ajout du dossier talk

diff --git a/talk/#assync.tex# b/talk/#assync.tex#
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+\begin{itemize}
+\item Calculs effectués sur chaque n{\oe}ud: réécriture de 
+$$
+w_l^{(k+1)} = w_l^{(k+1)} +  \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right)
+$$
+\item Conditions pour la convergence asynchrone du calcul\footnote{Nedić, A., Bertsekas, D. P., \& Borkar, V. S. (2001). Distributed asynchronous incremental subgradient methods. Studies in Computational Mathematics, 8, 381-407.}
+\begin{itemize}
+\item Majoration des sous-gradients par une constante $C$
+\item $\theta$: sous la forme $\omega / t^{q}$ avec $3/4 < q \leq 1$ 
+ \end{itemize}
+\item Premières expérimentations:
+\begin{itemize}
+\item Pour chaque variable $X$ à modifier:
+  $X^{k+1} =  f(X^{k}) if \textit{random()}<T else X^{k}$
+\item Convergence pour 
+Synchrone
+1 746
+2 1580
+3 4490
+4 7257
+5 8541
+6 10000
+7 9189
+8 10000
+9 10000
+10 10000
+
+
+
+\end{itemize}
+\end{itemize}

diff --git a/talk/#problematique5.tex# b/talk/#problematique5.tex#
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+
+
+\begin{minipage}{0.45\textwidth}
+\includegraphics[width=4cm]{lunettes.png}
+\end{minipage}
+\hspace{0.5cm}
+\begin{minipage}{0.45\textwidth}
+\includegraphics[width=3cm]{iwatch.jpg}
+\end{minipage}
+
+\begin{itemize}
+
+\item Nouvelles technologies (lunettes, montres) et nouvelles 
+  fonctionnalités 
+
+
+
+% Utilisabilité 
+\item Analyse d'utilisation d'applications, d'interfaces utilisateur, 
+  convivialité des applications e t des systèmes mobiles
+\item \ldots 
+\end{itemize}
+
+

diff --git a/talk/Hexagones.jpg b/talk/Hexagones.jpg
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diff --git a/talk/I03_couchot.pdf b/talk/I03_couchot.pdf
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diff --git a/talk/Linear_congruential_generator_visualisation.svg b/talk/Linear_congruential_generator_visualisation.svg
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+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><!--\r
+""" To recover the Python script to generate this SVG, delete the line above -->\r
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+ <title>Linear congruential generator visualisation</title>\r
+ <desc>Visualisation of generation of pseudo-random numbers in [0, 8] using a linear congruential generator by CMG Lee. The top two rows show a generator with m = 0, a = 2 and c = 0 outputting numbers from left to right. A seed of 1 gives a cycle length of 6 but a seed of 3 gives a cycle length of only 2. Using a = 4 and c = 1 (bottom row) gives a full cycle length of 9 with any seed.</desc>\r
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+    <g stroke="#009900">\r
+     <path d="M -8.66,5.00 L -11.26,6.50 A 13.00,13.00 0 0 1 -0.00,-13.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-0,-13) rotate(360)"/>\r
+    </g>\r
+    <text x="19" y="7" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M -0.00,-13.00 L -0.00,-16.00 A 16.00,16.00 0 0 1 13.86,8.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(14,8) rotate(480)"/>\r
+    </g>\r
+    <text x="14" y="19" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M 13.86,8.00 L 15.59,9.00 A 18.00,18.00 0 0 1 6.16,16.91"/>\r
+     <use xlink:href="#arrowhead" transform="translate(6,17) rotate(520)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(400,0)">\r
+    <use xlink:href="#radius" transform="rotate(320)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output 8</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(160)" stroke="#666666" stroke-dasharray="3,2"/>\r
+    <text x="7" y="10" font-size="8" font-weight="bold" fill="#999999">1</text>\r
+    <g stroke="#999999">\r
+     <path d="M 0.00,-0.00 L 0.00,-7.00 A 7.00,7.00 0 0 1 2.39,6.58"/>\r
+     <use xlink:href="#arrowhead" transform="translate(2,7) rotate(160)"/>\r
+    </g>\r
+    <text x="-12" y="-3" font-size="8" font-weight="bold" fill="#cc0000">2</text>\r
+    <g stroke="#cc0000">\r
+     <path d="M 2.39,6.58 L 3.42,9.40 A 10.00,10.00 0 0 1 -6.43,-7.66"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-6,-8) rotate(320)"/>\r
+    </g>\r
+    <text x="16" y="6" font-size="8" font-weight="bold" fill="#009900">3</text>\r
+    <g stroke="#009900">\r
+     <path d="M -6.43,-7.66 L -8.36,-9.96 A 13.00,13.00 0 0 1 11.26,6.50"/>\r
+     <use xlink:href="#arrowhead" transform="translate(11,6) rotate(480)"/>\r
+    </g>\r
+    <text x="-19" y="5" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M 11.26,6.50 L 13.86,8.00 A 16.00,16.00 0 0 1 -15.76,-2.78"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-16,-3) rotate(640)"/>\r
+    </g>\r
+    <text x="-19" y="-8" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M -15.76,-2.78 L -17.73,-3.13 A 18.00,18.00 0 0 1 -11.57,-13.79"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-12,-14) rotate(680)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(480,0)">\r
+    <use xlink:href="#radius" transform="rotate(240)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output 6</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(320)" stroke="#666666" stroke-dasharray="3,2"/>\r
+    <text x="-10" y="-1" font-size="8" font-weight="bold" fill="#999999">1</text>\r
+    <g stroke="#999999">\r
+     <path d="M 0.00,-0.00 L 0.00,-7.00 A 7.00,7.00 0 1 1 -4.50,-5.36"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-4,-5) rotate(320)"/>\r
+    </g>\r
+    <text x="-13" y="6" font-size="8" font-weight="bold" fill="#cc0000">2</text>\r
+    <g stroke="#cc0000">\r
+     <path d="M -4.50,-5.36 L -6.43,-7.66 A 10.00,10.00 0 1 1 -9.85,-1.74"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-10,-2) rotate(640)"/>\r
+    </g>\r
+    <text x="-11" y="15" font-size="8" font-weight="bold" fill="#009900">3</text>\r
+    <g stroke="#009900">\r
+     <path d="M -9.85,-1.74 L -12.80,-2.26 A 13.00,13.00 0 1 1 -11.26,6.50"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-11,7) rotate(960)"/>\r
+    </g>\r
+    <text x="-1" y="22" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M -11.26,6.50 L -13.86,8.00 A 16.00,16.00 0 1 1 -5.47,15.04"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-5,15) rotate(1280)"/>\r
+    </g>\r
+    <text x="-14" y="19" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M -5.47,15.04 L -6.16,16.91 A 18.00,18.00 0 0 1 -15.59,9.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-16,9) rotate(1320)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(560,0)">\r
+    <use xlink:href="#radius" transform="rotate(280)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output 7</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(240)" stroke="#666666" stroke-dasharray="3,2"/>\r
+    <text x="-6" y="12" font-size="8" font-weight="bold" fill="#999999">1</text>\r
+    <g stroke="#999999">\r
+     <path d="M 0.00,-0.00 L 0.00,-7.00 A 7.00,7.00 0 1 1 -6.06,3.50"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-6,4) rotate(240)"/>\r
+    </g>\r
+    <text x="13" y="4" font-size="8" font-weight="bold" fill="#cc0000">2</text>\r
+    <g stroke="#cc0000">\r
+     <path d="M -6.06,3.50 L -8.66,5.00 A 10.00,10.00 0 1 1 8.66,5.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(9,5) rotate(480)"/>\r
+    </g>\r
+    <text x="-5" y="-13" font-size="8" font-weight="bold" fill="#009900">3</text>\r
+    <g stroke="#009900">\r
+     <path d="M 8.66,5.00 L 11.26,6.50 A 13.00,13.00 0 1 1 -0.00,-13.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-0,-13) rotate(720)"/>\r
+    </g>\r
+    <text x="-13" y="17" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M -0.00,-13.00 L -0.00,-16.00 A 16.00,16.00 0 1 1 -13.86,8.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-14,8) rotate(960)"/>\r
+    </g>\r
+    <text x="-21" y="6" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M -13.86,8.00 L -15.59,9.00 A 18.00,18.00 0 0 1 -17.73,-3.13"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-18,-3) rotate(1000)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(640,0)">\r
+    <use xlink:href="#radius" transform="rotate(80)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output 2</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(280)" stroke="#666666" stroke-dasharray="3,2"/>\r
+    <text x="-10" y="6" font-size="8" font-weight="bold" fill="#999999">1</text>\r
+    <g stroke="#999999">\r
+     <path d="M 0.00,-0.00 L 0.00,-7.00 A 7.00,7.00 0 1 1 -6.89,-1.22"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-7,-1) rotate(280)"/>\r
+    </g>\r
+    <text x="0" y="16" font-size="8" font-weight="bold" fill="#cc0000">2</text>\r
+    <g stroke="#cc0000">\r
+     <path d="M -6.89,-1.22 L -9.85,-1.74 A 10.00,10.00 0 1 1 -3.42,9.40"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-3,9) rotate(560)"/>\r
+    </g>\r
+    <text x="16" y="6" font-size="8" font-weight="bold" fill="#009900">3</text>\r
+    <g stroke="#009900">\r
+     <path d="M -3.42,9.40 L -4.45,12.22 A 13.00,13.00 0 1 1 11.26,6.50"/>\r
+     <use xlink:href="#arrowhead" transform="translate(11,7) rotate(840)"/>\r
+    </g>\r
+    <text x="8" y="-15" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M 11.26,6.50 L 13.86,8.00 A 16.00,16.00 0 1 1 10.28,-12.26"/>\r
+     <use xlink:href="#arrowhead" transform="translate(10,-12) rotate(1120)"/>\r
+    </g>\r
+    <text x="19" y="-8" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M 10.28,-12.26 L 11.57,-13.79 A 18.00,18.00 0 0 1 17.73,-3.13"/>\r
+     <use xlink:href="#arrowhead" transform="translate(18,-3) rotate(1160)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(720,0)">\r
+    <use xlink:href="#radius" transform="rotate(0)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output 0</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(80)" stroke="#666666" stroke-dasharray="3,2"/>\r
+    <text x="8" y="-3" font-size="8" font-weight="bold" fill="#999999">1</text>\r
+    <g stroke="#999999">\r
+     <path d="M 0.00,-0.00 L 0.00,-7.00 A 7.00,7.00 0 0 1 6.89,-1.22"/>\r
+     <use xlink:href="#arrowhead" transform="translate(7,-1) rotate(80)"/>\r
+    </g>\r
+    <text x="9" y="13" font-size="8" font-weight="bold" fill="#cc0000">2</text>\r
+    <g stroke="#cc0000">\r
+     <path d="M 6.89,-1.22 L 9.85,-1.74 A 10.00,10.00 0 0 1 3.42,9.40"/>\r
+     <use xlink:href="#arrowhead" transform="translate(3,9) rotate(160)"/>\r
+    </g>\r
+    <text x="-11" y="15" font-size="8" font-weight="bold" fill="#009900">3</text>\r
+    <g stroke="#009900">\r
+     <path d="M 3.42,9.40 L 4.45,12.22 A 13.00,13.00 0 0 1 -11.26,6.50"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-11,7) rotate(240)"/>\r
+    </g>\r
+    <text x="-16" y="-8" font-size="8" font-weight="bold" fill="#0000ff">4</text>\r
+    <g stroke="#0000ff">\r
+     <path d="M -11.26,6.50 L -13.86,8.00 A 16.00,16.00 0 0 1 -10.28,-12.26"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-10,-12) rotate(320)"/>\r
+    </g>\r
+    <text x="-7" y="-17" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+     <path d="M -10.28,-12.26 L -11.57,-13.79 A 18.00,18.00 0 0 1 -0.00,-18.00"/>\r
+     <use xlink:href="#arrowhead" transform="translate(-0,-18) rotate(360)"/>\r
+    </g>\r
+   </g>\r
+   <g transform="translate(-80,-30)" fill="#000000">\r
+    <text x="0" y="20" font-style="italic">m</text><text x="25" y="20">= 9</text>\r
+    <text x="0" y="40" font-style="italic">a</text><text x="25" y="40">= 4</text>\r
+    <text x="0" y="60" font-style="italic">c</text><text x="25" y="60">= 1</text>\r
+   </g>\r
+   <use xlink:href="#hr"/>\r
+  </g>\r
+ </g>\r
+<!-- } END_DYNAMIC_SVG -->\r
+</svg>\r
+<!-- Please retain this and other comments, which contain Python code to generate this SVG. """\r
+import math\r
+import re\r
+def join(list, delimiter = ''): return delimiter.join(list).strip('\n')\r
+def x_to_deg(x): return x * 360 / m\r
+def path_arc(radius, radius_previous, deg_begin, deg_step):\r
+ deg_end   = deg_begin + deg_step\r
+ rad_begin = deg_begin * math.pi / 180\r
+ rad_end   = deg_end   * math.pi / 180\r
+ return ('''     <path d="M %.2f,%.2f L %.2f,%.2f A %.2f,%.2f 0 %d 1 %.2f,%.2f"/>\r
+     <use xlink:href="#arrowhead" transform="translate(%.0f,%.0f) rotate(%d)"/>''' %\r
+         (radius_previous * math.sin(rad_begin), radius_previous * -math.cos(rad_begin),\r
+          radius * math.sin(rad_begin), radius * -math.cos(rad_begin),\r
+          radius                      , radius , 0 if (deg_step < 180) else 1,\r
+          radius * math.sin(rad_end  ), radius * -math.cos(rad_end  ),\r
+          radius * math.sin(rad_end  ), radius * -math.cos(rad_end  ), deg_end))\r
+\r
+outs          = []\r
+out_frames    = []\r
+out_numbers   = []\r
+m             = 9\r
+radius_number = 29\r
+frames        = [ {'a':2,'c':0,'seed':1},\r
+                  {'a':2,'c':0,'seed':3},\r
+                  {'a':4,'c':1,'seed':0} ]\r
+colours = ['#999999','#cc0000','#009900','#0000ff']\r
+\r
+if (0):\r
+ for    m    in range(8, 100):\r
+  for   a    in range(2, m):\r
+   for  c    in range(0, m):\r
+    print ('m=%d a=%2d c=%2d:' % (m, a, c)),\r
+    for seed in range(0, m):\r
+     if (seed % 5 == 0): print ' ',\r
+     is_seen = [False] * m\r
+     count   = 0\r
+     x       = seed\r
+     while (not is_seen[x]):\r
+      is_seen[x] = True\r
+      x          = (x * a + c) % m\r
+      count += 1\r
+     print ('%2d%s' % (count, '*' if (count == m) else ' ')),\r
+    print\r
+   print\r
+  raw_input('Press Enter ')\r
+ quit()\r
+\r
+for i_frame in range(0, len(frames)):\r
+ frame      = frames[i_frame]\r
+ a          = int(frame['a'])\r
+ c          = int(frame['c'])\r
+ seed       = int(frame['seed'])\r
+ x          = seed\r
+ x_previous = x\r
+ i_cycle    = 0\r
+ out_cycles = []\r
+ out_cycles.append('''\r
+   <g>\r
+    <text x="0" y="48" fill="#000000">seed = %d</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(%d)" stroke="#000000"/>\r
+   </g>''' % (x, x_to_deg(seed)))\r
+ is_seen = [False] * m\r
+ while (not is_seen[x]):\r
+  print x,\r
+  is_seen[x]      = True\r
+  deg             = x_to_deg(x)\r
+  x               = (x * a + c) % m\r
+  radius_previous = 0\r
+  out_arcs        = []\r
+  for i_a in range(0, a):\r
+   radius = i_a * 3 + 7\r
+   rad    = (i_a + 1) * deg * math.pi / 180\r
+   radius_text = radius + 4\r
+   rad_text    = rad - 0.6 + 0.02 * radius\r
+   if (x_previous > 0):\r
+    out_arcs.append('''\r
+    <text x="%d" y="%d" font-size="8" font-weight="bold" fill="%s">%d</text>''' %\r
+     (radius_text * math.sin(rad_text), 3 - radius_text * math.cos(rad_text), # 3 to valign text\r
+      colours[i_a % len(colours)], i_a + 1))\r
+   out_arcs.append('''\r
+    <g stroke="%s">\r
+%s\r
+    </g>''' % (colours[i_a % len(colours)], path_arc(radius, radius_previous, i_a * deg, deg)))\r
+   radius_previous = radius\r
+  radius_c = radius + 6\r
+  rad_c    = x_to_deg(x - 0.5) * math.pi / 180\r
+  out_c = '' if (c == 0) else '''\r
+    <text x="%d" y="%d" font-size="8" font-weight="bold" fill="#000000">c</text>\r
+    <g stroke="#000000">\r
+%s\r
+    </g>''' % (radius_c * math.sin(rad_c), 3 - radius_c * math.cos(rad_c), # 3 to valign text\r
+               path_arc(radius + 2, radius, (i_a + 1) * deg, x_to_deg(c)))\r
+  out_cycles.append('''\r
+   <g transform="translate(%d,0)">\r
+    <use xlink:href="#radius" transform="rotate(%d)" stroke="#000000"/>\r
+    <text x="0" y="48" fill="#000000">output %d</text>\r
+    <use xlink:href="#common"/>\r
+    <use xlink:href="#radius" transform="rotate(%d)" stroke="#666666" stroke-dasharray="3,2"/>\r
+%s%s\r
+   </g>''' % ((i_cycle + 1) * 80, x_to_deg(x), x, deg, join(out_arcs), out_c))\r
+  i_cycle += 1\r
+  x_previous      = x\r
+ print\r
+ out_frames.append('''\r
+  <g transform="translate(0,%d)">\r
+%s\r
+   <g transform="translate(-80,-30)" fill="#000000">\r
+    <text x="0" y="20" font-style="italic">m</text><text x="25" y="20">= %d</text>\r
+    <text x="0" y="40" font-style="italic">a</text><text x="25" y="40">= %d</text>\r
+    <text x="0" y="60" font-style="italic">c</text><text x="25" y="60">= %d</text>\r
+   </g>\r
+   <use xlink:href="#hr"/>\r
+  </g>''' % (i_frame * 100, join(out_cycles), m, a, c))\r
+\r
+for i_number in range(0, m):\r
+ rad = i_number * 2 * math.pi / m\r
+ out_numbers.append('''\r
+   <text x="%.0f" y="%.0f" dy="0.7ex">%d</text>''' %\r
+  (radius_number * math.sin(rad), radius_number * -math.cos(rad), i_number))\r
+\r
+outs.append('''\r
+  <circle cx="0" cy="0" r="20" fill="#cccccc"/>\r
+  <g id="common" font-size="12" fill="#000000">\r
+%s\r
+  </g>\r
+ </defs>\r
+ <circle cx="0" cy="0" r="99999" fill="#ffffff"/>\r
+ <g font-family="Arial,Helvetica,sans-serif" font-size="16" text-anchor="middle" stroke-linejoin="round" stroke="none" fill="none">\r
+%s\r
+ </g>\r
+''' % (join(out_numbers), join(out_frames)))\r
+\r
+# Code to compile everything into a .svg file follows...\r
+double_dash = '-' + '-' # because SVG comments cannot have 2 consecutive '-'s\r
+myself      = open(__file__, 'r').read() # the contents of this very file\r
+file_out    = open(re.sub(r'\..*?$', '', __file__) + '.svg', 'w') # *.* -> *.svg\r
+try: # use try/finally so that file is closed even if write fails\r
+ file_out.write(\r
+  '''<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><!%s\r
+%s''' % (double_dash,\r
+         re.sub(r'(BEGIN[_]DYNAMIC_SVG.*\n?)[\s\S]*(\n.*END[_]DYNAMIC_SVG)',\r
+                r'\1\n%s\2' % (join(outs)),\r
+                myself))) # replace dynamic SVG year with contents of "outs"\r
+finally:\r
+ file_out.close()\r
+# SVG-Python near-polyglot framework by CMG Lee (Oct 2014) -->\r

diff --git a/talk/asynchrone.tex b/talk/asynchrone.tex
new file mode 100644 (file)
index 0000000..3a83f7e
--- /dev/null
+++ b/talk/asynchrone.tex
@@ -0,0 +1,26 @@
+\begin{itemize}
+ \item Deux modes: 
+\begin{itemize}
+ \item 
+   \emph{synchrone}: chaque élément attend la valeur des élements dont il dépend;
+ \item 
+  \emph{asynchrone}: chaque élément met  à jour sa valeur sans attendre.
+ \end{itemize}
+
+\item  $(D^{t})^{t \in  \Nats}$:  suite de matrices de taille $\mathsf{N}  \times \mathsf{N}$ t.q.  
+  $$
+  D_{ij}^{t} =\textrm{ date où $x_j$ est  disponible au composant $i$}
+$$ 
+
+
+\item  $x^{t+1}_i= \left\{
+    \begin{array}{l}
+      f_i( x_1^{D_{i1}^t},\ldots, x_{\mathsf{N}}^{D_{i{\mathsf{N}}}^t})
+      \textrm{ si } i \in s^t\\ 
+      x^{t}_i  \textrm{ sinon }
+    \end{array} 
+  \right.$
+
+
+%\item Délai : $\delta_{ij}^t  = t  - D_{ij}^{t} \le \delta_0\, \forall i, j$  
+\end{itemize}

