309. From (1) it is clear that $\hat{\sigma}_{\lambda}^+|_{C_+}.$ Since $u_{\lambda}^* \in D_+$, from (4) it follows that
  310. \begin {align*}
  311. & u_\lambda^*\mathrm{\,\,is\,\,a\,\,local\,\,} C^1(\overline\Omega) - \mathrm{\,\,minimizer\,\,of\,\,} \hat\sigma_\lambda, & \\
  312. \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}}).
  313. \end{align*}
  314.  
  315. Similarly for $v_\lambda^* \in -D_+$ using this time the functional $\hat\sigma_\lambda^-$
  316.  
  317. This proves Claim 2.
  318.  
  319. Without any loss of generality, we may assume that
  320. \begin{align}
  321. \tag{7'}
  322. \hat\sigma_\lambda(v_\lambda^*) \leq \hat\sigma(u_\lambda^*).
  323. \end{align}
  324. \setcounter{equation}{7}
  325.  
  326. From (3) and Claim 2, we see that we can find $p\in (0,1)$ small such that
  327. \begin{align}\label{asta2}
  328. \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
  329. \end{align}
  330. (see Aizicovici-Papageorgiou-Staicu [1], proof of Proposition 29). The functional $\hat\sigma_\lambda$ is coercive (see(1)). Hence
  331. \begin{align}
  332. \label{asta3}
  333. \hat\sigma_\lambda \mathrm{\,\,satisfies \,\,the \,\,PS-condition}
  334. \end{align}
  335.  
  336. 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
  337. \begin{align}
  338. y_{\lambda}\in K_{\hat{\sigma}_{\lambda}} \mathrm{\,\,and\,\,}\hat m_{\lambda} \leq \hat{\sigma}_{\lambda}(y_\lambda)
  339. \end{align}
  340.  
  341. \newpage
  342. \pagestyle{fancy}
  343. \renewcommand{\headrulewidth}{0pt} % no line in header area
  344.  
  345. \fancyhead{}
  346. \rhead{IADIS International Conference WWW/Internet 2006}
  347. \cfoot{}
  348. \lfoot{}
  349. \rfoot{287}
  350. \setcounter{figure}{1}
  351. \begin{figure}[th]
  352.     \centering
  353.         \includegraphics[width=0.73\textwidth]{C:/Users/Frentescu Stefan/Desktop/clustersgraph.jpg}
  354.     \caption{Cluster's graph}
  355.     \label{fig:clusters graph}
  356. \end{figure}
  357. %\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
  358.  
  359. \textbf{Algoritm 1:} PatternMatch\vspace{-3mm}\\
  360. \hspace{-10mm}\noindent\rule{15.2   cm}{0.4pt}\\
  361. \tab\textbf{Input}:patern - the root of the pattern tree (TreeNode)\newline
  362. \tab\tab\hspace{4,8mm}input - the root of the data tree\newline
  363. \tab\textbf{Output}: the list of data extracted\newline
  364. \tab\textbf{Local}: data - a temporary list\newline
  365. \tab\tab \tab[6mm] k-the index of child node from where searching starts\newline
  366. \tab1: \textbf{procedure} ExtractData(pattern,input)\newline
  367. \tab2:\tab\tab if pattern.getLabel() \,\,$\joinrel=\joinrel=$ input.getLabel() \textbf{then}\newline
  368. \tab3:\tab\tab\tab[2.5mm]data $\leftarrow$ ExtractDataFromList(pattern.getChilds(),input.getChilds()); \newline
  369. \tab4:\tab\tab\tab[2.5mm]k $\leftarrow$ 1\newline
  370. \tab5:\tab\tab\tab[2.5mm]\textbf{for\,\,} i $\leftarrow$ 1, pattern.getChildrenCount() \textbf{do}\newline
  371. \tab6:\tab\tab\tab\tab\hspace{1mm} label $\leftarrow$ pattern.getChildAt(i).getLabel();\newline
  372. \tab7:\tab\tab\tab\tab\textbf{if} label is not wildcard \textbf{then}\newline
  373. \tab8:\tab\tab\tab\tab\tab node $\leftarrow$ GetChild(input, label, k);\newline
  374. \tab9:\tab\tab\tab\tab\tab \textbf{if} node $\neq$ null \textbf{then}\newline
  375. \tab10:\tab\tab\tab\tab\tab\tab k $\leftarrow$ input.getChildIndex(node) +1;\newline
  376. \tab11:\tab\tab\tab\tab\tab\tab data.addAll(ExtractData(pattern.childAt(i),node));\newline
  377. \tab12:\tab\tab\tab\tab\tab \textbf{end if}\newline
  378. \tab13:\tab\tab\tab\tab \textbf{end if}\newline
  379. \tab14:\tab\tab\tab[2.5mm]\textbf{end for}\newline
  380. \tab15:\tab\tab\tab[2.5mm]\textbf{return} data;\newline
  381. \tab16:\tab\tab \textbf{else}\newline
  382. \tab17:\tab\tab\tab[2.5mm]\textbf{return $\varnothing$};\newline
  383. \tab18:\tab\tab \textbf{end if}\newline
  384. \tab19:\textbf{ end procedure}\vspace{-3mm}\newline
  385. \hspace{-10mm}\noindent\rule{15.2cm}{0.4pt}
  386.  
