lectures/lecture4/lecture4.ipynb

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   "source": [
    "# Собственные векторы, собственные значения. Разложение Шура и QR-алгоритм "
   ]
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   "source": [
    "## На прошлой лекции\n",
    "\n",
    "- Матричное умножение\n",
    "- Иерархия памяти\n",
    "- BLAS\n",
    "- Алгоритм Штрассена\n",
    "- Вычисление QR разложения"
   ]
  },
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   "source": [
    "## План на сегодня\n",
    "\n",
    "- Собственные векторы и их приложения (PageRank)\n",
    "- Круги Гершгорина\n",
    "- Степенной метод вычисления собственных векторов и значений\n",
    "- Теорема Шура\n",
    "- Нормальные матрицы\n",
    "- QR алгоритм"
   ]
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   "source": [
    "## Что такое собственный вектор?\n",
    "\n",
    "**Определение.** Вектор $x \\ne 0$ называется **собственным** для квадратной матрицы $A$, если найдётся такое число $\\lambda$ что\n",
    "\n",
    "$$\n",
    "   Ax = \\lambda x.\n",
    "$$\n",
    "\n",
    "Число $\\lambda$ называется **собственным значением**.\n",
    "\n",
    "Так как матрица $A - \\lambda I$ должна иметь нетривиальное ядро (что такое ядро?), \n",
    "собственные значения являются корнями характеристического полинома\n",
    "\n",
    "$$ \\det (A - \\lambda I) = 0.$$"
   ]
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   "source": [
    "## Разложение по собственным векторам (eigendecomposition)\n",
    "\n",
    "Если матрица  $A$ размера $n\\times n$ имеет $n$ собственных векторов $s_i$, $i=1,\\dots,n$:\n",
    "\n",
    "$$ As_i = \\lambda_i s_i, $$\n",
    "\n",
    "то это может быть записано в виде\n",
    "\n",
    "$$ A S = S \\Lambda, \\quad\\text{где}\\quad S=(s_1,\\dots,s_n), \\quad \\Lambda = \\text{diag}(\\lambda_1, \\dots, \\lambda_n),$$\n",
    "\n",
    "или эквивалентно\n",
    "\n",
    "$$ A = S\\Lambda S^{-1}. $$\n",
    "\n",
    "Такая форма представления матрицы $A$ называется разложением по собственным векторам (**eigendecomposition**). Матрицы, которые могут быть представлены в таком виде, называются **диагонализуемыми**."
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   "source": [
    "#### Существование\n",
    "\n",
    "Какие матрицы являются диагонализуемыми?\n",
    "\n",
    "Простым примером являются матрицы, у которых все собственные значения различны.\n",
    "Более общий факт звучит так: \n",
    "\n",
    "матрица диагонализуема тогда и только тогда, когда **алгебраическая кратность** каждого собственного значения (кратность корня характеристического многочлена) совпадает с его **геометрической кратностью** (размерностью собственного подпространства). \n",
    "\n",
    "Для наших целей наиболее важный класс диагонализуемых матриц – это класс **нормальных матриц**: \n",
    "\n",
    "$$AA^* = A^* A.$$\n",
    "\n",
    "Почему такие матрицы диагонализуемы станет ясно чуть позже."
   ]
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   "source": [
    "#### Пример\n",
    "\n",
    "* Легко проверить, что матрица $$A = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$$ имеет одно собственное значение  $1$ кратности $2$ (поскольку характеристический многочлен имеет вид $p(\\lambda)=(1-\\lambda)^2$), но только один собственный вектор  $\\begin{pmatrix} c \\\\ 0  \\end{pmatrix}$ и следовательно такая матрица недиагонализуема."
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   "source": [
    "## Почему важны собственные векторы и собственные значения?\n",
    "\n",
    "- Собственные векторы имеют как важное теоретическое значения, так и многочисленные приложения.\n",
    "- Весь микромир подчиняется законам следующим из **уравнения Шрёдингера**, которое является задачей на поиск собственного вектора и собственного значения:\n",
    "\n",
    "$$\n",
    "    H \\psi = E \\psi,\n",
    "$$\n",
    "\n",
    "где $E$ – нижний уровень энергии, $\\psi$ – волновая функция и $H$ – гамильтониан.  \n",
    "- Более половины вычислительных мощностей в мире тратится на решение задач такого типа для задач дизайна материалов и разработки новых лекарств "
   ]
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   "source": [
    "## Собственные значения – это частоты выбраций\n",
    "\n",
    "Обычно вычисление собственных значений и собственных векторов необходимо для изучения\n",
    "\n",
    "- вибраций в механических структурах\n",
    "- снижения сложности моделей сложных систем"
   ]
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   "execution_count": 1,
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\n",
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/xKGA3RNzvKg\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x10da18630>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo \n",
    "YouTubeVideo(\"xKGA3RNzvKg\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Google PageRank\n",
    "\n",
    "Одна из самых известных задач, сводящихся к вычислению собственного вектора, – это задача вычисления **Google PageRank**. \n",