diff --git a/talk/attracteurs.tex b/talk/attracteurs.tex
new file mode 100644 (file)
index 0000000..edc0088
--- /dev/null
+++ b/talk/attracteurs.tex
@@ -0,0 +1,46 @@
+\begin{itemize}
+ \item  $x$ \emph{point fixe} de $f$ si  $x = f (x)$
+ \item $A$ \emph{attracteurs} du graphe si
+\begin{itemize}
+ \item pour tout arc $x \rightarrow y$ , si $x \in A$, alors 
+   $y \in A$ et 
+\item $A$: le plus petit au sens de l'inclusion
+\end{itemize}
+\end{itemize}
+
+\vspace{-1em}
+\begin{exampleblock}{Attracteurs de 
+$(x_1, x_2, x_3) \mapsto 
+((\overline{x_1} + \overline{x_2}).x_3,
+x_1.x_3,
+x_1 + x_2 + x_3)$.}
+\vspace{-2em}
+\begin{figure}[h]
+  \begin{center}
+    \subfloat[$A_1=\{000\}$ et $A_2=\{011, 101, 111\}$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{fsig3}
+        \end{center}
+      \end{minipage}
+      \label{fig:fsigb}
+    }
+    \subfloat[$A_1=\{000\}$ et $A_2=\{001, 101,111, 011 \}$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{faig3}
+        \end{center}
+      \end{minipage}
+      \label{fig:faigb}
+    }   
+    \subfloat[$A_1=\{000\}$ et $A_2=\{001, 101,111, 011 \}$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{fgig3}
+        \end{center}
+      \end{minipage}
+      \label{fig:fgigb}
+    }
+  \end{center}
+\end{figure}
+\end{exampleblock}

diff --git a/talk/beamerthemefemto.sty b/talk/beamerthemefemto.sty
new file mode 100644 (file)
index 0000000..0ab88ce
--- /dev/null
+++ b/talk/beamerthemefemto.sty
@@ -0,0 +1,249 @@
+%-------------------------------------------------------------------------------
+% LaTeX Beamer theme "femto" from the femto-st evaluation model (2011) JM Nicod and PC Heam (LIFC)
+% 
+% author: Christophe Varnier et passage au ".sty" par G. Laurent
+%         Joel Abadie (12/06/2012) rajout de fonctionnalites tikzpicture
+%         Jean-Marc Nicod (logo + reglage pied de page)
+%-------------------------------------------------------------------------------
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{beamerthemefemto}[2012/06/05 beamer theme femto package]
+\RequirePackage{times}
+\RequirePackage{tikz}
+%-------------------------------------------------------------------------------
+
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+%          |___/                                                           
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+%                  |_|                                         
+
+%\pgfdeclareimage[height=\paperheight,width=\paperwidth]{titrefemto}{TitreFemto.png}
+
+\setbeamertemplate{title page}
+{
+\vskip 5mm 
+
+\usebeamerfont{title}\usebeamercolor[fg]{title}\Large\textcolor{BleuFemto}\inserttitle
+
+%\vspace{5mm} 
+\footnotesize
+\begin{center}
+\insertauthor\\
+\vspace{5mm}
+\insertdate
+\end{center}
+
+\begin{footnotesize}
+Rapporteurs~:
+\begin{tabular}{p{4cm}l}
+Olivier {\sc Bournez}& Professeur à l'Ecole Polytechnique. \\
+Jean-Paul {\sc Comet}&  Professeur à l'Université de Nice Sophia Antipolis\\
+Juan-Pablo {\sc Ortega} & Professeur à l'Université de St. Gallen--Suisse \\
+\end{tabular}
+
+
+\vspace{5mm} 
+Examinateurs~:
+\begin{tabular}{p{4cm}l}
+Sylvain {\sc Contassot-Vivier} & {Professeur à l'Université de Lorraine} \\
+Raphaël {\sc Couturier}& {Professeur à l'Univ. Bourgogne Franche-Comté} \\
+Christophe {\sc Guyeux}& {Professeur à l'Univ. Bourgogne Franche-Comté}
+\end{tabular}
+\end{footnotesize}
+
+% Examinateurs:
+
+% \begin{footnotesize}
+% \begin{tabular}{rll}
+% 
+% 
+% 
+% \end{tabular}
+% \end{footnotesize}
+
+}
+
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+
+%-------------------------------------------------------------------------------
+

diff --git a/talk/conclusion.tex b/talk/conclusion.tex
new file mode 100644 (file)
index 0000000..960b079
--- /dev/null
+++ b/talk/conclusion.tex
@@ -0,0 +1,25 @@
+\begin{itemize}
+\item Peut-on prédire le comportement de réseaux bouléens (Sec.~\ref{sec:sdd})
+\begin{itemize}
+\item Théorique: \alert{conditions théoriques}
+  nécessaires/suffisantes de convergence/divergence? 
+\item Pratique: vérification par simulation (\alert{exhaustive?})
+\end{itemize}  
+\item Itérations divergentes \alert{$ \leftrightarrow$ comportement chaotique} (Sec.~\ref{sec:sddchaos})?
+\begin{itemize}
+\item Caractérisation des réseaux bouléens chaotiques.
+\item Génération et prédiction.
+\end{itemize}
+\item Générateurs de nombres pseudo-aléatoires (Sec.~\ref{sec:prng}):
+\begin{itemize}
+\item Caractérisation d'un PRNG chaotique.  
+\item Générations et qualité.
+\end{itemize}
+\item Du Chaos au masquage d'information (Sec.\ref{sec:ih}).
+\begin{itemize}
+\item D'un point de vue chaotique.
+\item D'un point de vue analyse vectorielle discrète.
+\end{itemize}
+\end{itemize}
+
+

diff --git a/talk/couchot.pdf b/talk/couchot.pdf
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diff --git a/talk/ctx.tex b/talk/ctx.tex
new file mode 100644 (file)
index 0000000..a003116
--- /dev/null
+++ b/talk/ctx.tex
@@ -0,0 +1,19 @@
+\begin{exampleblock}{Utilisation des nombres aléatoires}
+\begin{itemize}
+\item Vie courante: localisation GPS, cryptage de données (RSA);
+\item Science: statistiques, en probabilité, en simulation numérique.
+\end{itemize}
+\end{exampleblock}
+
+\begin{exampleblock}{Génération}
+\begin{itemize}
+\item Réellement aléatoire: processus physique $\leadsto$ non reproductible.
+\item Pseudo-aléatoire (PRNG): algorithme $\leadsto$ reproductible, quid de la sécurité?
+\end{itemize}
+\end{exampleblock}
+
+\begin{alertblock}{Problématique}
+Les systèmes dynamiques sont-ils un bon outil mathématique pour 
+générer des nombres pseudo-aléatoires? 
+Quelles sont les qualités minimums d'un PRNG?
+\end{alertblock}

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diff --git a/talk/figreg-eps-converted-to.pdf b/talk/figreg-eps-converted-to.pdf
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diff --git a/talk/figreg.eps b/talk/figreg.eps
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index 0000000..336e714
--- /dev/null
+++ b/talk/figreg.eps
@@ -0,0 +1,601 @@
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diff --git a/talk/figreg.tex b/talk/figreg.tex
new file mode 100644 (file)
index 0000000..17f137a
--- /dev/null
+++ b/talk/figreg.tex
@@ -0,0 +1,11 @@
+\psfrag{fW}[cc][cc]{$a_k$}
+\psfrag{wdjW}[cc][cc]{$a_{k+1}$}
+\psfrag{OFFY}[cc][cc]{$a_{k+2}$}
+\psfrag{uH8PA}[cc][cc]{$a_{k+r-1}$}
+\psfrag{x_oF}[cc][cc]{$a_{k+r}$}
+\psfrag{al}[cc][cc]{$q_0$}
+\psfrag{Lt}[cc][cc]{$q_1$}
+\psfrag{kG}[cc][cc]{$q_2$}
+\psfrag{QMeO}[cc][cc]{$q_{r-1}$}
+\psfrag{MG}[cc][cc]{$\sum$}
+\psfrag{Hvb}[cc][cc]{$\mod p$}

diff --git a/talk/fsig2.pdf b/talk/fsig2.pdf
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diff --git a/talk/fsig3.pdf b/talk/fsig3.pdf
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diff --git a/talk/gcldef.tex b/talk/gcldef.tex
new file mode 100644 (file)
index 0000000..5e1078d
--- /dev/null
+++ b/talk/gcldef.tex
@@ -0,0 +1,27 @@
+\begin{definition}[]%[Générateur congruentiel linéaire]
+Pour $m$, $a$, $c \in \Nats$ t.q. $m \ge a,c$.
+La suite  d'entiers
+
+\begin{equation}
+\tag{1}
+\left\{
+\begin{array}{l}
+\textrm{$x_0 \in \Nats$ t.q. $0 \le x_0 \le m-1$}\\
+x_i = a \times x_{i-1} +c \mod m.
+\end{array}
+\right.
+\label{eq:glc}
+\end{equation}
+est un \emph{générateur congruentiel linéaire} 
+de \emph{graine} $x_0$. 
+\end{definition}
+
+\vspace{-2em}
+\begin{center}
+\includegraphics[scale=0.2]{lcgexemple.png}
+\end{center}
+
+\vspace{-2em}
+\begin{itemize}
+\item \alert<1> {Bien choisir $m$, $a$, $c$ et $x_0$: primordial!}
+\end{itemize}

diff --git a/talk/gcldef.tex.bak b/talk/gcldef.tex.bak
new file mode 100644 (file)
index 0000000..b000cd6
--- /dev/null
+++ b/talk/gcldef.tex.bak
@@ -0,0 +1,27 @@
+\begin{definition}[]%[Générateur congruentiel linéaire]
+Pour $m$, $a$, $c \in \Nats$ t.q. $m \ge a,c$.
+La suite  d'entiers
+
+\begin{equation}
+\tag{1}
+\left\{
+\begin{array}{l}
+\textrm{$x_0 \in \Nats$ t.q. $0 \le x_0 \le m-1$}\\
+x_i = a \times x_{i-1} +c \mod m.
+\end{array}
+\right.
+\label{eq:glc}
+\end{equation}
+est un \emph{générateur congruentiel linéaire} 
+de \emph{graine} $x_0$. 
+\end{definition}
+
+\vspace{-2em}
+\begin{center}
+\includegraphics[scale=0.2]{lcgexemple.png}
+\end{center}
+
+\vspace{-2em}
+\begin{itemize}
+\item \alert<1> {Bien choisir $m$, $a$, $c$ et $x_0$: primoridial!}
+\end{itemize}