  387. The current routine returns a list of mappings. If the two labels are identical (see line \textbf{\textit{2}}) then we shall try
  388. to find a matching between the sequence of children of node \textit{pattern} and the sequence of children of node \textit{input.} The procedure \textit{ExtractDataFromList} receives two list of nodes as input: the algorithm is similar to that of aligning a string with wildcards with a string composed only of regular characters and returns the matchings. Then for each data node from children list of node \textit{pattern} we try to find the first mapping that matches in labels( lines \textbf{\textit{5-14}}).  \textit{\,GetChild(input, \,label, \,index)} gets the child of the node \textit{input} whose information is \textit{label}. The search for the child with this property starts from the position \textit{index}.
  389. \newpage
  390. \rhead{}
  391. \rfoot{}
  392. \begin{center}
  393. \footnotesize{\hspace{10mm}AUTHENTIFICATION PROTOCOL BASED ON ELLIPTIC CURVE CRYPTOGRAPHY\hspace{12mm} 3}
  394. \end{center}
  395. \begin{center}
  396. \begin{tabular}{||c | c | c | c | c||}
  397. \hline
  398. \hline
  399. Symmetric & ECC & RSA/DH/DS & MIPS Yrs to attack & Protection Lifetime \\ [0.4ex]
  400. \hline
  401. 80 & 160 & 1024 & $10^{12}$ & Until 2010  \\
  402. \hline
  403. 112 & 224 & 2048 & $10^{24}$ & Until 2020 \\
  404. \hline
  405. 128 & 256 & 3072 & $10^{28}$ & Beyond 2031\\
  406. \hline
  407. 192 & 384 & 7680 & $10^{47}$ & Beyond 2031\\
  408. \hline
  409. 256 & 512 & 15360 & $10^{66}$ & Beyond 2031\\ %[1ex]
  410.  \hline \hline
  411. \end{tabular}
  412. \end{center}
  413. \begin{center}
  414. \begin{minipage}[t]{0.68\textwidth}
  415. \textbf{2. Related Work}
  416.  
  417. \small{\tab In [4, 19] the authors present an authentification protocol for exchanging encrypted messages via an authentification server based on elliptic curve cryptography using El Gamal's algorithm. They chose El gamal's algorithm for encryption/decryption and, also, for the authentication. The authors motivated this by the fact that using two or more algorithms in the same protocol makes it more vulnerable. They, also, pointed out that using a digital signature algorithm is more secure, but they preferred not to use one because the presented protocal has a big advantage: the receiver does not need to know the sneders public key [2]. A short description of the protocol is presented below.}
  418. \begin{center}
  419. \begin{tabular}{|c|}
  420. \hline
  421. \textit{SenddingProcess(From = Alice, To = Bob) :}\\
  422.      $\mathit{A \rightarrow S : C_S(From,To,N_1)}$\\
  423.      $\mathit{S \rightarrow A : C_A(C_B,N_1,N_2,Id)}$\\
  424.      $\mathit{A \rightarrow S : C_B(M,N_2), Id}$ \\
  425. \hline
  426. \textit{ReceivingProcess(bob receives the message) :}\\
  427. $\mathit{B \rightarrow S : C_S (To, N_3)}$\\
  428. $\mathit{S \rightarrow B : C_B(M,N_2) , C_B(N_2,N_3)}$ \\
  429. \hline
  430.  
  431. \end{tabular}
  432. \end{center}
  433.  