    "\n",
    "- Задача состои в ранжировании веб-страницы: какие из них являются важными, а какие нет\n",
    "- В интернете страницы ссылаются друг на друга\n",
    "- PageRank определяется рекурсивно. Обозначим за $p_i$ **важность** $i$-ой страницы. Тогда определим эту важность как усреднённую важность всех страниц, которые ссылаются на данную страницу. Это определение приводит к следующей линейной системе\n",
    "\n",
    "$$ p_i = \\sum_{j \\in N(i)} \\frac{p_j}{L(j)}, $$\n",
    "\n",
    "где $L(j)$ – число исходящих ссылок с $j$-ой страницы, $N(i)$ – число соседей $i$-ой страницы. Это может быть записано следующим образом\n",
    "\n",
    "$$ p = G p, \\quad G_{ij} = \\frac{1}{L(j)} $$\n",
    "\n",
    "или как задача на собственные значения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\n",
    "   Gp = 1 p,\n",
    "$$\n",
    "\n",
    "то есть мы уже знаем, что у матрицы $G$ есть собственное значение равное $1$. \n",
    "Заметим, что $G$ – **левостохастичная** матрица, то есть сумма в каждом столбце равна $1$. \n",
    "Проверьте, что любая левостохастичная матрица имеет собственное значение равное $1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "## Демо\n",
    "\n",
    "Мы можем вычислить PageRank с помощью библиотек на Python. \n",
    "Будем использовать бибилотеку ```networkx``` для работы с графами, она может быть установлена с помощью следующей команды\n",
    "\n",
    "```conda install networkx```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Возьмём простой пример графа [Zachary karate club](https://en.wikipedia.org/wiki/Zachary%27s_karate_club). \n",
    "Этот граф был собран вручную в 1977, и является классическим графом для анализа соцсетей."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/lib/python3.6/site-packages/networkx/drawing/nx_pylab.py:611: MatplotlibDeprecationWarning: isinstance(..., numbers.Number)\n",
      "  if cb.is_numlike(alpha):\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import networkx as nx\n",
    "kn = nx.read_gml('karate.gml')\n",
    "#nx.write_gml(kn, 'karate2.gml')\n",
    "nx.draw_networkx(kn) #Draw the graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Сейчас мы можем вычислить PageRank, используя функцию, встроенную в NetworkX. Мы также изобразим вершины графа пропорционально тому, наскольку они важны в смысле величины PageRank'a."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/lib/python3.6/site-packages/networkx/drawing/nx_pylab.py:611: MatplotlibDeprecationWarning: isinstance(..., numbers.Number)\n",
      "  if cb.is_numlike(alpha):\n"
     ]
    },
    {
     "data": {
      "image/png": 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9mitXruDRrh16U1PiMfSDB0gA4lM+J1asSLyzMy9jDccrQUReus3b21sU5aURHS1y9apIVFS63du2bRNXV1cJCgp6ZhKJiYmya9cuGTFihFSrVk0KFiwobdq0kblz58rly5efef3hw4eluKenvPnGG2JpYiK2ZmZia2kpZUqXztatJCUlSWhoqEyfPl3at28vhQsXFg8PD3nvvfdk5syZcvjwYUlKSko9f8yYMYKhu3zqNhpEQIo+th+Qv//+O1vleRHMmjVLvL29xcLCQrp165a6f//+/dK4cWNxdHQUFxcXad++vVy/fj3/CioiQKjk4Jmr2igUJR8dOnQIf39/Jk6cyAcZjNR+mlu3brFlyxY2b97Mli1bcHV1pVmzZjRr1oz69eunH7kNcPUq10uVolNSEl1F6AWGBndbW9i5E8qWzTCf6OhoQkJCUgcDHjx4kNdffz11MGDdunUpVqxYlntUTWjalPcuX6ZEdDTY20OvXvDRR4bPL6k1a9ZgYmJiGCsSF2cYVKjX88evvxJtYkLTli0xMzOjf//+XL9+nc2bN+dbWfNtZHZ+UIFC+S85c+YMTZs2ZeDAgXz++efPvuAxer2eI0eOsHnzZjZv3syxY8eoV69eauAoU7o02htvoD9/nkZ6PV3AECjA0B3V3R2uXAFzc65fv55uXqmzZ89StWrV1MBQu3ZtnJyccnSfd+7coUyZMly4cOE/ORnhqFGjuHbtGos6doQPP4TISMOBAQPg++85cvQovr6+RKV0csgP+TmOQlGUXPDy8mL37t00adKEe/fu8c0332RrzIOJiQk+Pj74+PgwatQoHjx4wF9//cXmzZuZPHky9ZOTWXjvHlYpa2x/CQwHygLfilDn/n3mNm7MtKtXefjwIXXr1qVevXrMmjULb2/vLE/X8Szz58+nbdu2/8kgkSomBtq1Sz99y+zZUKIEu2JjKV++fP6VLRdUoFCUF8Drr7/O7t27efvtt7l79y5z5sx59vxOT+Ho6Ej79u1p3749IsKd/v2xDAgAYCLwBmABLAdaAmEJCTQWofHvv1O2bNk8WcI1KSmJH3/8kQ0bNhg97RfK339DcnL6fTExHJ88mXH37rE+G7MGv0hUrydFeUG4urqyfft2zp8/z/vvv09iYmKu09Q0DbfChdFSgk5NDN10LTEMAKyLYWGjcpUqUa5cuTxb53vdunUUL16cKlX+2ysK6JKS0D0WKC4AzS9fZsaMGdSvXz9/CpZLKlAoygukQIEC/P777yQlJdGyZUuio6Nzn+jbb8NTqo80QCwsoHXr3OeTiVmzZvHpp5/maR756ebNm2zfvp3Np0+TnKba8ArQWNP4qmVLunbtmn8FzCUVKBTlBWNlZcXKlSvx9PSkcePG3MvtEqbVqoGPDxEWFvwJqWMZfgV2AU2LFcvTEdFhYWFcunSJ1nkcjPLDpUuX6NOnD+XKlSMhIYH6LVogc+aQbG3Nv3Z2NNI0PqlUib4rVuR3UXNFBQpFeQGZmZkxf/58fH19efPNN/n3339zl+CGDSTVqMEoTcMVw3raM4DVpUtTds+ePJ2Mb9asWXz88ceYm5vnWR7P24kTJ+jcuTM1atQgPDyciIgIjhw5wtq1a7Hu25fxAwfyU8eOXBJh7IUL2Dk6Ymdnh51dttZoe3HkZPBFfm9qwJ3yKpk4caIUK1ZMzp49m/vETpwQmTJFdrVsKU1dXGTBggW5TzMTd+/eFQcHB7l9+3ae5vO87Nu3T1q2bCnu7u4yYcIEiYiIyO8iZQs5HHCX7w/9nGwqUCivmp9++kkKFy4sR44cMUp6Dx48EFtbW8nrf0sTJkxIN1r5ZaTX62Xz5s3i6+srxYoVk4CAAImNjc3vYuVITgOF6h6rKC+Bnj174uTkRNOmTfntt9/w9fXNVXoODg507NiRlStXcurUqTzp35+cnExAQABr1qwxetrPg06nY82aNUyYMIGEhASGDx9Ox44dMTN79R6bqo1CUV4Sbdq0Yfny5bz77rsEBQXlOr1HvZDmzp2b67QyEhQUxGuvvYa3t3eepJ9XEhMT+fnnn3njjTeYOnUqY8aM4fjx43Tp0uWVDBKgAoWivFQaNWrEpk2b6NOnD4sXL85VWpUrV6ZMmTIsWrSI+Pj4Z1+QTbNmzWLAgAFGTzevxMTEMG3aNEqWLMnKlSuZO3cu+/btw9/fP8/Gl7wsXu27V5SXkI+PD8HBwYwePZqpU6fmKq1BgwZhbm5u9OqhEydOcO7cOdq1a2fUdPPC/fv3GTduHMWLF2ffvn2sW7eOP//8kwYNGhht+diXnQoUivIS8vLyYs+ePcyfP58RI0YYeqbkQLt27dDr9cyYMcOo5Zs1axZ9+vR5obvEXr9+ncGDB1OqVCkuX77M7t27+e233166qrLnQQUKRXlJeXp6snv3brZu3Urfvn3RZbYm9VNYWFjw8ccfc+LECS5cuGCUct2/f5/ffvuNPn36GCU9Y7tw4QK9e/emQoUKJCcnc+zYMRYsWEDZp0y1rqhAoSgvNRcXF7Zv386FCxfo2LEjCQkJ2U6jX79+iAhz5swxSpkWLFhAixYtKFSokFHSM5awsDA6duxI7dq1cXd359y5c0yfPh1PT8/8LtoLzyiBQtO0ZpqmndU07YKmacMzOP6hpml3NE0LS9nSTIevddM07XzK1s0Y5VGUV0mBAgXYtGkTer0+R/NDFSlShHr16vHzzz9na4nVjOh0OubMmfNCzev0aFbed955Bx8fHy5dusS4ceNwcXHJ76K9NHIdKDRNMwXmAM0xzGDcSdO0NzI4dYWIVEnZfkq51gkYg2FSyxrAGE3THHNbJkV51VhaWrJy5UqKFi2Kn59ftueHGj58ODqdLtfdbjdu3Iibmxs1atTIVTq5JSL8/vvv1KtXj+7du9O6dWsuXrzI4MGDKVCgQL6W7WVkjDeKGsAFEbkkIokYprlvlcVrmwJbReS+iDwAtgLNjFAmRXnlmJqaMm/ePBo2bEj9+vW5du1alq9t1KgRBQsWZNKkSbkqQ353iU1OTmb58uVUqVKFESNG0L9/f86cOUPv3r2xsrJ65vWzZ8/Gx8cHS0tLPvzww9T94eHh+Pj44OjoiKOjI40bNyY8PDwP7+TFYoxA8RpwNc33ayn7HtdO07Tjmqat0jTtUaVgVq9F07TemqaFapoWeufOHSMUW1H+ezRNY8KECXTv3p169epx7ty5LF83ePBgwsLCuHLlSo7yDg8P59SpU7z77rs5uj43EhISmDdvHl5eXsyePZvvvvsutU0iO4PkPDw8GDVqFD169DDsCAqCChXwqFiRVX//zf3PP+fujRv4+/vTsWPHPLqbF48xAkVGHY0f76u3ASgmIpWAbcCjkUJZudawU2SeiPiIiI+rq2uOC6sor4IhQ4YwevRofH19OXLkSJau6dmzJ5qm5bir7OzZs+nduzcWFhY5uj4noqKi+OGHHyhRogTr1q1j4cKF7Nmzh3feeSdHYyDatm1L69atDcu1XrwInTrBqVM46PUUu38fbcIEpEMHTE1NjdZL7GVgjEBxDUjbbaAIcD3tCSJyT0QedceYD3hn9VpFUXKmR48eBAQE0KxZM3bs2PHM8wsWLIi/vz8///xztrvaRkREsGzZsufWJfbu3buMHj2aEiVKEBoayu+//86mTZuMt4KcXg+HDqVf+xpwiIvDKiiITz/9lBEjRhgnr5eAMQLFIaC0pmnFNU2zADoC6VrENE0rnOarP3A65fOfQBNN0xxTGrGbpOxTFMUI2rRpw4oVK3jvvfdYt27dM88fPXo0cXFxbNy4MVv5LFy4kObNm+Ph4ZHTombJ1atXGThwIGXKlOHmzZvs378/tU3CqB48gAyCZQTw0MKC2W3bUrVqVePm+SLLyZSzj2/A28A54CIwMmXfOMA/5fP3wCngGBAMeKW5tgeGZWUvAN2zkp+aZlxRsic0NFTc3d2ztP5EmTJlxMfHJ8tpJycnS4kSJWTfvn25KWKmzpw5I927dxdHR0cZNGiQXLt2LU/y+fvvv2XBggXi4+UlXcCwEsPjm62t6ObNEycnJ7l161aelCOvoNajUBQlM2fOnJGiRYvKDz/8kOl5ixcvFlNTU7l+/XqW0t2wYYN4e3uLXq83RjHTOXz4sLRv315cXV1l7Nixcu/ePaOmf+3aNQkMDJQePXpI8eLFxc3NTd577z2pWLGitNQ00WUUKKysJOnmTbGysjLa+iDPS04DhRqZrSiviLJly7J7925++uknvvzyy6fOD9WpUycsLS2ZMGFCltKdNWsWn376qdEm0BMRduzYQdOmTfH396dOnTpcunSJ0aNH4+TklKu0b926xYoVK+jbty9ly5alcuXKrF27lmrVqrFx40aWLVvGuXPnuHfvHvp69Uh0dCTZ2pqtwFEzM3RWVkROn84X336Lo6Mj5cqVM8o9v/ByEl3ye1NvFIqSc3fu3JHq1atLr169JDk5OcNzevbsKfb29qLT6TJN68yZM+Lm5iZxcXG5LpdOp5P169dLrVq1pHTp0vLTTz9JfHx8rtK8e/eurF69Wvr37y/ly5eXggULSsuWLWXq1KkSFhaWen+XLl2Sd999V+zt7QVDz8vUbUyTJrKyRg0p6+gotjY24uLiIs2bN5djx47l+p6fN1TVk6IoWRUZGSl+fn7Srl27DB/GV69eFVNTU1m/fn2m6fTv319GjhyZq7IkJSVJYGCglC9fXqpWrSorV658agB7loiICAkKCpLPP/9cqlSpIgUKFJBmzZrJxIkT5eDBg5KUlJTu/MjISPnyyy/FyclJxo0bJzExMbm6lxedChSKomRLfHy8tGvXTvz8/CQyMvKJ41WqVJFq1ao99fqHDx+Ko6OjXL16NUf5x8bGypw5c6RYsWLi6+srmzdvznY7R1RUlGzevFmGDh0q1atXFzs7O/Hz85Px48fL3r17JTExMcPrdDqdLFiwQAoXLixdu3bN8T28bHIaKF7Ndf0URcHS0pIVK1bw8ccf4+fnx6ZNm9JNlDdu3Dhat27NnTt3cI2MhFu3ICkJHB3By4tFixbRuHFjihQpkq18IyMj+fHHH5k+fTrVq1dn6dKl1K5dO0vXxsXFsX//frZv305wcDDHjh3D29ubhg0bMnnyZGrVqoWlpWWmaezevZuBAwdiaWnJunXr8n1eqpeBChSK8gozNTVl7ty5jBgxgvr167Nly5bUabdbNGpEX0tLdGXKQEICPBpxrdcjJiZYmpgweP78LOd1+/ZtZsyYwdy5c2natClbtmyhYsWKmV6TmJjIgQMHUgNDaGgolSpVomHDhowdO5Y6depgY2OTpfwvX77M0KFDCQkJYeLEiXTs2FGtYJdVOXkNye9NVT0pivH98MMPUrRoUTlz5ozI5s0idnYSZ24uGY4lAInXNNFbWop89plIJo3ely9flv79+4ujo6P07dtXLl68+NRzk5KSZP/+/fLdd99J48aNxc7OTry9vWXIkCGyadOmDKvIniUyMlJGjBghTk5OMnbs2P98O0RmUFVPiqLkxqBBg3BycuKHmjX5X3w8pgkJZDbfqqWI4U1j/nz4919YsQJM/r/HfXh4OBMnTmTjxo306tWL8PBw3N3d06Wh0+kICwsjODiY7du3s3fvXooVK0ajRo349NNP+e2333BwcMjR/ej1ehYvXszIkSPx8/Pj2LFj2a4mUwxUoFAUJVV3Ly+6xsVhmpiY9YtiY2HTJhg2DCZP5uDBg3z//ffs27ePAQMGcPHixdSHvV6v5+TJkwQHBxMcHMzOnTspXLgwDRs2pGfPnixZssQoCwo9aoewsLBg7dq11KxZM9dpvso0w9vIy8XHx0dCQ0PzuxiK8t9TuzaEhDyxezawCDgBdEr5/DidhQXv1axJ6JUrDB48mJ49e2Jtbc2ZM2dSA8OOHTtwcHCgYcOGNGrUiAYNGjzxlpEbly9fZtiwYezfv1+1Q2RA07TDIuKT3evUG4WiKAbnzsGxYxke8gBGYZixM+4plycmJTHC1RWbuXPZs2cPvXr1Ijg4GEtLSxo2bIi/vz9Tp07NkzWqo6Oj+f777/nf//7HZ599xsKFC7PcyK08m5rCQ1EUgxkzIDk5w0NtgdaAcyYT2kyLAAAgAElEQVSXW4tQbN06mvj5sXPnTvz8/Ni3bx+XL19m4cKFdO3a1ehBQq/Xs2jRIsqWLcs///zDsWPHGD169AsXJBISEujZsydFixalQIECVK1alT/++AOAkJAQ3nrrLZycnHB1deXdd9/lxo0b+Vzi9NQbhaIoBn/9ZRgnkQuOFhb8c+AAWh68NTxuz549DBw4EHNzc9asWfNCt0MkJyfj6enJzp07ef3119kUFMR7bdpwokQJHjx8SO9q1Wh64ABmr71G//796d69O5s3b87vYqdSbRSKohh4eMAz/pIdhWG1sUVPOR5vYcGCnj156OmJpaUllpaWWFhYZPg5s2OPPltYWDzRxpC2HWLChAl06tTp5WqH0OngrbeotGMHY0RoB4YxKvb2cOQIR+7cwdfXl6ioKKNnrdooFEXJHSMsYSo6Hf/evUuclRUiQkJCAomJiSQkJGT6+WnHkpKSsLCwSN0SExOJjY3F0dGRwoULM23aNAICAnIUhHJ6zaPvJiY5rLlfv55bBw5wToTyj/YlJhoWSxo5kl3VqlG+fPnMUnjuVKBQFMXAwwOuXMlVEmYi7DpzhsMbN+Lm5sYbb7zxxGZvb5/l9PR6PQkJCSxevJixY8fSoEEDvvjiC5ycnHIUhKKionIVuNJ+NjMzy1EQ+mTPHr6KjaUb4JX2ZnU6jq9Zw7iNG1m/fn2ufg/GpgKForzidDod27dv56KJCZ2BAhmck5yy6VK2eAwPj7QPEAHMGzVi99at6HQ6Ll++zKlTpwgPD2fHjh0EBARw+vRpnJyc0gWO8uXLU65cORwdHZ/Id9++fantEOvWrXth2iFEhKSkpCwHq0ef4+Li+HbTJiwwdDlO6wLQPDaWGT/+aLy1v41EBQpFeUUdP36cwMBAli5dSuHChenWoQO2YWEQE/PEueOBsWm+/wKMAb5Osy8GCC5XjubJyZiZmVGyZElKliyJv79/6jn6Cxe4u2ED169e5bhOR/Devfz000+Eh4dToECB1ODh7u7Otm3bOHfuHJMmTXrh2iE0TUutDssqEeH999/nvKaxBzBPc+wK0Bj4qlYtunbtauTSGkFO5v3I703N9aQoOXPt2jWZNGmSVKpUSTw9PeXLL7+UU6dO/f8JgwdLsoXFU+d3etqmA7llaSmW5ubi4eEha9asST9leHi4SL16IlZWInZ2cs7GRixBOhcoIPLnn6LX6+XKlSuyZs0aady4sVhaWkqRIkWkQIEC4ubmJg0aNJBPPvlE5syZI8HBwS/dWtVxcXFSs2ZNMTMzk74ffSTxDRuK2NqKgFwDKQEyyc5O5N9/87Qc5Od6FEAz4CyGt6fhGRz/AggHjgN/AUXTHNMBYSlbUFbyU4FCUbIuKipKFi9eLI0bNxYHBwfp0aOHBAcHZ7h63byZM+WYmZnoMpkMMKMt1tRUulapIhs3bpRSpUqJjY2NVKhQQYKDg0WOHxcpUEBE01LPfwukHkhnELG2Ft3y5bJo0SLx8PCQLl26pK4Podfr5dq1a7JlyxaZPn26fPTRR1K3bl1xcHAQFxcXefPNN6Vv374yc+ZM2bZtm1y/fj1P1u7OKZ1OJ7/88ot4eHgIIBYWFmJra2vYLC3ll1Kl5Gs3NwHE1sbm/4/Z2uZJefItUACmwEWgBGABHAPeeOychoBNyuePgRVpjkVnN08VKBQlc0lJSfLHH39I586dpWDBgtKiRQtZvny5xMbGZnh+cnKyDBw4UMqWLSsXQ0NFvL0lKSvBwsRExN5edHv2yKhRo6RYsWJy9OhRmTdvnhQsWFAK2NnJNSsr0ae5ZhnIuyBjHgUKkDhNk7e8vWX//v1Zuj+9Xi83btyQv/76S2bOnCl9+/aVN998U5ydncXR0VHq1q0rH330kUybNk3+/PNPuXbt2nMPIDt27BBvb2+pXr267Nq167nm/TT5GShqA3+m+f4l8GUm51cF9qb5rgKFohiBXq+XI0eOyOeffy7u7u5SvXp1mTlzpty+fTvT6yIjI6VFixbSqFEjuX//voiIxNy/L+MtLCTS3NzwNvB4gLC2NlQjdeggcuFCalpLly4VFxcXWbdunTx48ECmt2snUWmuewhSGuSfxwJFkoWF6CZPNsrP4fbt2xIcHCxz5syRTz75RBo2bCiFChUSe3t7qVWrlvTo0UOmTJkif/zxh1y5csXoAeT06dPi7+8vxYoVk2XLlj1z3fHnKaeBwhiN2a8BV9N8vwZk1jWhJ/BHmu9WmqaFYuhUMUFE1hmhTIryyrh69SpLly4lMDCQ6OhounTpQnBwMF5eXlm6tkWLFtSsWZM5c+Zgbm5oYh05bhwBQM9LlygQGpo6lfiNf/5BHB3x6NsXevYEJ6d06XXq1ImSJUvStm1b+vfvz7DixdMd/wrDA+DxcdtmiYmGacoHD875DyKFq6srDRo0oEGDBun237t3j/Dw8NTtjz/+IDw8nMjISMqVK0f58uXT9cYqWrRotsZK3L59m7Fjx7Jy5UqGDRvGihUrsLLKbKL2l4cxAkVGXREyHO6taVoXwAfwTbP7dRG5rmlaCWC7pmknRORiBtf2BnoDvP7667kvtaK8xCIjI1m9ejWBgYGEhYXRvn17fvzxR+rWrZvlh9uhQ4do3bo1X3zxBV988UVqr6J//vmHgIAAevfujXuRIlCkCLRuDcC8sWPR6XSMGzLkqenWqFGDkJAQWrVqRY2oKBql7A8DtgFHn3ZhbGyWyp1Tzs7O1K9f/4mupw8ePOD06dOpAeSvv/7i1KlTPHjwAC8vryfGgRQvXhxTU9PU6+Pi4pg+fTpTpkyhS5cunDlzBmfnzGbFegnl5DUk7UYWq54w9P46DbhlktYioP2z8lRVT8qrKDExUTZu3CgdO3YUe3t78ff3l99++03i4uKyndZvv/2WWkX0uKZNm4qtra1EREQ8cSwgIED69OmTpTyio6NlUcWKEpfSiD0NxAakUMpmC2IFUvVR1dQ772T7PvLSw4cPJSQkRBYsWCCDBw+W5s2bS9GiRcXa2lqqVKkinTp1knbt2omLi4s0bdpUTp8+nav8Zs2aJd7e3mJhYSHdunVLd2zFihXi5eUldnZ2Uq5cOVm7dm2O8iAf2yjMgEtAcf6/Mbv8Y+dUxdDgXfqx/Y6AZcpnF+A8jzWEZ7SpQKG8KvR6vRw6dEgGDBggbm5uUqtWLZkzZ47cuXMnx+l9++23UqRIETly5MgTx3fs2CFWVlby/fffZ3j9qlWrpE2bNs/MJzExUZYuXSrNqlSRuJRAEANyI802CKQdyG2QBEtL0f3xR47u6XmLioqSgIAAKVq0qHh4eEidOnWkRIkSYmVlJRUrVpQOHTrIuHHjZNWqVRIeHi6JiYlZSnf16tWydu1a6du3ryFQJCSILF8u1957T8xNTGRTQIDo9XrZuHGjWFtb56iLcE4DRa6rnkQkWdO0/himqjcFFojIKU3TxqUUKgiYDNgBv6W83v4jIv5AOWCupml6DFOeTxCR8NyWSVFedleuXOHXX38lMDCQxMREunTpwp49eyhdunSO00xISKBPnz6cPHmSAwcO4OHhke64TqejZ8+e2NnZMXDgwAzTcHNz49atW0/N48GDB8ybN4/Zs2dTuHBh4pKT2eLsTPPISGySkkg7+bcdYAUUBC4kJdFt5EiWlSpFqVKlcnyPeS08PJyhQ4dy+vRpJk+eTPv27VOr7GJjYzl79mzqaPTAwEDCw8O5evUqJUqUeKINpHTp0lhaWqam3bZtWwBCQ0O5dukSVKkCV69yLToaB6D5oEEQEcE7X36Jra0tFy9exM3N7fnceE6iS35v6o1C+S+KiIiQ+fPnp3bz7Nu3r+zdu9covXLu3Lkj9evXlzZt2kh0dHSG5wQEBIi9vb0sXLjwqemcPXtWSpUqleH+fv36iYODg7z//vvSr18/cXZ2lunTp0tybKxI8+aSbG0tj/ee0tvYyBULC/mgaVOxs7MTKysr+eabbyQ5OflRwvJ3hw7S3MJCHDRNCllYyCetWklSUlKufybZcfPmTenTp4+4urrK1KlTJT4+PsvXxsXFSVhYmCxdulRGjRolbdu2FS8vL7GyshIvLy9p27atjBo1SpYuXSphYWEybNgw6VamjEjKwMdkkDdB1oMkW1nJ2hkz5LXXXnvq7zEz5OeAu+e9qUCh/FckJiZKUFCQvPvuu2Jvby9t2rSRNWvWZOtB9CynT5+WkiVLyrBhw57aVfP+/ftSsGBBKVWq1P8/pDPw4MEDsbe3FxFDNda2bdvknXfeEVdXVxk5cqSsX79eKlSoIG+//bZcvnz5/y/U6USCgiS+enWJ1zRJMjERXZkyIj/9JP9euCBFixaV//3vf9K1a1cxNzeX4sWLy99z54rY2EhzTZNuIHEpVVYVNE1mNG5stJ9PZmJiYuSbb74RZ2dn+eKLL+TevXtGSzs+Pl5OnDghK1askDFjxsi7774r5cuXF1NT09Ruw4+2n1LadExBrM3MZOPGjTnKUwUKRXlJ6PV6CQkJkf79+4urq6vUrVtXfvzxR6M+hB7Ztm2buLm5yc8//5zpef379xdHR0cJCgrK9Dy9Xi9mZmYyd+5cqVSpkrzxxhsyb948uXnzpnzyySdSuHBhWbFiRaZvQTExMdKhQwepUaOGXL9+XURETp48KW5ubvLXX39JWFiYVC1RInX8hRfI72kemoNBepuZiWzblv0fSBYlJyfLwoULpUiRIvLee+/JxYsX8yyvtKKioqRDhw7yQZr73QriBHIoZaqUgy1biru7uxw9ejTb6atAoSgvuEuXLsm4ceOkTJkyUqpUKRk7dmyePoDmzZsnbm5uhmk0MnHy5Emxs7OTGjVqZPqAv3nzpnz99ddiYmIivr6+snnzZtHr9bJ27VopUqSI9OrVK3XA3rPo9XoZN26ceHp6yuHDh0VEJDg4WFxdXeX48eOinzFDElOqXn4E6ZrSGH4NpDzIGhDJo7eKrVu3SuXKlaVu3bpZHimeUw8ePJANGzbIkCFDpEaNGmJjYyOenp7SPk013WSQ1o8Ch52dyMqV0qpVK5mcgwGKKlAoygvo/v37MnfuXKlXr564uLhIv379ZP/+/Xk6nURycrIMGjRISpcuLWfPns30XL1eLw0bNhRHR0fZs2dPhuccP35cunfvLg4ODvLRRx9JuXLl5NChQ/Lvv/9KmzZtpEyZMrJjx44clfVRN91Vq1aJiGFkt6enp8Q0a5b6oAwHqZZS7QJINzBMCVKwYI7yfJoTJ05I8+bNpVSpUrJ69eo8+R3dvn1bVq9eLZ999plUqVJF7OzsxM/PT8aOHSvbtm2T+/fvy/Dhw6VLkyYSZ20tSSA7QJxBjpqZibzxhhw5cECcnJzkzz//zHb+KlAoygsiISFB1q5dK23bthV7e3tp3769rF+/XhISEvI876ioKPH39xdfX98sVWWtXbtWChUqJO88NoZBp9PJxo0bxc/PTwoXLizjx49P7ZLbtGlT6devn7i4uMhXX32Vo3EcaR0+fFg8PT1l7NixEhMTIx999JGssrSUR7PSeoKMB4kHuQviDzIERFxccpXvI9evX5ePPvpI3NzcZMaMGUb9Pf3777+ybNky6du3r7zxxhtib28vzZs3l++//1727duXLq8xY8YIKcHw0TbGw0PExERmWVhISXt7sbOzk+LFi8sPP/yQo/KoQKEo+Uiv18u+ffvk448/FmdnZ6lfv77MmzdPHjx48NzKcPXqValSpYp07949Sw+7uLg4KVasmDg4OMiJEydExDBILiAgQMqWLStVq1aVJUuWpEvr5MmT4urqKqVKlZKTJ0/muszR0dGyZcsW+eyzz8TOzk7MzMykRo0a8kXx4hJtYiJ3Uh6YEWnq7NemVD9Jz565znvs2LHi7OwsQ4YMMcrv6u+//5bFixdLjx49pFSpUuLk5CStWrWSKVOmSGho6HPvrfW4nAYKtXCRouTChQsX+OWXX/jll18wMzOja9euhIaGUqxYsedajsOHD9OqVSsGDBjAkCFDsrTIz7Rp0zA3N6d169Y4ODgwfPhwfv75Z+rVq8e8efOoX79+ajrx8fF89913/Pjjj1SpUgU/P78crescFRXF3r172blzJzt37uT48eNUrVoVX19fli1bRmBgIJcuXWLgX38RU748zvHxFBfhR2AwEA0sBiqbmkIm04hkRqfTsWjRIkaPHo2vr2+Of18iwvnz59m1axc7d+5k165dJCQk8Oabb+Lr68vAgQMpX758ztfWfpHkJLrk96beKJT8dPfuXQkICJDatWuLq6urfPrpp3Lw4MF8Wwdh9erV4