diff --git a/talk/gclth1.tex b/talk/gclth1.tex
new file mode 100644 (file)
index 0000000..4652f6e
--- /dev/null
+++ b/talk/gclth1.tex
@@ -0,0 +1,21 @@
+\begin{theorem}[Théorème 1: Hull \& Dobell 1962]\label{theo:hull:dobell}
+La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si 
+\begin{enumerate}
+\item\label{item:gcl:c1} $c$ et $m$: premiers entre eux,
+\item\label{item:gcl:c2}  $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et 
+\item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si  $4$ divise $m$.
+\end{enumerate}
+\end{theorem}
+
+\begin{exampleblock}{Analyse de l'exemple de la Figure~1}
+On a $m=180$, $a=61$ et $c=7$. Or $m=180 = 2^2 \times 3^2 \times 5$.
+\begin{enumerate}
+\item $c$ et $m$: premiers entre eux,
+\item $61 \equiv 1 \mod 2$,  $61 \equiv 1 \mod 3$,  $61 \equiv 1 \mod 5$ et 
+\item $61 \equiv 1 \mod 4$.
+\end{enumerate}
+La période d'un tel générateur est donc maximale. 
+\end{exampleblock}
+
+
+

diff --git a/talk/gclth1demo.tex b/talk/gclth1demo.tex
new file mode 100644 (file)
index 0000000..09d708d
--- /dev/null
+++ b/talk/gclth1demo.tex
@@ -0,0 +1,46 @@
+% \only<1>{
+% \begin{theorem}[Théorème 1: Hull \& Dobell 1962]
+% La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si 
+% \begin{enumerate}
+% \item\label{item:gcl:c1} $c$ et $m$: premiers entre eux,
+% \item\label{item:gcl:c2}  $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et 
+% \item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si  $4$ divise $m$.
+% \end{enumerate}
+% \end{theorem}
+% }
+
+\begin{theorem}[]%[Théorème 1: Hull \& Dobell 1962]\label{theo:hull:dobell}
+La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si 
+\begin{enumerate}
+\item\label{item:gcl:c1} $c$ et $m$: premiers entre eux,
+\item\label{item:gcl:c2}  $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et 
+\item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si  $4$ divise $m$.
+\end{enumerate}
+\end{theorem}
+
+
+\begin{enumerate}
+\item {\sc Lemme}~1 $\Rightarrow$ la période est l'entier $n$ minimum qui établit
+\begin{equation}
+\tag{4}
+\frac{a^n-1}{a-1}\equiv 0 \mod m.
+\end{equation}
+
+\item \textbf{Cas $n=m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~2):
+\begin{enumerate}
+\item binôme de Newton pour réécrire $a^n$;
+\item analyse de $\frac{p^{\alpha}}{j}$ dans $(6)$ $\leadsto$ (4) établie.
+\end{enumerate}
+\item \textbf{Cas $n<m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~3):
+\begin{enumerate}
+\item (4) établie $\Rightarrow$  $n$   puissance de $p$,  
+\item or (4) pas établie pour $n=p^{\alpha-1}$.
+\end{enumerate}
+\item \textbf{Cas $m=p_1^{\alpha1}p_2^{\alpha2}\dots p_s^{\alpha_s}$  et 
+$a = 1 + kp_1^{\beta_1}p_2^{\beta_2}\dots p_s^{\beta_s}$}:
+\begin{enumerate}
+\item preuve similaire (cas 2. puis  3.).
+\end{enumerate}
+
+\end{enumerate}
+

diff --git a/talk/gclth2.tex b/talk/gclth2.tex
new file mode 100644 (file)
index 0000000..0919f91
--- /dev/null
+++ b/talk/gclth2.tex
@@ -0,0 +1,35 @@
+\begin{theorem}[Théorème 2]
+La période de la suite produite par le générateur congruentiel linéaire est $m-1$ si %et seulement si 
+\begin{enumerate}
+\item\label{item:gcl2:c1}  $x_0$ n'est pas nul;
+\item\label{item:gcl2:c2}  $m$ est un nombre premier;
+\item\label{item:gcl2:c4}  $a$ est d'ordre ${m-1}$
+\end{enumerate}
+\end{theorem}
+
+
+\begin{theorem}[Théorème 3]
+Soit $m$ un nombre premier. On peut toujours trouver $a$ d'ordre $m-1$,
+$1 \le a \le m-1$.
+\end{theorem}
+
+
+\begin{exampleblock}{Analyse de l'exemple de la Figure~2}
+On a $m=181$ et $a=19\times80\times125 \equiv 131 \mod 181$ et 
+
+\begin{enumerate}
+\item  $x_0=1$ n'est pas nul;
+\item  $m=181$ est un nombre premier;
+\item 
+\begin{itemize}
+\begin{footnotesize}
+\item 19 est d'ordre $4$, 80 est d'ordre $9$ et 125 est d'ordre $5$;
+\item 4, 5 et 9 premiers entre eux $\leadsto$ $a=19\times80\times125$ est d'ordre 180;
+\end{footnotesize}
+\end{itemize}
+\end{enumerate}
+La période d'un tel générateur est 180. 
+\end{exampleblock}
+
+
+

diff --git a/talk/gclth2demo.tex b/talk/gclth2demo.tex
new file mode 100644 (file)
index 0000000..2f16489
--- /dev/null
+++ b/talk/gclth2demo.tex
@@ -0,0 +1,43 @@
+% \only<1>{
+% \begin{theorem}[Théorème 1: Hull \& Dobell 1962]
+% La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si 
+% \begin{enumerate}
+% \item\label{item:gcl:c1} $c$ et $m$: premiers entre eux,
+% \item\label{item:gcl:c2}  $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et 
+% \item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si  $4$ divise $m$.
+% \end{enumerate}
+% \end{theorem}
+% }
+
+\begin{theorem}[Théorème 3]
+Soit $m$ un nombre premier. On peut toujours trouver $a$ d'ordre $m-1$,
+$1 \le a \le m-1$.
+\end{theorem}
+
+
+
+\begin{enumerate}
+\item {\sc Lemme}~1 $\Rightarrow$ recherche de l'entier $n$ minimum qui établit
+\begin{equation}
+\tag{4}
+\frac{a^n-1}{a-1}\equiv 0 \mod m.
+\end{equation}
+
+\item \textbf{Cas $n=m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~2):
+\begin{enumerate}
+\item binôme de Newton pour réécrire $a^n$;
+\item analyse de $\frac{p^{\alpha}}{j}$ dans $(6)$ $\leadsto$ (4) établie.
+\end{enumerate}
+\item \textbf{Cas $n<m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~3):
+\begin{enumerate}
+\item (4) établie $\Rightarrow$  $n$   puissance de $p$,  
+\item or (4) pas établie pour $n=p^{\alpha-1}$.
+\end{enumerate}
+\item \textbf{Cas $m=p_1^{\alpha1}p_2^{\alpha2}\dots p_s^{\alpha_s}$  et 
+$a = 1 + kp_1^{\beta_1}p_2^{\beta_2}\dots p_s^{\beta_s}$}:
+\begin{enumerate}
+\item preuve similaire (cas 2. puis  3.).
+\end{enumerate}
+
+\end{enumerate}
+

diff --git a/talk/gclth3.tex b/talk/gclth3.tex
new file mode 100644 (file)
index 0000000..57d81f0
--- /dev/null
+++ b/talk/gclth3.tex
@@ -0,0 +1,20 @@
+
+\begin{exampleblock}{Analyse de l'exemple de la Figure~2}
+On a $m=181$ et $a=19\times80\times125 \equiv 131 \mod 181$ et 
+
+\begin{enumerate}
+\item  $x_0=1$ n'est pas nul;
+\item  $m=181$ est un nombre premier;
+\item 
+\begin{itemize}
+\begin{footnotesize}
+\item 19 est d'ordre $4$, 80 est d'ordre $9$ et 125 est d'ordre $5$,
+\item Lemme~5: 4, 5 et 9 premiers entre eux donc $a=19\times80\times125 \equiv 131 \mod 181$ est d'ordre 180.
+\end{footnotesize}
+\end{itemize}
+\end{enumerate}
+La période d'un tel générateur est 180. 
+\end{exampleblock}
+
+
+

diff --git a/talk/gclth3demo.tex b/talk/gclth3demo.tex
new file mode 100644 (file)
index 0000000..a0d780c
--- /dev/null
+++ b/talk/gclth3demo.tex
@@ -0,0 +1,13 @@
+\begin{theorem}[]%[Théorème 3]
+Soit $m$ un nombre premier. On peut toujours trouver $a$ d'ordre $m-1$,
+$1 \le a \le m-1$.
+\end{theorem}
+
+Preuve constructive:
+\begin{enumerate}
+\item Décomposition de $m-1  = p_1^{k_1} \dots  p_s^{k_s}$  en produit de facteurs premiers;
+\item $x^{p_i^{k_i}}  -1 \equiv 0 \mod m$ admet une solution $g_i$ sur $\{1,\dots,m-1\}$ d'ordre $p_i^{k_i}$ ({\sc Lemme} 4);
+\item $a = g_1\dots g_s$ est d'ordre  $p_1^{k_1}\dots p_s^{k_s} = m-1$ ({\sc Lemme} 5).
+
+\end{enumerate}
+

diff --git a/talk/gf.pdf b/talk/gf.pdf
new file mode 100644 (file)
index 0000000..6d588d4
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diff --git a/talk/graphes.tex b/talk/graphes.tex
new file mode 100644 (file)
index 0000000..621c886
--- /dev/null
+++ b/talk/graphes.tex
@@ -0,0 +1,38 @@
+\begin{exampleblock}{Graphes des itérations de  
+$(x_1, x_2, x_3) \mapsto 
+((\overline{x_1} + \overline{x_2}).x_3,
+x_1.x_3,
+x_1 + x_2 + x_3)$.}
+\vspace{-2em}
+\begin{figure}[ht]
+  \begin{center}
+    \subfloat[$\textsc{gis}(f)$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{fsig2}
+        \end{center}
+      \end{minipage}
+      \label{fig:fsig}
+    }
+    \subfloat[$\textsc{giu}(f)$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{faig2}
+        \end{center}
+      \end{minipage}
+      \label{fig:faig}
+    }   
+    \subfloat[$\textsc{gig}(f)$]{
+      \begin{minipage}{0.33\textwidth}
+        \begin{center}
+          \includegraphics[scale=0.25]{fgig2}
+        \end{center}
+      \end{minipage}
+      \label{fig:fgig}
+    }
+  \end{center}
+  %\caption{}
+\end{figure}
+\end{exampleblock}
+% On remarque le cycle $((101,111),(111,011),(011,101))$ 
+%  à la \textsc{Figure}~(\ref{fig:fsig}).
\ No newline at end of file