  434. First, the user A sends a request to the server S consisting in the nickname of the sender (From), the nickname of the receiver (To) and a random message ($\mathit{N_1}$). This random message is used to verify the server. These elements are encrypted with the servers' public key and send to the server. These elements are encrypted with the servers' public key and send to the server. After decryption, the server encrypts a mail Id, $\mathit{N_1}$, a new $\mathit{N_2}$, the receiver's (B) public key with A's public key, and sends the result to the sender (A). The sender verifies $\mathit{N_1}$ to see if the message received from the server is valid. Next, A encrypts the clear message \textit{M} and $\mathit{N_2}$ with B's public key and sends the encryption to S along with the Id. to download his message, b send his nickname and a new random message $\mathit{N_3}$ encrypted with S's key. The server S sends back to B \textit{M} and $\mathit{N_2}$ encrypted with B's key along with $\mathit{N_2}$ and $\mathit{N_3}$, also encrypted with B's key. To verify if the encryption received is from the server S, B checks $\mathit{N_3}$. In [1] are introduced two authentication protocols which use two different algorithms: one for encrypting the message and one for generating the key. In both cases, the encryption algorithm is a symmetric one. The authors recommend the protocol presented in the first paper to be use for controller-pilot data link communications, while the one presented in the second paper is recommended for the wireless communication, but it does not mean that they cannot be adapted for other communications. They both use the Elliptic Curve Digital Signature Algorithm (ECDSA) for authentication. This algorithm is briefly described below. In [5] it is presented an authentication rank protocol based on elliptic curve infrastructure.\\
  435. \textbf{ECDSA}. The communicating parties choose an elliptic curve \textit{E} defined over $\mathit{F_p}$ or $\mathit{F_{2^m}}$ with large group of order \textit{n} and a point \textit{P}. All this information are made public. The algorithm, like all the digital signature algorithms, has three steps:
  436.  
  437. \end{minipage}
  438. \end{center}
  439.  
  440. \newpage
  441. %\restoregeometry
  442. \pagestyle{fancy}
  443. \begin{multicols}{2}
  444. \section*{SIMULATION AND FINAL CONCLUSIONS}
  445. In this section we perform some numerical computations to compare the results given by formula (3) and (6). This simulation study will be done for \textbf{AR}
  446. (1) model with $\theta=\upvarphi\in$ \{0.2,0.5,0.75,0.97\}. \tab There \tab where performed 1000 simulations of AR model and for every model where recorded 200 observations. The forecast parameters, K and L where set to 1. The implementation of the simulation process was performed with \textbf{S-PLUS} package.
  447.  
  448. We can simulate a double exponentially random variables by X-Y where X and Y are independent random variables exponentially distrubted or by $\mathrm{N_1N_2 + N_3N_4}$ where $\mathrm{N_i}$  are independent normal variables (see \textit{Devroye} [5], \textit{Vaduva} [11]).
  449.  
  450. In figure 1 are presented the esimators for density of $\updelta$ in case of the normality assumption and in case of the double exponential distribution (dotted curves). We see that for $\theta = \phi$ close to 1 the estimators of the distributions are very close.
  451.  
  452. In cryptography one of the most important problem is the key problem(\textit{Maurer}[6], \textit{Simion}[9], \textit{Simion} and \textit{Preda}[10]) once we generate using a pseudorandom generator or an hardware generator we must proceed to statistical correlation test. This new test can be performed at any codification level L$\geq$2 bits of the source [6]. The output of the key generator are thus numbers between 0 and $\mathrm{2^L}$-1. The probability Q defined by equations (3) and (6) must be equal, if the generated keys are not correlated, with $\mathrm{P2^{-L}}$\newline
  453.  
  454. \begin{figure}[H]
  455.         \includegraphics[width=0.44\textwidth]{C:/Users/Frentescu Stefan/Desktop/graph.jpg}
  456.  
  457. \end{figure}
  458.  
  459. Comparative study of the results obtained using normality assumption and double exponentially assumption\\ \\
  460.  
  461. \section*{AKNOWLEGMENTS}
  462.  
  463. \tab The authors wish to thank to and Ph. D. Grigore Stolojanu and Dipl. eng. Gheorghe Muresanu from Advanced Technologies institute for stimulating discussions during the elaboration of this work and for their generous supports.
  464. %\hphantom{cite{1}\cite{2}\cite{3}\cite{4}\cite{5}\cite{6}\cite{7}\cite{8}\cite{9}}
  465. \hphantom{cite{1,2,3,4,5,6,7,8,9,10,11}}
  466.  
  467.  
  468. \begin{thebibliography}{99}
  469.  
  470. \bibitem{1}
  471. \textit{M. Aminzadeh, Estimation of P(Z<Y) for correlated stochastic time series models\}. Applied Mtch Compt., 104, (1999), 179-189}
  472.  
  473. \bibitem{2}
  474. \textit{G.E.P. Box and G.M Jenkins, Time series Analysis: Forecasting and Control. San Francisco: holden-Day, (1974).}
  475.  
  476. \bibitem{3}
  477. \textit{G.E.P. Box, Some theorems on quadratic forms applied in the study of analysis of variance problem, I: Effects of inequality of variance in the one-way classifications. Ann. Math. Statist., 25, (1959), 290-302 }
  478.  