uLiImvWrMnyNdeuXRMHBwexs7MTf39/cXR0lAEDBsiFNDPBPrJz504pW7astG3bNnXhoy+++CJL+URERMjGjRtTG2ttbW3F19dXRo8eLX/99ZfExMSkOz/tyPGDv/0m1y0sJMTMTHxBHFJ6/7QzMZFby5dn+V7T2rx5s1SsWFHq168vBw4cyNa1Op1OTpw4IbNnz5b33ntP3N3dpUiRItK5c2eZN2+enDlz5oVaCyMjqKonRck78fHxsmrVKmnVqpXY29tLhw4dZOPGjVmeniEv6PV6mTBhgrz22msSGhqa5euSkpKkfv36YmtrK/b29jJlypQM53W6f/++9OrVS1577bV0cwstXrxYunTpkmHa9+/fl/Xr18sXX3wh3t7eYmdnJ40aNZKxY8fKjh07styesWbNGnFxcZH5P/4oH7u7y/nq1UVat5a1DRvK2L59RVauFJk1S2TePJGUXlOZOXbsmDRp0kRKly4ta9euzdIDPTk5WQ4fPixTp06V1q1bi7Ozs5QsWVK6d+8uixYtkkuXLr3wgeFxKlAoipHp9XrZvXu39O7dW5ycnKRBgwby888/Z/hQfd4SEhKke/fuUqVKldTV4J4lIiJCpkyZIu7u7mJubi52dnZy8+bNJ87T6/WyYsUKKVy4sPTr1++J+/3jjz+kSZMmImJ4u1qzZk1qL54CBQrIW2+9JePHj5fdu3fnauDg0aNH5fXXX5f+/fuLu7u7/Ll0qTzo2FFiQPQFChjWw7CxMWzlyomk9JxK699//5UePXpIoUKFZNasWZkG9sTERNm/f79MmDBB3n77bSlYsKCUK1dO+vTpI0uXLpVr167l+F5eFCpQKIqRnD17NnW1tnLlysl3330nV65cye9ipbp79674+vqKv7+/REVFPfP8ixcvymeffSaOjo7SoUMH8fLykpo1a8r48eOfOPfKlSvyzjvvSPny5WXv3r1PHL9165ZMnDhRXFxcpGLFimJvby/NmjVL7cVj7DesmzdvSu3ateXd2rXlXxMT0ZmZSTxID5DXQexAqoBsAkPAGDlSRAy9v0aPHi1OTk4yfPjwDIN7XFyc7Ny5U8aNGyeNGzcWOzs7qVy5sgwYMEBWrVr10q3LnRUqUChKLty5c0dmzZolNWrUkEKFCsnAgQPl8OHDL1zVwtmzZ6V06dIyePDgTKfa0Ov1snPnztQqk2HDhsk///wjCxYskAoVKkjhwoXTzRWUnJws06dPF2dnZxk/fnxqT6cbN27I8uXL5eOPP5Zy5cpJwYIFxc/PT+zt7eXgwYPPpRdPXFSU3LSzk6SUXk/RKavj/Z3SfXZDSsD4O2XuqOAPP5TChQtLly5d0k0j8qiH1ahRo1Kr3qpXry6DBw+WoKCgLA8WfJmpQKEo2RQXFycrV66Uli1bir29vXTq1Ek2bdqU710YnyY4OFjc3Nxk/vz5Tz0nISFBPvjgA7GxsRFN06SOt7fEjx8v4u0tyV5estrKSrq98Ya88847AsjWrVslLCxMqlevLr6+vrJr1y759ddfpXfv3lKmTBlxdHQUf3//1O6dycnJkpCQIObm5s9vic9VqwxVTY/Nf5R2qwiyKuXzbQsLCT14MLUhfejQoVKzZk2xtbWVevXqyYgRI2Tz5s0SGRn5fMr/AlGBQlGyQKfTyY4dO6Rnz57i6Ogofn5+smjRohf+ofHzzz+Lm5ubbHvK/EZ3796Vb7/9Vjw8PKRixYoyYsQI6dOihXQzMzNUyaQ8RJNBToBUKFhQ3N3dpWXLllKgQAGpV6+elChRQpydnaVNmzYyY8YMCQsLe2owcHBwyJO5qTJUvXqmQeImiCXI6ZTvcebm0rdkydSG9K+//lqCg4MlNjb2+ZT3BaYChaJk4vTp0zJixAgpWrSoVKhQQSZOnJjlRuD8pNPpZOjQoVKyZMkMV1ALDw+XPn36iIODg3Tv3l2OHTtmOHD+vIw0N5duGTxYm4Gs1TRxBDE3N5cWLVrI7Nmz5cSJE1l+SyhTpkyuV3TLEr1exNT0qUEiEcQPpHeafXqQf7p1ey4j4V82OQ0Ur/yAu4SEBPr168e2bdu4f/8+pV57je9KlqS5iQmXixal+Jw52Nrapp4/bNgwvvrqq3wssZJVt2/fTh3Edf36dd5//33Wr19PpUqVsjQgLc/o9bB1K0yeDGFhkJQETk7Qowf07g2FCgEQExND165duXfvHgcOHEhdh1lE2Lp1K9OmTePo0aP07duXM2fOUCjlOgAmT0aSk5/I+jcMy1C2FmEgsGLVKt7y98/2Lbi5uXH79m28vLxSyxQfH09sbGyGW0xMzFOPxcbGEh0dTUREBA8fPiQyMpLo6GhiYmJIjovjpk5HRkPW9EDXlPuZnWa/BngWLAgWFtm+LyVjr3ygSE5OxtPTk53Bwbz+zTdsWraM986e5QSAuTkAEZMnY/bxx/laTiVrYmNjCQoKIjAwkL1799KyZUu+++47/Pz8MDU1ze/iwblz0KQJ3LsH0dH/vz8yEr77zrANGcL1Pn1o6e9PxYoVWbZsGZaWlsTFxfHrr78yffp0TExMGDhwIGvXrsXKygowPKzPnTvHjh07+GDBAjSRdFlHAyOALY92aBrnly0jIiEh04d4Rtv58+dp164dYPiZx8XFYWlpiY2NTepmZWWFubk5JiYmaJqGiKDT6UhOTiYxMTE1sMTExBAXF4etrS2Ojo44OTlRtGhR3NzcKFSoEPrZszF5LOgJ0BO4BWwCzNMeNDNLDbaKcWjy2P9MLwMfHx8JDQ01bqIzZ8KXX0JsLJWAMYA3hoXAk6ytMdu5E6pXN26eilHo9Xp27NhBYGAg69ato0aNGnTt2pXWrVtjZ2eX38X7fxcugI+PISiI0AX4C4gB3IGhQC9AZ2XFQjMz7owYwfDhw7l58yYBAQHMmzeP6tWr8/nnn9OoUSMATp8+zY4dO1KnxLC0tKROnTr8unw5o4FrwKKU7AdhWHZ0dMr3okCLokW5Xb16ugf80zZzc3MSEhKIj49n3rx5ODs7U6lSJSIiInjw4AF37tzh9u3bqZu5uTlubm6pW6FChdJ9T7s5Ozs/NZBHNGtGgS1bME3zrOoLhAHbMCypmo61NRw9CmXL5u739R+kadphEfHJ9nUqUGCo2fTwgJs3uYXhH1AYhvV8iwMegGZtzVsdOjB58mRcXFyMl7eSY6dOnSIwMJBff/0VZ2dnunbtSqdOnZ5YC/qFUb48nDljqHoCTgGlAEvgDNAA+B3DHyjJlpZcnj6db/bvJygoiE6dOvHpp5+SlJSUGhQeBYbSpUtToECB1L/079+/z9X4eH5ITk4XKKpgCByPqhHuAPa2tvTo0wd/f/90D/nHt1u3bhEbG4urqytubm5ERUVhY2PDW2+9leGD39XVFRubtCtkZ9/Bgwf59ttvSdi7l6DISCySkgC4AhRL+bmlrRKZC3QGqFkTQkJylfd/VU4DxStf9QTAgwdw/z5JGP5H6wZ4YXhVP4ThH9hVc3M6nz3LO++8w5IlS3B2dsbR0fHFqM54hdy8eZOlS5cSGBjI7du36dy5M5s2baJixYr5XbTMHToEV66kBgmAtFPqaSnbRQyBwiQhgRuffUbBPn0YMmQIe/fupVatWlhaWuLo6EhSUhIxMTHY29vj4OBApUqVqFy5MqVLl8bS0pI748aR+Ouv6IB4DP/Q/wKS0uTpDUSLsGHDBg4dOpTuQV+xYsUn3gIcHBxS23YCAgI4ceIEU6ZMMfqPateuXXz77becPn2aoUOH0nP5cuLbtyd50yZsMPwh99Q/b21sICDA6GV61RklUGia1gyYAZgCP4nIhMeOWwJLMPy/eQ/oICKXU459iaG6UQcMEJE/jVGmbLGyQq/TPdEwZgc8Cr3mCQlYWFiwc+dO3n777dSGt4IFC+Ls7IyLiwvOzs7P/Ozk5ITFq9jIduUKbNoEUVFQuDC0bg0FCmTp0piYGNatW0dgYCAhISG0bt2aH374gQYNGrw8gXr6dIiLe2J3Pwx/8ccBVYG3U/abANUTE+n+00/c0DR0Oh1FixbF09MTR0dHrK2tAYiIiODGjRscO3aMKVOmoNPpMDMzIzpN+8cvGKpSv06Tr97aGnNra1avWEHjxo2zfTuPGrON5VED/fjx47l+/TrDhw/ngw8+wMLCgvPnz9Pk2DG2e3tT/MwZiIl5MgFLS0Ob4oYNUK2a0cqlGOQ6UGiaZgrMAd7C8GZ7SNO0IBEJT3NaT+CBiJTSNK0jMBHooGnaG0BHDH9ceQDbNE0rIyK63JYrO8Tamp4uLty6devJhjEg2dycvSVLcvjwYQD69evH+++/j5ubGw8ePODevXvcvXuXe/fupft86dKlJ/bfv38fa2vrLAeWR59z+xqfH7p06cJfW7cSc+8e7no9Q83M6CVCiLk5X334IYfNzTEtUIAGDRowc+ZMChcunHqtTqdj+/btBAYGEhQURO3atfnggw9YvXp1ul5oL43jx9O9TTwSAMwC9gM7MFSnPBIPFImPJ8LJCXd3dwoVKpRa9fO06p41a9YwZMgQ+vbty94FC9hjY4PJw4eGAA2G+nsRTMaN4/LgwTm+HTc3N27dupXj6x/R6/Vs3LiR8ePHEx0dzciRI+nQoQNmZgDtf90AACAASURBVIZH08WLF/Hz8+OrMWMo3qsXbNkCkybBnj2GwCBiCBKffAL9+hn+CFGMzhhvFDWACyJyCUDTtOVAKyBtoGjF//9BswqYrRneYVsBy0UkAfhb07QLKentN0K5suzjjz/mtIsL2yIjsU7zV98BwEHTKF2wII3XrqXpiBGcPXuWsLAwvv32W8qVK0e7du1o27YtdevWzVJeIkJkZGSGgeXevXv/x955h0VxfW/83aUuHZYmICigCCqKBWsAsWCLDVvEkij2GiuiBjWxJcZo7CUW9GsvsRcQW34WYixRFE0UAUWKYqOzu+/vj4UNKyALghizn+eZh53ZO/feGWDeuefccy5u3bpV6PizZ88gFApLJSxisRhGRkaVOg10emAgfgkLg45AgGgSPrm58ADwQiLBMAB+QiE03d0xxsAAX331