diff --git a/talk/jacobienne.tex b/talk/jacobienne.tex
new file mode 100644 (file)
index 0000000..b76dbc7
--- /dev/null
+++ b/talk/jacobienne.tex
@@ -0,0 +1,53 @@
+\begin{itemize}
+\item Mat. de $\{-1,0,1\}^{{\mathsf{N}}^2}$ des \og dérivées partielles\fg{} 
+$f'_{ij}=\dfrac{f_i(\overline{x}^j){-}f_i(x)}{\overline{x_j}{-}x_j}$.
+\item Représentée par un \emph{graphe des interactions} orienté:
+\begin{itemize}
+\item Sommets: $\{1, \ldots, {\mathsf{N}}\}$ 
+\item Arcs: $j \xrightarrow{s} i$ si $\exists x \in \Bool^{\mathsf{N}}$ tq. 
+  $f'_{ij}(x)=s$, 
+ $s\in\{-1,1\}$
+\end{itemize}
+\end{itemize}
+
+\begin{exampleblock}{Graphes des interractions de 
+$(x_1, x_2, x_3) \mapsto 
+((\overline{x_1} + \overline{x_2}).x_3,
+x_1.x_3,
+x_1 + x_2 + x_3)$.}
+\noindent
+%\vspace{-2em}
+\begin{minipage}{0.6\textwidth}
+$
+\arraycolsep=0.1pt\def\arraystretch{0.4}
+f' = \left(
+  \begin{array}{ccc}
+            \frac{   
+   (x_1 + \overline{x_2}).x_3
+              {-}  
+             (\overline{x_1} + \overline{x_2}).x_3
+            }{\overline{x_1}{-}x_1}
+            &
+            \frac{   
+             (\overline{x_1} + x_2).x_3
+              {-}  
+             (\overline{x_1} + \overline{x_2}).x_3
+            }{\overline{x_2}{-}x_2}
+            & \dots
+    \\
+\\
+            \frac{\overline{x_1}.x_3 {-} x_1.x_3}{\overline{x_1}{-}x_1}
+             & 0 &
+\dots
+ \\
+\\
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+          \dots & 
+            \dots 
+          \end{array}
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diff --git a/talk/main.bbl b/talk/main.bbl
new file mode 100644 (file)
index 0000000..0765a59
--- /dev/null
+++ b/talk/main.bbl
@@ -0,0 +1,29 @@
+\begin{thebibliography}{ABCVS05}
+
+\bibitem[ABCVS05]{abcvs05}
+A.~Abbas, J.~M. Bahi, S.~Contassot-Vivier, and M.~Salomon.
+\newblock Mixing synchronism / asynchronism in discrete-state discrete-time
+  dynamic networks.
+\newblock In {\em 4th Int. Conf. on Engineering Applications and Computational
+  Algorithms, DCDIS'2005}, pages 524--529, Guelph, Canada, July 2005.
+\newblock ISSN 1492-8760.
+
+\bibitem[BCVC10]{BCVC10:ir}
+J.~M. Bahi, S.~Contassot-Vivier, and J.-F. Couchot.
+\newblock Convergence results of combining synchronism and asynchronism for
+  discrete-state discrete-time dynamic network.
+\newblock Research Report RR2010-02, LIFC - Laboratoire d’{I}nformatique de
+  l'{U}niversit\'{e} de {F}ranche {C}omt\'{e}, May 2010.
+
+\bibitem[Cou10]{Cou10:ir}
+J.-F. Couchot.
+\newblock Formal {C}onvergence {P}roof for {D}iscrete {D}ynamical {S}ystems.
+\newblock Research Report RR2010-03, LIFC - Laboratoire d’{I}nformatique de
+  l'{U}niversit\'{e} de {F}ranche {C}omt\'{e}, May 2010.
+
+\bibitem[Hol03]{Hol03}
+Gerard~J. Holzmann.
+\newblock {\em The SPIN Model Checker: Primer and Reference Manual}.
+\newblock Addison-Wesley, Pearson Education, 2003.
+
+\end{thebibliography}