  479. \bibitem{4}
  480. \textit{E. Damsleth and A.H. El-Shaarawi, (1989) ARMA Models with Double-exponetially Distributed Noise, J.R. Statist. Soc. B, , (1989), 61-69}
  481.  
  482. \bibitem{5}
  483. \textit{L. Devroye Non-Uniform Random Variate Generation Springer Verlag, (1986).}
  484.  
  485. \bibitem{6}
  486. \textit{U. Maurer, A Universal Test for Random BIt Generators. J. of Cryptology 5(2), (1992).}
  487.  
  488. \bibitem{7}
  489. \textit{A. Pankratz , Forecasting with Dynamic Regression Models. Wiley\&Sons, (1991).}
  490.  
  491. \bibitem{8}
  492. \textit{V. Preda and D. Enachescu Autoregressive models with mixed double exponentially distributed noise. Proc. Math. Meth. on Reliability, bucharest,(1997), 44-50.}
  493.  
  494. \bibitem{9}
  495. \textit{E. Simion, Truncated Bayesian estimation of Renyi entropies and cryptographic applications. Math Reports, 52(2), (2000).}
  496.  
  497. \bibitem{10}
  498. \textit{E. Simion, V. Preda Correlated Stochatic time series and cryptographic applications, Special Issue of the annual Meeting of University of bucharest, (in press) (2000).}
  499.  
  500. \bibitem{11}
  501. \textit{I. Vaduva, Modele de Simulare cu calculatorul, Ed. tehnica, (1971).}
  502.  
  503. \end{thebibliography}
  504.  
  505. \end{multicols}
  506. %\end{figure}
  507. \newpage
  508. %$$ \int_{0}^\infty\frac{\int_{0}^1 f(\substack{x\\ y^2})dx}{\left\{_{\sum\nolimits_{0}^1 x_iy^i \xleftarrow[wi]{m} }^{x-y}  dx} $$
  509. \[ \int_{0}^\infty\frac{\int_{0}^1 f(\substack{x\\ y^2})dx}{\left\{
  510.  \begin{array}{lr}
  511.         x-y\\
  512.         \sum\nolimits_{0}^1 x_iy^i \xleftarrow[m]{wi}
  513.   \end{array}  
  514.     \right. dx}
  515. %\right.
  516. \]
  517.  
  518.  
  519. %\multirow{2}{*}{Text}&\multicolumn{1}{|c|}{}\\%%&\multicolumn{1}{c|}{}&\\\cline{3-3}
  520. \begin{center}
  521. \begin{tabular}{|cccc|ccccccc|}  
  522. \hline
  523. \multirow{2}{*}{Text} &\multicolumn{1}{c|}{}&\multicolumn{1}{c|}{} &\multicolumn{1}{|c|}{} & \multicolumn{1}{c}{\multirow{4}{*}{$\int_0^1\frac{x}{y^2}dx$}}&\multicolumn{1}{|c}{} & & &\multicolumn{1}{|c}{} & & \\ \cline{3-4}
  524. &\multicolumn{1}{c|}{} &\multicolumn{1}{c}{} & & &\multicolumn{1}{|c}{} & & &\multicolumn{1}{|c}{} & &\\ \cline{1-4}\cline{6-8}
  525. & & &\multicolumn{1}{|c|}{\multirow{2}{*}{text}} & & \multicolumn{1}{|c}{\multirow{2}{*}{$\int_y^x$}} &\multicolumn{1}{c}{} & &\multicolumn{1}{|c}{} & &\\
  526. & & &\multicolumn{1}{|c|}{} & &  \multicolumn{1}{|c}{} & & &\multicolumn{1}{|c}{} & &\\
  527. \hline
  528. & & & \multicolumn{1}{c|}{} & & &\multicolumn{1}{|c}{} & & & &\\
  529. & & & \multicolumn{1}{c|}{} & & &\multicolumn{1}{|c}{} & & & &\\
  530. \hline
  531. & & \multicolumn{1}{|c}{} &\multicolumn{1}{c|}{}&text& &\multicolumn{1}{|c}{} & & & &\\ \cline{5-6}
  532. & & \multicolumn{1}{|c}{} &\multicolumn{1}{c|}{}& & &\multicolumn{1}{|c}{} & & & &  \\\cline{5-6}
  533. & & \multicolumn{1}{|c}{} &\multicolumn{1}{c|}{}& & &\multicolumn{1}{|c}{} & & & & \\
  534. \hline
  535. \end{tabular}
  536. \end{center}
  537.  
  538.  
  539.  
  540. \end{document}
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