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+   defining Unicode char U+00DF (decimal 223)
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+   defining Unicode char U+00E1 (decimal 225)
+   defining Unicode char U+00E2 (decimal 226)
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+   defining Unicode char U+00E5 (decimal 229)
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+   defining Unicode char U+00E7 (decimal 231)
+   defining Unicode char U+00E8 (decimal 232)
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+   defining Unicode char U+00EA (decimal 234)
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+   defining Unicode char U+00F2 (decimal 242)
+   defining Unicode char U+00F3 (decimal 243)
+   defining Unicode char U+00F4 (decimal 244)
+   defining Unicode char U+00F5 (decimal 245)
+   defining Unicode char U+00F6 (decimal 246)
+   defining Unicode char U+00F8 (decimal 248)
+   defining Unicode char U+00F9 (decimal 249)
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+   defining Unicode char U+00FB (decimal 251)
+   defining Unicode char U+00FC (decimal 252)
+   defining Unicode char U+00FD (decimal 253)
+   defining Unicode char U+00FE (decimal 254)
+   defining Unicode char U+00FF (decimal 255)
+   defining Unicode char U+0100 (decimal 256)
+   defining Unicode char U+0101 (decimal 257)
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+   defining Unicode char U+0103 (decimal 259)
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+   defining Unicode char U+0105 (decimal 261)
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+   defining Unicode char U+0107 (decimal 263)
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+   defining Unicode char U+0109 (decimal 265)
+   defining Unicode char U+010A (decimal 266)
+   defining Unicode char U+010B (decimal 267)
+   defining Unicode char U+010C (decimal 268)
+   defining Unicode char U+010D (decimal 269)
+   defining Unicode char U+010E (decimal 270)
+   defining Unicode char U+010F (decimal 271)
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+   defining Unicode char U+0115 (decimal 277)
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+   defining Unicode char U+0117 (decimal 279)
+   defining Unicode char U+0118 (decimal 280)
+   defining Unicode char U+0119 (decimal 281)
+   defining Unicode char U+011A (decimal 282)
+   defining Unicode char U+011B (decimal 283)
+   defining Unicode char U+011C (decimal 284)
+   defining Unicode char U+011D (decimal 285)
+   defining Unicode char U+011E (decimal 286)
+   defining Unicode char U+011F (decimal 287)
+   defining Unicode char U+0120 (decimal 288)
+   defining Unicode char U+0121 (decimal 289)
+   defining Unicode char U+0122 (decimal 290)
+   defining Unicode char U+0123 (decimal 291)
+   defining Unicode char U+0124 (decimal 292)
+   defining Unicode char U+0125 (decimal 293)
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+   defining Unicode char U+0129 (decimal 297)
+   defining Unicode char U+012A (decimal 298)
+   defining Unicode char U+012B (decimal 299)
+   defining Unicode char U+012C (decimal 300)
+   defining Unicode char U+012D (decimal 301)
+   defining Unicode char U+012E (decimal 302)
+   defining Unicode char U+012F (decimal 303)
+   defining Unicode char U+0130 (decimal 304)
+   defining Unicode char U+0131 (decimal 305)
+   defining Unicode char U+0132 (decimal 306)
+   defining Unicode char U+0133 (decimal 307)
+   defining Unicode char U+0134 (decimal 308)
+   defining Unicode char U+0135 (decimal 309)
+   defining Unicode char U+0136 (decimal 310)
+   defining Unicode char U+0137 (decimal 311)
+   defining Unicode char U+0139 (decimal 313)
+   defining Unicode char U+013A (decimal 314)
+   defining Unicode char U+013B (decimal 315)
+   defining Unicode char U+013C (decimal 316)
+   defining Unicode char U+013D (decimal 317)
+   defining Unicode char U+013E (decimal 318)
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+   defining Unicode char U+0142 (decimal 322)
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+   defining Unicode char U+0144 (decimal 324)
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+   defining Unicode char U+0148 (decimal 328)
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+   defining Unicode char U+014B (decimal 331)
+   defining Unicode char U+014C (decimal 332)
+   defining Unicode char U+014D (decimal 333)
+   defining Unicode char U+014E (decimal 334)
+   defining Unicode char U+014F (decimal 335)
+   defining Unicode char U+0150 (decimal 336)
+   defining Unicode char U+0151 (decimal 337)
+   defining Unicode char U+0152 (decimal 338)
+   defining Unicode char U+0153 (decimal 339)
+   defining Unicode char U+0154 (decimal 340)
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+   defining Unicode char U+0156 (decimal 342)
+   defining Unicode char U+0157 (decimal 343)
+   defining Unicode char U+0158 (decimal 344)
+   defining Unicode char U+0159 (decimal 345)
+   defining Unicode char U+015A (decimal 346)
+   defining Unicode char U+015B (decimal 347)
+   defining Unicode char U+015C (decimal 348)
+   defining Unicode char U+015D (decimal 349)
+   defining Unicode char U+015E (decimal 350)
+   defining Unicode char U+015F (decimal 351)
+   defining Unicode char U+0160 (decimal 352)
+   defining Unicode char U+0161 (decimal 353)
+   defining Unicode char U+0162 (decimal 354)
+   defining Unicode char U+0163 (decimal 355)
+   defining Unicode char U+0164 (decimal 356)
+   defining Unicode char U+0165 (decimal 357)
+   defining Unicode char U+0168 (decimal 360)
+   defining Unicode char U+0169 (decimal 361)
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+   defining Unicode char U+016B (decimal 363)
+   defining Unicode char U+016C (decimal 364)
+   defining Unicode char U+016D (decimal 365)
+   defining Unicode char U+016E (decimal 366)
+   defining Unicode char U+016F (decimal 367)
+   defining Unicode char U+0170 (decimal 368)
+   defining Unicode char U+0171 (decimal 369)
+   defining Unicode char U+0172 (decimal 370)
+   defining Unicode char U+0173 (decimal 371)
+   defining Unicode char U+0174 (decimal 372)
+   defining Unicode char U+0175 (decimal 373)
+   defining Unicode char U+0176 (decimal 374)
+   defining Unicode char U+0177 (decimal 375)
+   defining Unicode char U+0178 (decimal 376)
+   defining Unicode char U+0179 (decimal 377)
+   defining Unicode char U+017A (decimal 378)
+   defining Unicode char U+017B (decimal 379)
+   defining Unicode char U+017C (decimal 380)
+   defining Unicode char U+017D (decimal 381)
+   defining Unicode char U+017E (decimal 382)
+   defining Unicode char U+01CD (decimal 461)
+   defining Unicode char U+01CE (decimal 462)
+   defining Unicode char U+01CF (decimal 463)
+   defining Unicode char U+01D0 (decimal 464)
+   defining Unicode char U+01D1 (decimal 465)
+   defining Unicode char U+01D2 (decimal 466)
+   defining Unicode char U+01D3 (decimal 467)
+   defining Unicode char U+01D4 (decimal 468)
+   defining Unicode char U+01E2 (decimal 482)
+   defining Unicode char U+01E3 (decimal 483)
+   defining Unicode char U+01E6 (decimal 486)
+   defining Unicode char U+01E7 (decimal 487)
+   defining Unicode char U+01E8 (decimal 488)
+   defining Unicode char U+01E9 (decimal 489)
+   defining Unicode char U+01EA (decimal 490)
+   defining Unicode char U+01EB (decimal 491)
+   defining Unicode char U+01F0 (decimal 496)
+   defining Unicode char U+01F4 (decimal 500)
+   defining Unicode char U+01F5 (decimal 501)
+   defining Unicode char U+0218 (decimal 536)
+   defining Unicode char U+0219 (decimal 537)
+   defining Unicode char U+021A (decimal 538)
+   defining Unicode char U+021B (decimal 539)
+   defining Unicode char U+1E02 (decimal 7682)
+   defining Unicode char U+1E03 (decimal 7683)
+   defining Unicode char U+200C (decimal 8204)
+   defining Unicode char U+2010 (decimal 8208)
+   defining Unicode char U+2011 (decimal 8209)
+   defining Unicode char U+2012 (decimal 8210)
+   defining Unicode char U+2013 (decimal 8211)
+   defining Unicode char U+2014 (decimal 8212)
+   defining Unicode char U+2015 (decimal 8213)
+   defining Unicode char U+2018 (decimal 8216)
+   defining Unicode char U+2019 (decimal 8217)
+   defining Unicode char U+201A (decimal 8218)
+   defining Unicode char U+201C (decimal 8220)
+   defining Unicode char U+201D (decimal 8221)
+   defining Unicode char U+201E (decimal 8222)
+   defining Unicode char U+2030 (decimal 8240)
+   defining Unicode char U+2031 (decimal 8241)
+   defining Unicode char U+2039 (decimal 8249)
+   defining Unicode char U+203A (decimal 8250)
+   defining Unicode char U+2423 (decimal 9251)
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+   defining Unicode char U+00C6 (decimal 198)
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+   defining Unicode char U+00E6 (decimal 230)
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+   defining Unicode char U+00ED (decimal 237)
+   defining Unicode char U+00EE (decimal 238)
+   defining Unicode char U+00EF (decimal 239)
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+   defining Unicode char U+0219 (decimal 537)
+   defining Unicode char U+021A (decimal 538)
+   defining Unicode char U+021B (decimal 539)
+   defining Unicode char U+2013 (decimal 8211)
+   defining Unicode char U+2014 (decimal 8212)
+   defining Unicode char U+2018 (decimal 8216)
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+   defining Unicode char U+200C (decimal 8204)
+   defining Unicode char U+2026 (decimal 8230)
+   defining Unicode char U+2122 (decimal 8482)
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+********************************************************
+Package `algorithm2e' Release 5.1 -- october 19 2015 --
+- algorithm2e-announce@lirmm.fr mailing list for announcement about releases
+- algorithm2e-discussion@lirmm.fr mailing list for discussion about package
+subscribe by emailing sympa@lirmm.fr with 'subscribe <list> <firstname name>'
+- Author: Christophe Fiorio (christophe.fiorio@umontpellier.fr)
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+\newcommand{\Z}[0]{\ensuremath{\mathbb{Z}}}
+\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}}
+\newcommand{\rel}[0]{\ensuremath{{\mathcal{R}}}}
+\newcommand{\Gall}[0]{\ensuremath{\mathcal{G}}}
+\newcommand{\F}[0]{\ensuremath{\mathbb{F}}}
+\newcommand{\StratSet}[0]{\ensuremath{\mathbb{S}}}
+\newcommand{\class}[1]{\ensuremath{\langle #1\rangle}}
+
+
+ \newcommand{\eqNode}[0]{\ensuremath{{\mathcal{R}}}}
+
+
+\setbeamertemplate{itemize/enumerate body begin}{\footnotesize}
+\setbeamertemplate{itemize/enumerate subbody begin}{\footnotesize}
+
+
+\newcommand{\fc}[1]{\ensuremath{\overline{#1}}}
+%\renewcommand\footnotelayout{\tiny}
+\let\oldfootnotesize\footnotesize
+\renewcommand*{\footnotesize}{\oldfootnotesize\scriptsize}
+
+\usepackage{graphicx}
+\DeclareGraphicsExtensions{.jpg, .png , .pdf, .bmp, .pdftex}
+\begin{document}
+
+
+\title[Modèles discrets pour la sécurité]{Modèles discrets pour la sécurité informatique: des méthodes itératives à l'analyse vectorielle.}
+
+
+
+\author[J.-F. Couchot]{Jean-Fran\c{c}ois {\sc Couchot}}
+\date[HDR, le 30/01/16]{\vspace{-1.5em}Soutenance d'HDR: le 30/01/16}
+
+
+
+\institute[Institut FEMTO-ST]{FEMTO-ST Institut, Université de Franche-Comté, France}
+
+
+% \newtheorem{theorem}{Théorème}
+% \newtheorem{lemma}{Lemme}
+% \newtheorem{corollary}{Corollaire}
+% \newtheorem*{xpl}{Exemple}
+
+% \newtheorem{Def}{Définition}
+%  ____  _____ ____  _   _ _____ 
+% |  _ \| ____| __ )| | | |_   _|
+% | | | |  _| |  _ \| | | | | |  
+% | |_| | |___| |_) | |_| | | |  
+% |____/|_____|____/ \___/  |_|  
+% 
+
+%\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}}
+
+\setbeamertemplate{background}{\titrefemto}
+
+\begin{frame}[plain]
+\titlepage
+\end{frame}
+ 
+\setbeamertemplate{background}{\pagefemto}
+
+% \begin{frame}{Outline}
+\setbeamertemplate{section in toc}[sections numbered] 
+% \tableofcontents
+% \end{frame}
+
+
+
+
+%\frame{\frametitle{Plan}\tableofcontents[hideallsubsections]}
+
+\section{Introduction: iterations de réseaux booléens}
+\inputFrameb{Réseau booléen (définition)}{rb}
+\inputFrameb{3 schémas $\leadsto$ 3 graphes d'itérations}{graphes}
+\inputFrameb{Attracteurs}{attracteurs}
+\inputFrameb{Dépendance entre éléments}{jacobienne}
+\inputFrameb{Mode asynchrone}{asynchrone}
+\inputFrameb{Un exemple motivant}{xplsddasync}
+\inputFrameb{Problématique}{sddproblematique}
+
+\section{Réseaux booléens: des preuves de convergences}\label{sec:sdd}
+\frame{\frametitle{Plan}\tableofcontents[currentsection]}
+\inputFrameb{Un peu de Synchronisme}{synchronisons}
+\inputFrameb{Mode mixte avec $g$}{mixteexample}
+\inputFrameb{Convergence par la pratique}{preuveconvmotiv}
+\inputFrameb{Du système booléen au modèle PROMELA}{sdd2promela}
+
+
+
+\section{Des systèmes dynamiques discrets au chaos}\label{sec:sddchaos}
+\frame{\frametitle{Plan}\tableofcontents[currentsection]}
+
+
+\section{Applications à la génération de nombres pseudo-aléatoires}\label{sec:prng}
+\frame{\frametitle{Plan}\tableofcontents[currentsection]}
+
+
+
+\section{Application au masquage d'information}\label{sec:ih}
+\frame{\frametitle{Plan}\tableofcontents[currentsection]}
+
+\section{Conclusion}
+\frame{\frametitle{Plan}\tableofcontents[currentsection]}
+\inputFrameb{Synthèse scientifique}{conclusion}
+
+
+
+% \section{Formalisation}
+% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]}
+% \inputFrameb{Algèbre de Boole}{ab}
+% \inputFrameb{Réseaux booléens (définition)}{rb}
+% \inputFrameb{Trois modes $\leadsto$ trois graphes d'itérations}{graphes}
+% \inputFrameb{Attracteurs}{attracteurs}
+% \inputFrameb{Matrice Jacobienne $f'$}{jacobienne}
+
+
+% \section{Quelques résultats théoriques}
+% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]}
+% \subsection{Conditions de convergence}
+% \inputFrameb{François Robert 1980}{robert80}
+% \inputFrameb{François Robert 1995}{robert95}
+% \subsection{Nombre d'attracteurs}
+% \inputFrameb{Absence de cycles}{attracteursth}
+
+% \section{Application à l'étude du bacille de pyo}
+% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]}
+% \inputFrameb{Retour aux réseaux de régulation de gènes}{pyo}
+
+% \section{Des questions?}
+% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]}
+% \inputFrameb{Un résumé}{../abstract}
+% \inputFrameb{Mettre en confiance le candidat}{confiance}
+% \inputFrameb{Similarités}{similarites}
+% \inputFrameb{Exemple d'itérations asynchrones}{asynchroneex}
+% \inputFrameb{Montrer que $d$ (Sect. 3) est une distance}{dv} 
+% \inputFrameb{Application du théorème 2}{th2} 
+% \inputFrameb{Théorème 4: illustrez le dans $\Bool^2$}{th4} 
+
+
+
+% \section{Conclusion et perspectives}
+% \frame{\frametitle{Outline}\tableofcontents[currentsection,hideallsubsections]}
+  
+
+\bibliographystyle{alpha}
+\bibliography{abbrev,biblioand}
+\end{document}
+%  _____ ___ _   _ 
+% |  ___|_ _| \ | |
+% | |_   | ||  \| |
+% |  _|  | || |\  |
+% |_|   |___|_| \_|
+%

diff --git a/talk/main.thm b/talk/main.thm
new file mode 100644 (file)
index 0000000..5278c70
--- /dev/null
+++ b/talk/main.thm
@@ -0,0 +1 @@
+\contentsline {\let N}{{\translate {Theorem}}{1}{~\cite {BCVC10:ir}}}

diff --git a/talk/main.toc b/talk/main.toc
new file mode 100644 (file)
index 0000000..fbd10b8
--- /dev/null
+++ b/talk/main.toc
@@ -0,0 +1,8 @@
+\beamer@endinputifotherversion {3.36pt}
+\select@language {french}
+\beamer@sectionintoc {1}{Introduction: iterations de r\IeC {\'e}seaux bool\IeC {\'e}ens}{2}{0}{1}
+\beamer@sectionintoc {2}{R\IeC {\'e}seaux bool\IeC {\'e}ens: des preuves de convergences}{9}{0}{2}
+\beamer@sectionintoc {3}{Des syst\IeC {\`e}mes dynamiques discrets au chaos}{14}{0}{3}
+\beamer@sectionintoc {4}{Applications \IeC {\`a} la g\IeC {\'e}n\IeC {\'e}ration de nombres pseudo-al\IeC {\'e}atoires}{15}{0}{4}
+\beamer@sectionintoc {5}{Application au masquage d'information}{16}{0}{5}
+\beamer@sectionintoc {6}{Conclusion}{17}{0}{6}

diff --git a/talk/mixteexample.tex b/talk/mixteexample.tex
new file mode 100644 (file)
index 0000000..0617945
--- /dev/null
+++ b/talk/mixteexample.tex
@@ -0,0 +1,19 @@
+\begin{exampleblock}{}
+\begin{figure}
+\subfloat[Parallèle synchrone]{
+  \includegraphics[scale=0.2]{../images/eval_sync}
+}
+
+\vspace{-1em}
+\subfloat[Asynchrone]{
+  \includegraphics[scale=0.2]{../images/eval_async}
+}
+
+\vspace{-1em}
+\subfloat[Mixte tq.\class{1} $=\{1,2\}$, \class{3} $=\{3\}$,
+\class{4} $=\{4,5\}$.]{
+  \includegraphics[scale=0.2]{../images/eval_mixte}
+}
+\caption{Graphes des itérations de $g$} 
+\end{figure}
+\end{exampleblock}

diff --git a/talk/preuveconvmotiv.tex b/talk/preuveconvmotiv.tex
new file mode 100644 (file)
index 0000000..b3e38d4
--- /dev/null
+++ b/talk/preuveconvmotiv.tex
@@ -0,0 +1,18 @@
+
+\begin{itemize}
+\item Conditions suffisantes de convergence: facile à appliquer, 
+  domaine restreint
+\item Recherche d'une métrique decroissante minorée: difficile
+\item Simulations:  
+\begin{itemize}
+\item Non exhaustives pour les schémas généralisés et asynchrones.
+\item Verdict $\leftrightarrow$ vérité ssi divergence (contre-exemple).
+\end{itemize}
+\item Souhait: exploiter un outil qui traiterait toutes les transitions 
+\begin{itemize}
+\item explosion combinatoire: par abstraction,
+  quotientage, ordre partiel\ldots
+\item Model checker: SPIN~\cite{Hol03}.
+\item Correction et complétude de la démarche.
+  \end{itemize}
+\end{itemize}

diff --git a/talk/q2.tex b/talk/q2.tex
new file mode 100644 (file)
index 0000000..8a2c11b
--- /dev/null
+++ b/talk/q2.tex
@@ -0,0 +1,5 @@
+Dans quel(s) contexte(s) avez vous déjà rencontré des générateurs de nombres pseudo-aléatoires.
+
+Réponses: les probabilités, les statistiques, dans des simulations (méthodes de Monte-Carlo),
+le chiffrage (algorithme RSA par exemple), notamment pour  générer des grands nombres aléatoires
+premiers, la localisation par GPS.
\ No newline at end of file

diff --git a/talk/q3.tex b/talk/q3.tex
new file mode 100644 (file)
index 0000000..13b29b6
--- /dev/null
+++ b/talk/q3.tex
@@ -0,0 +1,6 @@
+A la ligne~30, il est écrit qu'\og un générateur congruentiel linéaire
+produit une suite  périodique dont la période ne peut pas excéder $m$\fg{}. Pourquoi?
+
+Réponse: En construisant $m+1$ termes successifs, il est nécessaire de rencontrer 
+au moins deux termes $x_i$ et $x_j$ égaux puisqu'il n'y a que $m$ termes différents dans
+$\{0,\dots,m-1\}$. Quel que soit $k \in \Nats$, on a  $x_{i+k} = x_{j+k}$.
\ No newline at end of file

diff --git a/talk/q4.tex b/talk/q4.tex
new file mode 100644 (file)
index 0000000..b316811
--- /dev/null
+++ b/talk/q4.tex
@@ -0,0 +1,9 @@
+A la ligne~204, on prétend savoir calculer 
+\og l'inverse de $x_i-x_{i+1}$ modulo $m$\fg{}. Comment faire ceci? Ce problème a-t-il toujours une solution?
+
+Réponses:
+Il s'agit d'appliquer l'algorithme d'Euclide étendu: on cherche $u$ (et $v$)
+tels que $(x_i-x_{i+1}).u + m.v = 1$. Or ceci n'admet une solution que si 
+$(x_i-x_{i+1})$ et $m$ sont premiers entre-eux. 
+(Par exemple, impossible de trouver l'inverse de 
+2 modulo 6).
\ No newline at end of file

diff --git a/talk/q6.tex b/talk/q6.tex
new file mode 100644 (file)
index 0000000..2988081
--- /dev/null
+++ b/talk/q6.tex
@@ -0,0 +1,13 @@
+Que dire des générateurs suivants:
+\begin{itemize} 
+\item $x_i = (129x_{i-1} + 907633385) \mod 2^{32}$ (générateur du Turbo Pascal) 
+\item $x_i = 69069x_{i-1} \mod 2^{32}$  (générateur de Marsaglia) 
+\end{itemize}
+
+Réponses:\begin{itemize}
+\item Le premier vérifie toutes les hypothèses du théorème~1
+($907633385$ et $2^{32}$ sont premiers entre eux, $2$ est le seul facteur premier de $2^{32}$ et comme 129 est impair, $129 \equiv 1 \mod 2$, et, enfin, $129 \equiv 1 \mod 4$). Sa période est donc $2^{32}$.
+\item $2^{32}$ n'est pas premier. On ne peut pas directement appliquer 
+le théorème~2 et on ne peut donc rien conclure. 
+Remarque: cette condition n'est en effet pas nécessaire car la période de ce générateur est cependant $2^{32}$.
+\end{itemize} 

diff --git a/talk/q8.tex b/talk/q8.tex
new file mode 100644 (file)
index 0000000..8b2552a
--- /dev/null
+++ b/talk/q8.tex
@@ -0,0 +1,9 @@
+Que démontre la section~2.4? Quelle est la portée de ce résultat?
+
+Réponse. Cette section montre qu'un générateur de nombres pseudo-aléatoires 
+peut certes produire des sorties qui vérifient les critères statistiques 
+de bonne distribution (période longues, sous-suites équitablement réparties),
+mais une personne peut cependant deviner les paramètres et donc 
+\og casser \fg{} l'aléa.
+
+Dans une simulation physique, cela n'est pas nécessairement grave. Lorsqu'on génère des grands nombres pseudo aléatoirement  pour de la cryptographie ou dans un casino, connaître l'algorithme sous-jacent peut être une faille de sécurité.

diff --git a/talk/q9.tex b/talk/q9.tex
new file mode 100644 (file)
index 0000000..e57eee3
--- /dev/null
+++ b/talk/q9.tex
@@ -0,0 +1,22 @@
+Supposons qu'on dispose d'une fonction qui génère un nombre pseudo-aléatoire
+dont la sortie $j$ est uniformément distribuée sur l'intervalle $[0,1[$. 
+Comment peut-on faire 
+pour disposer d'une sortie uniformément distribuée sur l'ensemble des entiers 
+$\{0,\dots, m\}$.
+
+Réponse: On commence par diviser $[0,1[$ en $m+1$ intervalles de même 
+longueurs: 
+$I_0= [0, \dfrac{1}{m+1}[$, 
+$I_1= [\dfrac{1}{m+1}, \dfrac{2}{m+1} [$, \ldots 
+$I_m = [\dfrac{m}{m+1}, 1[$. 
+Si la sortie $j$ appartient à $I_0$, on retourne 0,\ldots. 
+
+Ceci revient à d'abord multiplier $j$ par $m+1$ et à prendre ensuite sa partie entière, ce qui est fait dans le code Python suivant.
+La fonction Python \verb+randint(a,b)+ génère un entier 
+entre $a$ et $b$, les deux nombres compris.  
+
+\begin{lstlisting}
+def genereAleaEntier(m):
+  j= random()
+  return int(j*(m+1))
+\end{lstlisting}
\ No newline at end of file

diff --git a/talk/rb.tex b/talk/rb.tex
new file mode 100644 (file)
index 0000000..ad0d751
--- /dev/null
+++ b/talk/rb.tex
@@ -0,0 +1,45 @@
+\begin{itemize}
+\item Une fonction
+$
+f:\Bool^{\mathsf N}\to\Bool^{\mathsf N}$, $x=(x_1,\dots,x_{\mathsf N})\mapsto (f_1(x),\dots,f_{\mathsf N}(x))
+$
+\item Un schéma de mise à jour de la suite $(x^{t})^{t
+  \in  \Nats}$ des  configurations:
+\begin{itemize}
+\item \emph{Parallèle synchrone:} 
+  $x^{t+1}=f(x^t)$. 
+\item \emph{Unaire:}  à partir de la \emph{stratégie unaire}  
+  $S = \left(s^t\right)^{t \in  \mathds{N}}$, 
+  modification de l'élément $s^t$ de $x^t$ 
+  $$  
+  \begin{array}{l}   
+    x^{t+1}= (x^{t+1}_1,\dots,x^{t+1}_n) \textrm{ où } 
+    x^{t+1}_i =  
+    \left\{
+      \begin{array}{l}
+        f_i( x^{t}) \textrm{ si $i = s^t$}  \\
+        x^t_i \textrm{ sinon.} 
+      \end{array} 
+    \right.
+  \end{array} 
+$$
+
+\item \emph{Généralisé:} à partir de la \emph{stratégie généralisée}
+  $\left(s^t\right)^{t \in  \mathds{N}}$, 
+  à  chaque  itération $t$, 
+  modification des éléments de $x^t$ dans $s^t\subset\{1,\ldots,n\}$ 
+  $$  
+  \begin{array}{l}   
+    x^{t+1}= (x^{t+1}_1,\dots,x^{t+1}_n) \textrm{ où } 
+    x^{t+1}_i =  
+    \left\{
+      \begin{array}{l}
+        f_i( x^{t}) \textrm{ si $i \alert{\in} s^t$}  \\
+        x^t_i \textrm{ sinon} 
+      \end{array} 
+    \right.
+  \end{array} 
+$$  
+
+\end{itemize}
+\end{itemize}

diff --git a/talk/regdef.tex b/talk/regdef.tex
new file mode 100644 (file)
index 0000000..a88cc0c
--- /dev/null
+++ b/talk/regdef.tex
@@ -0,0 +1,19 @@
+\begin{definition}[]%[A base de registres à décalage linéaire]
+Soit $p$ un nombre premier.
+\vspace{-2em}
+\begin{center}
+\includegraphics[scale=0.2]{regxpl.png}
+\end{center}
+\vspace{-2em}
+\noindent Un générateur \emph{à base de registres à décalage linéaires}
+produit $a_{k+r}$ de $\F_{p}$ selon  
+\begin{equation}
+\tag{9}
+a_{k+r} \equiv \sum\limits_{j=0}^{r-1}q_ja_{j+k} \mod p.
+\label{eq:reg:def}
+\end{equation}
+
+\end{definition}
+
+
+

diff --git a/talk/regth5.tex b/talk/regth5.tex
new file mode 100644 (file)
index 0000000..1cdb543
--- /dev/null
+++ b/talk/regth5.tex
@@ -0,0 +1,29 @@
+\begin{theorem}[Théorème 5]
+Soit $p$ premier,  $q_0, \dots , q_{r-1} \in \F_p$  tels que 
+$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif 
+sur $\F_p$ et soit 
+$a_0, \dots, a_{r-1}$  une  graine différente du vecteur nul.
+Le générateur à base de registres à décalage linéaires engendre une suite de 
+période $p^r-1$.
+\end{theorem}
+
+
+
+\begin{exampleblock}{Analyse de l'exemple de la Figure~5}
+On a $p=3$, $r=4$, la graine $1,0,0,0$ n'est pas nulle.
+Il suffi(rai)t de démontrer que
+
+\begin{enumerate}
+\item  $Q(x) = x^4 - 2 x^{3} - 1 \equiv x^4 + x^{3} + 2$ est irréductible 
+(facile)
+\item  chaque  élément de 
+$
+(\F_{p})_{r-1}(x) 
+$
+peut s'écrire sous la forme $x^i$ modulo $Q(x)$, $0 \le i \le p^r-2$ (long, mais vrai)
+\end{enumerate}
+
+La période d'un tel générateur est donc $3^4-1=80$. 
+\end{exampleblock}
+
+

diff --git a/talk/regth5.tex.bak b/talk/regth5.tex.bak
new file mode 100644 (file)
index 0000000..e9a2696
--- /dev/null
+++ b/talk/regth5.tex.bak
@@ -0,0 +1,29 @@
+\begin{theorem}[Théorème 5]
+Soit $p$ premier,  $q_0, \dots , q_{r-1} \in \F_p$  tels que 
+$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif 
+sur $\F_p$ et soit 
+$a_0, \dots, a_{r-1}$  une  graine différente du vecteur nul.
+Le générateur à base de registres à décalage linéaires engendre une suite de 
+période $p^r-1$.
+\end{theorem}
+
+
+
+\begin{exampleblock}{Analyse de l'exemple de la Figure~5}
+On a $p=3$, $r=4$, la graîne $1,0,0,0$ n'est pas nulle.
+Il suffi(rai)t de démontrer que
+
+\begin{enumerate}
+\item  $Q(x) = x^4 - 2 x^{3} - 1 \equiv x^4 + x^{3} + 2$ est irréductible 
+(facile)
+\item  chaque  élément de 
+$
+(\F_{p})_{r-1}(x) 
+$
+peut s'écrire sous la forme $x^i$ modulo $Q(x)$, $0 \le i \le p^r-2$ (long, mais vrai)
+\end{enumerate}
+
+La période d'un tel générateur est donc $3^4-1=80$. 
+\end{exampleblock}
+
+

diff --git a/talk/regth5demo.tex b/talk/regth5demo.tex
new file mode 100644 (file)
index 0000000..5aa50f7
--- /dev/null
+++ b/talk/regth5demo.tex
@@ -0,0 +1,20 @@
+\begin{theorem}[Théorème 5]
+Soit $p$ premier,  $q_0, \dots , q_{r-1} \in \F_p$  tels que 
+$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif 
+sur $\F_p$ et soit 
+$a_0, \dots, a_{r-1}$  une  graine différente du vecteur nul.
+Le générateur à base de registres à décalage linéaires engendre une suite de 
+période $p^r-1$.
+\end{theorem}
+
+
+\begin{enumerate}
+\item Construction d'un polynôme $b \in (\F_{p})_{r-1}(x)$ et d'une fonction 
+$T: (\F_{p})_{r-1}(x) \rightarrow  \F_p$ t.q. $T(b.x^n)=a_n$ ({\sc Lemme}~6) 
+\item Comme $x^{p^r-1}\equiv 1 \mod Q(x)$, d'après 1. la période de $a_n$ est au plus $p^r-1$ (l.~290)
+\begin{itemize}
+\item $a_{n+p^r+1} \equiv T(b.x^{n+p^r+1}) \equiv  T(b.x^{n}x^{p^r+1}) \equiv a_n$.
+\end{itemize}
+\item Si la période était inférieure à $p^r -1$, alors $Q(x)$ ne serait pas 
+primitif (l. 296) 
+\end{enumerate}

diff --git a/talk/regth6.tex b/talk/regth6.tex
new file mode 100644 (file)
index 0000000..f82225c
--- /dev/null
+++ b/talk/regth6.tex
@@ -0,0 +1,15 @@
+
+\begin{definition}[Fenêtre]
+Sous les mêmes conditions que celles du théorème~5,
+on appelle \emph{fenêtre} toute sous-suite de longueur 
+$p^r-1$.
+\end{definition}
+
+\begin{theorem}[Théorème 6]
+Sous les mêmes conditions que celle du théorème~5,
+soit $k$ un entier, $k \le r$. Alors dans chaque fenêtre, toute 
+sous-suite de longueur $k$ apparaît $p^{r-k}$ fois,
+sauf la sous-suite nulle  qui, elle, apparaît $p^{r-k}-1$ fois.  
+\end{theorem}
+
+

diff --git a/talk/regxpl.png b/talk/regxpl.png
new file mode 100644 (file)
index 0000000..b5e56d4
Binary files /dev/null and b/talk/regxpl.png differ

diff --git a/talk/sdd2promela.tex b/talk/sdd2promela.tex
new file mode 100644 (file)
index 0000000..4676dfa
--- /dev/null
+++ b/talk/sdd2promela.tex
@@ -0,0 +1,17 @@
+\begin{itemize}
+\item Points clefs de la traduction:
+\begin{itemize}
+\item Stratégie: pseudo périodicité garantie par le choix indéterministe de 
+  SPIN.
+\item Délais(bornés par construction): oubli de certaines valeurs grâce à l'indéterminisme de SPIN.
+\end{itemize}
+\item Convergence universelle:   $\diamond  (\Box  \verb+Xp+ = \verb+X+)$.
+\end{itemize}
+
+\begin{theorem}[Correction et complétude de la traduction vers Promela~\cite{Cou10:ir}]
+  Soit $\phi$ un modèle de système dynamique discret et $\psi$ sa traduction PROMELA.  
+  Les itérations de $\phi$ sont universellement convergentes si et 
+  seulement si
+  $\psi$ vérifie
+  la propriété  LTL  sous hypothèse d'équité faible.
+\end{theorem}

diff --git a/talk/sddproblematique.tex b/talk/sddproblematique.tex
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+\begin{itemize}
+\item Peut-on prédire le comportement de réseaux bouléens (Sec.~\ref{sec:sdd})
+\begin{itemize}
+\item Théorique: \alert{conditions théoriques}
+  nécessaires/suffisantes de convergence/divergence? 
+\item Pratique: vérification par simulation (\alert{exhaustive?})
+\end{itemize}  
+\item Itérations divergentes \alert{$ \leftrightarrow$ comportement chaotique} (Sec.~\ref{sec:sddchaos})?
+\begin{itemize}
+\item Caractérisation des réseaux bouléens chaotiques.
+\item Génération et prédiction.
+\end{itemize}
+\item Générateurs de nombres pseudo-aléatoires (Sec.~\ref{sec:prng}):
+\begin{itemize}
+\item Caractérisation d'un PRNG chaotique.  
+\item Générations et qualité.
+\end{itemize}
+\item Du Chaos au masquage d'information (Sec.\ref{sec:ih}).
+\begin{itemize}
+\item D'un point de vue chaotique.
+\item D'un point de vue analyse vectorielle discrète.
+\end{itemize}
+\end{itemize}
+
+

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diff --git a/talk/synchronisons.tex b/talk/synchronisons.tex
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+\begin{itemize}
+\item \emph{Mode mixte}~\cite{abcvs05}: regroupement
+des n{\oe}uds qui pourraient introduire des cycles.
+\begin{itemize}
+\item A l'intérieur de chaque groupe: mode synchrone.
+\item A l'extérieur de chaque groupe: mode asynchrone.
+\end{itemize}
+\item Relation de synchronisation:  $i \eqNode j$ 
+  si $i$ et $j$  dans la même CFC du graphe des interactions.
+\end{itemize}
+\begin{theorem}[~\cite{BCVC10:ir}]
+  Soit $f$ possédant un unique point fixe $x^*$ et une stratégie 
+  pseudo-périodique $s$.
+  Si les itérations synchrones convergent vers $x^*$ pour cette stratégie, 
+  alors les itérations mixtes à délai uniforme convergent aussi vers $x^*$
+  pour cette stratégie.
+\end{theorem}
+

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diff --git a/talk/xplsddasync.tex b/talk/xplsddasync.tex
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+\begin{exampleblock}{ $g(x_1,x_2,x_3,x_4,x_5)  = 
+(x_1.\overline{x_2} + \overline{x_1}.x_2,\overline{x_1 + x_2}, x_3.\overline{x_1}, x_5, \overline{x_3} + x_4)$}
+  \begin{figure}  
+    \centering
+    \subfloat[$\textsc{gis}(g)$]{
+        \includegraphics[scale=0.30]{../images/para_iterate_dec}
+    }\qquad
+    \subfloat[$\textsc{giu}(g)$ (extrait)]{
+        \includegraphics[scale=0.30]{../images/chao_iterate_excerpt}
+    }
+\caption{Graphes des itérations synchrones}
+    \begin{itemize}
+      \item Avec $D^t = t$ sauf $D^t_{12} = t-1$ pour  $t$ impair, 
+        $g$ oscille entre $(0,0,0,1,1)$ et $(0,1,0,1,1)$
+      \item Schéma parallèle: converge en synchrone, diverge en asynchrone 
+      \end{itemize}
+
+\end{figure}
+\end{exampleblock}