FU6cOIE///wTW7duxfbt22FtbY0BAwbg+++/h7W1daVdx/uQk5ODBw8ewPbFCxgVU0YDQEvI3/xXAxiXd9zIyAin9+6FRtu2Jbbz6tUrjBgxAjdv3kR4eDhGjRqFwYsWQTh4sPzhun8/kJEBeHgAX34J5E21LStWVlbvNaKQSqXYu3cv5s2bB01NTcycORPdunVTWrvh4cOH8PX1xYwZMzB06FD5QT8/+fb6NfDsmVwkrKzkM53UVBjlcXdtAcQX2H8MoElxZUhKBALBKwDivOOX3zrXthz6pDKxsbFYu3YtdHR0YC0QgJDbiteKRBBKJAgWCpGcng4jb2+0bdsWYWFhsLa2Rk5ODiIiIrBv3z54enqiatWq8Pf3h7+/P1zeMdtCIBDA2NgYxsbGcHJyUqmPJJGRkVGksDx79gwPHjxAZGRkoePZ2dkwMzMrlWnMxMSk3Mw5tZcsAVJTAYlEyQbfO79AZiZw+TIGubvDb8cO1KtXDy9evEBAQADCwsLg5uZWLv34ELx+/RrR0dGIjo7G3bt3cffuXURHRyMmJgbm5ubY8eIFvEqoQwL5/clHKJEA9vYltn3p0iX069cPHTp0wNWrV7F//35kZWXhq6++AoRCoH17+VaOlNX0lJubi//9739YsGABxGIxFi5ciA4dOhR6oXn06BF8fX0xbdo0DB8+vHBFRkbyTc0HoTyEoqhX1rd9TcWVUeVceQUCwTAAwwDAXoV/HlVxcHBA/syvZ8+ewdnJCbEHD8I4OxuoUQNfODoWeZ62tjbat2+P9u3bY/Xq1bhw4QL2798PX19fmJiYKESjPIK7BAIB9PX1oa+vX6prz87ORmpqapGjl6dPnxY5enn9+jWMjY1LbRrT0nrLYJeYCJw6hVESSZE2eAUZGQhbtgxG5uZYtmwZvLy8PtoVwUji6dOnChEo+PPly5dwcXFBrVq1YGFhATMzM1haWiIhIQF2dnaIb9EC2YcOQSc7GwCQDCACQGcAIsinee4AsL1gg/b275ziKZVKMX/+fKxcuRJr165F165dkZaWhmnTpmHXrl0V6r8xMTFBRkYGsrOzoaOjU2L5rKwsbN68GQsXLoSTkxPWrFkDHx+fIv83YmNj0apVK0yePBmjRo2qiO6rKSXlIRSPAVQtsG8HIKGYMo8FAoEm5NO5U1U8FwBAch2AdYB8emw59LsQW7ZsQddu3WDs41Oq8zQ1NdGqVSu0atUKy5Ytw5UrV7Bv3z5069YNGhoa6NGjB/z9/eHp6flBTUE6OjqoUqWKku2/JCQSicLvUtzo5e3jqamp0NPTUxKQz1+/xmCZ7J02eAD4E8AqEgfXr8dnpbzvFUVubi4ePnxYSBCio6Oho6MDV1dX1KpVC66urvj8889hY2ODe/fuISwsDCdPnoREIoGfnx/GjBmDNm3a4OHDh5g2aRI6SiSK6xdAbmYaAXmEsQOApZDbYgEA+vrA1KnF9jEuLg79+/eHpqYm/vjjD9jaygfiCxYsgLe3t8qm0LIiEAhgYWGBlJQU2NnZFVsuPT0d69atw+LFi+Hh4YEdO3agWbNmxZaPi4tDq1at8PXXX2PMmDEV0XU1ZaEseT8KbpCLzUPIQw60AdwEUPutMqMBrMn73BfA7rzPtfPK6+Sd/xCARkltVkSuJ5lMxho1ahSZg/996rx27RpnzJjBWrVq0c7OjuPGjePZs2ffmSL634ZMJmNKSgrPnTvH1atXc9KkSVzl4cEMgUApL89wgMsK7P8F0AZgqLY2+fDhB+/3mzdv+Pvvv3Pr1q0MDg5mjx496OrqSh0dHTo6OrJTp06cNGkSN2zYwN9++43Pnj0jKU/JffnyZc6ZM4fNmzengYEB/fz8uGTJEt6+fVuRmjwmJoZffPEFbWxsuG7dOkoOH1ZK0FfcJtHSIps1I4tZ22HPnj20sLDgggULlP6OHjx4QDMzs3LPYZWVlcXBgwfT3t6eBgYGrF+vHo/NnMnjJiZ83ro1w6dOpUvNmhSJRPTx8eGjR4/48uVLzp8/n5aWlvT39+e1a9dKbCcuLo6Ojo5csmRJufZfzT+gMpMCQm5RuA+5iXVG3rG5ALrkfdaFPM3M3wAiATgWOHdG3nn3AHRQpb2KEIrTp0+zTp06Fbr+wJ07d/jtt9/Sw8ODlpaWHDZsGE+cOFGpy2mqSlZWFu/fv8+wsDCuX7+eM2fO5IABA+jl5UUHBwdqaWnR1taWLVq0YL9+/bipd29m6+oqPQCHAByX9/kRQAeAqwFSR4dUYQGesiCTyfj06VNGRERw1apVHDt2LNu2bUs7OzuKRCLWq1ePffv25ezZs7lr1y7evHmzyOU64+PjuWHDBvbu3ZtmZmasXbs2J06cyJMnTxbKSvr8+XNOmjSJZmZmnD17ttLiQtI9e5ghFDK3mER3uSIRL2lpMf7OnUJ9SEtLY2BgIJ2cnIpc77l79+789ttvy+GuFW43JCSEMTExlKak8LC9vWL9hxSARgB3m5oyMzqao0ePpq2tLcViMfv378+oqCiV2oiPj6eTkxMXL15c7v1X8w9lFQp1ZHYeffr0gZeXF0aPHl2u9RbHw4cPsX//fuzbtw/3799H586d4e/vj7Zt2yrmyH9I0tLSEBsbi9jYWDx69EjxOX8/NTUVdnZ2cHBwUNqqVasGBwcH2NnZKcWHJD95gggXF3ROT1fY4HtAboNvBMALcrPLFIFAHlOxf/979V8ikSAmJqZIc5GGhoaSuSj/p4ODQ7H+kMzMTJw/fx4nT57EyZMnkZSUhDZt2sDPzw/t2rVTmHoKkp2djRUrVmDRokXo0aMHZs+eXWi21tq1a3F87Vrsb90awnXrAIlE7nDOyZGn95g2Dd9HRWH/wYM4f/684p5ev34dX3zxBZo0aYIVK1bA8K0YlNOnTyMwMBB37typ2L+fzp2BsDC45+QgBPKgqM0A/k8oRJKZGdwkErx+8wbHjh1Du3btVKoyISEBPj4+CAwMxNR3mNvUvD/qFB7vQVJSEmrVqoVHjx7B2Ni43OpVlSdPnuDAgQPYt28frl+/Dj8/P/j7+6Njx47lkquIJF68eFGsCMTGxiIzM7NYEXBwcECVKlVK5RxNSUlBzyZNcDMmRmGDHwdgKIA5kM+VVkRDiESAUKgUJFYc6enpuHfvXiFn8oMHD2BtbV2kIKiScoUkoqKiFMJw6dIl1K9fXyEMDRs2LPb6ZTIZdu3aheDgYNStWxeLFi2Cq6troXIJCQmoV68ezpw5gzp16sizxj59CmRnA+bmgKmpoi/du3dH1apVsWzZMixduhQLFizAsmXL0K9fv0L1SiQS1K9fH3PnzkWPHj1KvNYyk5AAODkhKStLkeZmNYCcvJ/Zmpp4eeAAWgcFYc6cOYqkge/i6dOn8PHxwVdffYWgoKCK67saAOoUHu/Fpk2b0KNHj0oRCQCwtbXFmDFjMGbMGCQnJ+PQoUPYtGkTAgMD0apVK/j7++Pzzz+Had6D5G1kMhmSk5OLFYHY2FhoaGgUEoEWLVoo9i0sLMrV0W5hYYFzDx7I02bv2KEUTRsCYKZAgGwSU83N8VNCgtKsKZJISUkpcnZRcnIyatSooRACf39/1KpVCzVr1ix1UOLz588RFhaGU6dO4dSpU9DS0oKfnx9GjhyJPXv2qPT3cPbsWUyZMgUCgQCbN2+Gt7d3sWXHjRuH4cOHy0UCkAeMFTGLLb+u+vXr4/z589DT00NkZCSqV69eZL2rV6+GlZUVunfvrtqFl5XoaORqayMgK0spzY1F3tc6OjqwSkmBsbEx3uQH+L2DxMREtGrVCgMHDlSLxEfOf14oZDIZ1q1bh127dlV2VwDI56cHBgYiMDAQL1++xJEjR7Bnzx6MHj0atWrVUuTgefbsmUIE4uPjYWhoqCQCtWrVgp+fn+KYiYnJh78YgQBYt05uWlq4ELh0CdDQgIzEAU1NtDx4EBu7dsWZ+vXx5ZdfKsUhAFAaFbRp0waurq6oVq1amad95ubm4sqVK4pRQ3R0NLy9veHn54fp06fD2dlZZbG8c+cOpk2bhqioKMyfPx+9e/d+57TegwcP4s8//8S2bdtUqv/ixYtIT09HcnIyLl26VKxIPHv2DHPnzsWZM2cqfEbd0+xsjE9LK5Tm5nV+AQ0NwMwMr1+/LmQae5ukpCT4+voiICAAM2bMqLhOqykfyuLYqOytPJ3ZJ06coIeHR4U6sUuioKN4w4YNhRzF2tratLGxoYuLC+3t7amjo0NnZ2eOGDGCERERTEtLK12DmZlkWpp89bAPQEZGBq9fv85doaFcMGUK+/bsSZFIRC0tLVpZWVEgENDV1ZVr1qzh2bNnmZSUVG6/j4cPH3L16tXs1q0bjY2N6eHhwWnTpjEiIoJZWVmlri8hIYFDhw6lhYUFlyxZolIdr169op2dHc+cOVNi2czMTI4fP55Vq1bluXPnuHnzZrq4uCgv1Vrg3owcOZKjR48u9XWoSnZ2Nvfs2cP27dtTW1ubLTQ0mFHA+b4WYPP8fUNDpj17RpFI9M7V75KSkujm5saQkJAK67eaooF6hbuiKbSCnaMj5jdvjg6RkUBmJjLS02Gel0MnNzcX9erVw/nz58u1DwUdxUX5CZ4/fw5bW1slv0CrVq0U+287irOyshAWFoZ9+/ahZ8+eqFGjhiJWo2C0d6Frt7TEfKEQHWJiAIEAu42NEaKpicfp6ahatSrmz5+Pbt26lfk6nz17VqS5KDExEU5OTv/EHnTvDudatXDjxg0cPnwYe/fuRZ8+ffDixYt3mm5Uvddnz55VjBpevXqFdu3awd/fH2vWrFFeBa6U9S5evBjLly/H4MGDce/evWJNgW8zY8YM+Pn5waeEOJE7d+6gX79+cHZ2xo0bN2BmZgYvLy/83//9H8YNGICNHh6I/vlnjE5NxR8AxCIR9AUCnIuPf2e9ZSEqKgq//PILtm3bBjc3NwiFQtSrVw8n5syBqGdPeToQAN0BTAGwT1sbnZYvx9zvv4e7u7ti5bu3SUlJQevWreHv74+QkJBy77eaCqIs6lLZW2lGFEpT++LjedjcXDG1jwC/ANhHQ4PJv/xCiUTCq1evqlw3KZ9++fz5c167do0HDhzgTz/9xAkTJrB79+5s0KABzczMKBKJ6OLiQj8/Pw4bNozz5s3jtm3b+NtvvzE+Pv69YipycnJ46tQpDh8+nFZWVqxXrx7nzp3L27dv882bN/9c+7ff8rCOjuLaHwPUAnhMR4eyTp145OBBikQiJiUlvbM9qVTKhw8f8tixY/zxxx85dOhQtmzZkmKxmMbGxmzatCm//PJLLlq0iAcPHuT9+/eZm5tbqJ7Xr1/TxMSEiYmJJMk5c+ZQQ0ODe/bsKdX1S6VSXrt2jQsWLKCPjw8NDAzYqlUrLly4kNeuXVN5DejiyM3N5dq1a1mlShUGBAQwJiamVOdfvHiRVapU4fPnz4stI5PJuGbNGpqbm3P9+vWFRlOZ9+8zSUuLGRoarAHwR4ASgKcA6gG8V05TSl+9esW1a9fS09OTNjY2DA4O5l9//cVHjx4RAHV0dKivr099kYj6GhrcJhCQQiHDXF3pUrUqdXV16e3tXew9SklJobu7O4ODgyt1BP9fBpUZR/GhtzKbnnx8SE1N1gW4F2A0QEOArwB5IFQRD8n8efiXL1/mzp07uWjRIo4aNYqdOnVi7dq1aWBgQCMjI7q7u/Pzzz/nmDFj+MMPP3DPnj2MjIwsVzNKSUgkEp4/f15hunBxcWFwcDBv795NmUhEAoprvwzQIt9koK9PbtpEc3NzXrx4kaTcBHLz5k3u2rWLs2fPZt++fVmvXj2KRCLa2dmxTZs2HDt2LFeuXMmIiAg+ffq0xOt8O3DL3MSECxo3JocMIefOpbe7OwFQJBLRz8+PT548KbKexMREhoaGMiAggJaWlqxZsybHjBnDw4cPK8UsFGT58uVs2LAhtbW1OWjQIKXvwsPD6eLiohQwJpPJeOjQIbq6urJVq1alfoEg5WabOnXqcOfOncWWefbsGbt378769esXb65xd6dUQ4O3AOoDlBUw/bQFOFNTk7x/v9T9I+V/3+fPn+egQYNobGzM7t2788iRI0WKexEnq2y+fPbsGevVq8egoCC1SFQiaqEoiUePSF1dJgLUAXgX4BaAdQBOACgG6GRkxL59+zIwMJBt2rRhjRo1qKurSwsLCzZq1Ij+/v6cOHEily1bxoMHD/LGjRt88eJF6fvyAZDJZIyMjOS0adO4w8iIOYDStUsAegE8mPd5q6mpIsLY0dGROjo6dHV1Zffu3RkcHMzQ0FD+/vvvyrbyUqI0utu9m79qaipGOGe1tGgBcKKlJUUiEQMCAujl5UVSLjARERGcNm0a69evr3igrVmzhg9VjOjet28fDxw4wBEjRsiFIi6OnDKFKU2a0EhLi7sXLWJmZiYnT57MOnXq0Nvbm25ubjxy5EiZH2zfffcdO3bsWOz5ERERtLOz48SJE4v3dURGyoUc4J9FCEUbgN0EAnLUqFL1LSEhgQsWLGCNGjXo6urKxYsXK0Z35c3z58/p4eHBKVOmqEWiklELRUmcOMEcIyO2Bjgs759snjwBIUMAZgNcZWFBLS0thoSE8Pjx47xz507pHcUfIbK6dZkDKF07AW7Ie/BoABQBHDRoEA8cOMDo6OiKjRa/d498a4QzCeAogDI9PfYwM6OpqSkB0MfHh4aGhvT09OSsWbN44cKF9+rbjBkzOKhTJ9LAgNTW5lqAzQBSJGLy99+zZ8+eBMC5c+eq9lZd7CXeo1gs5qNHjwp9l5OTw+DgYFapUoXHjx9/d0UhIaRQSALMAVgd4KK8zyfzzIftALJKlRL7lJOTw19//ZWff/45TUxMOGTIEF68eLFCH96pqals0KABJ06cqBaJj4CyCsUn78zOR2ZiggEZGUpT+0SQL1I0E4BQKMTInj1xLD4epqamaF/OaZkrE+rqFlq9LxzAVMgT9TUAcFVbG11PnsSECRPemSa9XFi6FMjNRRLkeV9qA7iIPNXOyMA4qRT787Ks1qtXD3v37oX4PddPUOLCBSAvuC8KQD0AyMyEwdSpaDh9OqJcXeHm5qZYPKe0kMTw4cMxc+ZMODg4KH334MED9OvXD2KxGNevXy/ZuZ6erljwSAvArwDGQr7yVyPIU7brAPLI7mKIjo7Gxo0bERoaCmdnZwwZMgTbt28vl2DOd/Hy5Uu0a9cOXl5eWLx4caWujaLm/fg48zmXMyQxZPVqJAmF2Id/VrBzL1Ami8TIq1eRmppaCT2sOEhiiEBQ6NpvQJ5GoxHkD+hnQiFMTExw4MCBiu/U//0fciUSpfXJOwLYDeA2gAa6uvj8888BABERESrPLlKJV6+Ugv/SIE9lDAA6BgYI8vCAqampSgFjxbFp0yakpaVh7NixSse3bduGpk2bol+/fjhy5IhKM7BYuzYkurqKfXcA5yBPnXES8iyangBQs6bSeWlpadi0aRNatmyJVq1aQSAQ4OzZs/jtt9/w1VdfVbhIvHr1Cn5+fmjevDmWLFmiFol/Of8JoRg5ciTuRkfj8I4dSnlwvADYA1igpQVNf38Y+/ri0qVL2LlzJy5cuFBp/S1PRo4cibsyGQ4bGUG3wPHGAC5ALhgaenrI/uYbxMTE4Mcff0Tfvn1x/vx5uW2yApCZmhYa4bSGPLWHPwCnN2/QrFkziEQiREdHl2+66beWIy0YMCYUCIDcXJUCxoojKSkJQUFB2LBhgyIw8PXr1+jfvz/mzZuH8PBwjB8/vsQ1N0giIiIC7davR3be6AqQp2XPApABYDGApwB6AYj09gZJXLp0CYGBgahatSoOHDiAKVOmIC4uDosWLSp2ymp58/r1a7Rv3x6NGzfG0qVL1SLxKVAWe1Vlb6XxURSa2qerS32hkNs0NEg9Pd42MmLTqlWpp6dHV1dX7t69mxs2bGC1atXo6+vL8+fPq9zWx4bStYtE1M/zSWzL81Es19Kik0BAA11dVq9enYsXL+aLFy+4bNkyuri4sE6dOly1atV7ObDfRiaT8UsvL/oIhUqBW4oU29ra5Hff8d69e9TT0+Po0aMpEonKLavojOBgDiqQ6lspYExXl2kxMSUGjL2Lvn37ctq0aYr9S5cu0dHRkcOGDWN6enqJ58tkMp4+fZqfffYZnZ2d2bZtW04RiynR0SEBTgZokvd7bA/wLx0dpri50VIspqGhIR0dHblw4UImJCSUqf/vy+vXr9msWTOOHDlS7ZP4CIHamV1KkpLkM6GKcVjm5OTwl19+YfXq1enr68tz5869f5uVyJEjR1jXyYk5a9eSHTqQrVuT331X5JRgUv7ACg8PZ48ePWhqasrRo0fz9u3b792P4cOHs4mnJ980akQWSEOeCfCmhgYfCgRcsWABvb29OX36dEqlUnbs2JEikYh79+4tc7u5ubnMzMxkUFAQ+/v6MlMkYjbAZMjTZO/V0WHmN99w6tSpbNKkSZnaOHr0KJ2cnJienk6JRMLvvvuOlpaW3LdvX4nnFhSIGjVq8IcffmD9+vXp7+8vn1kXGkqampKGhqSuLmUGBpRoaTGialVaGhmxf//+7N+/P83Nzblhw4ZKeUi/fv2aLVq04PDhw987fkVNxaAWigoiXzAcHR3ZqlWrf6VgZGZm0tHRkSdOnCjT+fHx8Zw1axatra3p4+PD3bt3l2nmUaHRnZaWfISjocEXurqsa2ZGka4uhUIhfX19FYGI6enpdHNzo56eHn/77bcyXUNISAiR5y/P34YYGPCNWMywOnXoYmtbYsDYu3jz5g3t7e0ZFhbG+Ph4+vj40Nvbu8RFhN4WiK1bt3Lr1q00NzfnypUrlR/4ubl8sm4dD7Vrx/EmJmzXoAHXrVvHV69eKYrcuHGDjRs3po+PD++XMbaiLLx584afffYZAwMD1SLxEaMWigomJyeHGzduVAjG2bNnP3gfysqcOXPYo0eP964nOzubO3fupJeXF6tUqcKQkBA+fvz4/SqVSslXr5RGdnFxcXR1dVWad5+QkEALCwsaGxszOjr6/drMo2rVqvzrr7/Kpa4JEyZw4MCB3LdvHy0tLTlv3rx3RtwXJRCvXr1iYGAga9SowevXryvKpqenMzQ0lN7e3rSwsODXX3/NW7duFVu3RCLhTz/9RLFYzPnz51f4wlhpaWn08vLi4MGD1SLxkaMWig9ETk4ON23aREdHR/r4+Hz0gvHw4UOKxWLGxsaWa723bt3iyJEjaWJiQn9/f54+fbpczR3Pnj1j06ZNOWjQIMWD7tq1azQwMKCNjc17B4c9fvyY5ubm5dLnyMhIWlpacuDAgaxevTovXbpUbNmiBCI3N5dRUVGsXbs2AwIC+Pr1a0XA5PDhw2lqasoOHTpw7969zM7OVrlfMTEx9PPzo7u7OyMjI9/7OosiPT2drVq14pdffqkWiX8BaqH4wOQLhpOTE318fFTKDFoZdOnShfPnz6+w+l+9esWVK1fSzc2NtWrV4s8//8yXL1+WS91paWns2LEjO3XqpHAEHzhwgIaGhqxbt26x6TpUYe/evezcufN79zEnJ4c1a9akjY0NAwIClMxABXlbIEJDQ5mbm0uZTMaNGzfS3NycGzduZHJyMpcuXcq6deuyevXq/Pbbb99rDWyZTMZt27bRysqKEyZMeK979jbp6en09fXlgAEDPqk14D9l1EJRSeTm5nLz5s10cnKit7f3RyUYhw8fZs2aNcuUTru0yGQynj17lr1796aJiQmHDx/Omzdvvne9OTk5HDhwIJs1a6ZIrLdgwQKKxWK2a9euzNHTkydP5nffffdefZPJZPz888+ppaXF0NDQYstERETQy8tLSSBIufO3f//+dHNz49q1a9mrVy8aGxszICCAERER5fqGnpKSwgEDBrBatWolR4OrQEZGBtu0acOAgAC1SPyLUAtFJZMvGM7OzvTy8mJERESlTg/MyMigo6MjT548+cHbfvLkCefMmUNbW1u2bNmS27dvL5XJ5G2kUiknT55MNzc3xsfHUyaTceDAgbS0tGRgYGCZ7nPLli0ZHh5e5j4lJibS29ubmpqajIiIKPT9uwSCJK9fv87q1avTw8ODdnZ2bNCgAVeuXFnhucNOnjzJatWqMSAggMnJyWWqIzMzk+3ateMXX3yhFol/GZUiFADMAIQB+Cvvp2kRZeoDuAR5toQ/AfQp8N1mADGQx33dAFBflXY/RqHIJzc3l1u2bKl0wZg9ezb9/f0/eLsFycnJ4b59++jr60srKyvOmDHjvXwlP/zwA+3t7Xnnzh1mZ2ezefPmtLS0LPXIIDs7m/r6+sWaiUri+PHjrFKlCqtVq8aFCxcqfVeSQGRkZPCrr76ilpYWDQwMOHbsWCXH9YcgLS2NkyZNoqWlJUNDQ0v195mZmcn27duzT58+75ULS03lUFlC8T2AoLzPQQAWFVGmJoAaeZ9tIA8mNeE/QtGztO1+zEKRT25uLkNDQ1mjRg1+9tln5e7sfRcPHjyoEAf2+3DngEMPDAAAIABJREFUzh2OHTuWpqam7Nq1K0+dOlUm08qWLVtoZWXFS5cuMSUlhQ4ODhSLxdy8ebPKdfz++++sW7duqdvOysri119/TTs7OwYFBdHDw0PxsCxJIK5du8ahQ4dSW1ubhoaGXLJkCTMzM0vdh/Lk999/Z/369dm2bVs+fPiQAQEBtLa2pqGhIWs4OHB9x47kiBHMXraM/l260N7engDo7e2tFol/KZUlFPcAVMn7XAXAPRXOuVlAOD5ZocinMgSjc+fOXLBgQYW2UVbevHnDtWvX0t3dnTVq1OCSJUuYmppaqjqOHj1KCwsLHjt2jHfu3KGZmRlNTEx46tQplc5fvnw5hw4dWqo27969y/r167Nbt26Mjo6mpaUlr169+k6BSE1N5YoVK+jh4UFra2uamJhw0KBBH8RnpCo5OTlcuHAhxWIxJ0+ezLSUFLJzZ97V0aEVwKsAs0UiLtHS4ihnZ+rq6jIsLKyyu62mjFSWULx8a/9FCeU9AdwFIOQ/QnEvzyT1EwCdd5w7DMBVAFft7e0r5CZWJLm5udy6dStr1KihsI9XhGAcOnSILi4u7+UT+BDIZDL+9ttv7NevnyLl9R9//KHy+RcvXqSVlRVDQ0N58uRJmpmZ0czMjDdu3Cjx3ICAAP7yyy8q93P9+vU0NzfnmjVrKJPJ2L9/f3799ddFCoRUKmV4eDj79etHY2Nj9unTh8OHD6eFhYVKEdqVxV9//UVfX1+GmZhQqqPDaIDWAHcVSK+SpaFBW3Pzj2rChprSUWFCAXlG6ttFbF1LIxT5Iw4ATd86JoA8U/IWAN+o0ul/04jibfIFo2bNmuUuGBkZGaxevbrKb9YfC4mJiZw3bx6rVq3Kpk2bMjQ0VCWzTFRUFO3t7bl48WKuWLGCtra2tLGxKdHk5ujoyKioqBLrf/78Of39/enu7q4of+LECVpZWbFFixZ0dnZWCERcXBznzp3L6tWr093dncuWLeP9+/fZuXNnNmnSpEzR3h8a2YMHHC4UUpQXue4B8M1bubhstbXVQvEv5qM2PQEwAnANQK931OUD4Igq7f6bhSIfiUTCbdu2sWbNmmzRogXDwsLeWzBCQkLYs2fPcurhh0cikfDgwYNs164dLSwsOG3atBJXsCsYxT169Gi6uLjQ1dVVbs6SSsljx8i2bUl7e9LcnBJbW4ZraFB69Kj8+2I4d+4cq1atyvHjxytE6+jRo9TV1aWNjQ23bNnCtLQ07t69m35+fjQzM+PIkSMV5qgLFy6watWqnDJlSoVHRpcbS5eSurqUALwA8Nu8BZKUhALgmaNHK7unaspIZQnFD285s78voow2gNMAJhTxXb7ICAAsBbBQlXY/BaHIRyKR8H//+x9dXFzeSzD+/vtvisVixsXFVUAvPzz379/nxIkTKRaL2blzZx47dqxY53d+FPeAAQPYrl07utety5+dnCiztJSvZFdElloaGJDW1vKHY4H7nZOTw5kzZ9La2ppH8x6IZ86cobe3N01MTNi0aVNev36dEyZMoIWFBX18fLh161ZFQKBUKuW8efNoZWWlOP9j4s2bN4yOjubp06cZGhrKBQsWcMyYMezUqRN/MDdXEobhAJcVJRQHDlT2ZagpI2UVCoH83LIhEAjEkK83Yw8gLm/EkCoQCBoBGEEyUCAQ9AewCfLpsfl8SfKGQCCIAGCRJxQ38s5JK6ndRo0a8erVq2Xu98eIVCrFrl27MHfuXIjFYsyePRtt2rQpOpd/ejqwbx8QGwsYGoJduqDz2LHw8vLCtGnTPnznK5CMjAzs3LkTK1euxMuXLzFixAgMHjy40Ip36enp6NWrF5Cbi3GXL8M7PR0iVf629fSATp2A//0PMY8fo1+/fjAyMsKWLVsQHR2N2bNn48mTJ+jVqxeWL18OZ2dnpKSk4Msvv8RXX30FJycnRVVJSUkYMGAAsrKysH37dtjZ2ZX37SgWqVSK5ORkPHny5J1bbm4ubGxsYGRkBKFQiIyMDDx79gxpaWkItLHB93Fx0M3NBQAEAtAHsKxAO3YCAbaFhcGndesPdm1qyg+BQPAHyUalPu99hKKy+BSFIh+pVIrdu3dj7ty5MDU1xezZs+Hl5YXRo0cjPDwcqYmJcM7NxXwdHXTIzga0tSGVyXBRSwtho0bh28WLERYWhjZt2lT2pZQrJBEZGYlVq1bh0KFD6Nq1K0aPHo3GjRsryuTm5OB3Fxd4xMaqJhL5iESIadAAntHRmB4cjPr162Pu3Ll4/Pgxevfujbi4OOzcuVNx3M/PT7EoUT6nT5/GwIEDMWTIEHzzzTdlXka1KNLT00sUgKSkJJiamsLW1lZpq1KlCgAgJSUFDx8+xM2bNxEVFYWaNWuiSZMm8PT0hKenJ8RiMc6dOYPOEyZAlJKCcAA9AGyH3BmZDfmSus5aWti4dy+8vLygo6OjXpToX4ZaKD4x8gXj22+/haGhIWrWrIm5BgZwCA3FsYwMfAHgFoBqeeXva2rCX0MDz01NEbp16ycnFAV59uwZNm3ahNWrV0MsFmPUqFHo27cvRIcOgUOGQJCejv6Q2zvTAVhDvj544Fv1zAEwG/JI0eYCAS4OHYrv7t1DbGwsGjVqhBs3bkBbWxvOzs5ITU3F+fPnCz0YJRIJ5s6di19++QWhoaFoXYo3bZlMptIoIDs7u5AAFCUI2traSEpKQmRkJK5cuYLIyEj8/vvvMDExURKFBg0aQE9PT6kvKSkp6NmzJ25euwZZWhocAIwDMDTv+2oCAWLfelbExMSgWrVqKl+vmspHLRSfKFKpFHv27MHaWbNw4sED6OT9vtwBhEC+dCgAdAAwVlsbowwMsGHXrk9aKPKRSqU4efIkVq1ahStXruCmQACblBQAcjunM+TT6aIhnylxFEDDvHMfAOgG+drToQDaALilqYmBderg0aNH6NWrFwYPHgxLS0t4enri8uXLcHZ2Vmr/yZMn6NevH7S1tbFt2zalNbAzMjJUGgUYGxuXKAKmpqZFvrmnp6fj2rVrClG4cuUK3rx5oxCE/M3S0rJ0N/b+fWDePGDPHiArC7C1Bb7+Ghg9GtDRKV1daj4q1ELxiSMLDobshx+gKZEgCYAD5E6dWgD2ANgG4CCAahoa2HDixH9CKAry+NgxWHbtCm2JpNB39yAXimUAeucd6wBgLIBRADZALhRZQiFOzJmDtl9/DX19fZBEx44d4e3tjaCgIADyUUBKSgp27dqFkJAQeHl5oW7dukhISFASgaysLNjY2JQ4CtBR8cErlUoRFRWFyMhIhSj8/fffqFu3rkIQmjRpAmdn5/I1B5GA2rz0yVBWoSg/Q6qaCkX4++8QSiTIBRAAYBDkIpEGIBjAqfyCUilWLl+Os2fPQltbG5qamtDS0lJ5K215LS0tCIXCSrdV2508KX+oFWAU5BGdmQA8AHTMO74H8ql4HaGMjkAA7wcPcPXqVTx58gRHjhzB1atXoaenh+bNm+PJkydITEyEhoYGJBIJGjZsCCsrK2hpaaF58+ZKImBmZlbme0IS8fHxSqJw7do12NjYKExIQ4cOhbu7u8pCU2bUIqEGaqH496CtDRmAAZA/5FbkHQ7JO1a9QNGjx4/j8NGj0NfXh76+PvT09CASiSASiaCjowNtbW2FiOQ/9HJzc5Gbm6v0WdWNZIWKkSrl2547BzupVOmWrQKwHPKMlGchN0MVEtYCCKRSMDQUJy9fxr2aNXEiPBxDhw5Fs2bNYGtrC5KYNGkSLC0tsXnzZpibm5fLr/bly5e4evWqkm9BJpMpRGHmzJlo1KgRTE1Ny6U9NWpKi1oo/iWwSxcMOXUKSRIJjgHQyjt+GsBjyB+KAJACwNjAABMnTkRAQACePn2qtCUmJirtP3/+HGKxGFWqVCl2s7a2RpUqVaCrq1tk32QyWamEpbRiVLB8VlZWkWU8nz5FUZNRNQC0hNw0txpALAoLa0HMZDLMf/IEaQ8f4oCPDwYsXQoA2L9/P0aMGIGgoCB8/fXXZR4t5OTk4M8//1QShfj4eDRo0ACenp7o378/fv75Z9jb21f6KE2NmnzUPop/CSOGDMGNzZsRLpPBoMDx5wBy83dEIjTW08OS1avRoUMHGBgYFK7oLSQSCZKTkwsJSlECo6+v/05Byd8MDQ0//EPO3x/Yv7/Yr/NjAs5BLqz5b0gpAIwBTMvbCkKRCJLgYExMSsLRo0exc+dOeHp6qtwlknjw4IGSKPz5559wcnJS+BQ8PT1Ru3btcp1Oq0ZNcah9FJ8wsbGxWLtxI3S0tGAtkymOr4XcXwEAudra0OrXDxrh4TA1NVVJJABAU1MTNjY2sLGxeWc5mUyG1NTUQgISFxeHK1euKB0jqdIIRSwWQygUlvGuyElOTsapU6fAmBj4A9ADkAwgAkBnACLIk5XtgDwm4BvIs0teAJABQAi5g3s0gDsABkI+IwoAGmZm4vtZs1C1aVNcu3YNJiYm7+xLSkoKfv/9d4UoREZGQl9fX+FsXrhwIRo2bKjy70aNmo8F9Yji30ZkJDBtGnDpEqCrC0gkyDEwQHBGBmbGxsLkI7Bjv3nzpsQRytOnT5GWlgYrK6sSRyhWVlaKN26pVIrIyEgcP34cx48fx19//QV3d3dop6fj8LVrEEE+SugJeT57GVAoJqDg1FlbyAUjHIATgJd55WUAVkI+I+qmkxMEf/2l5NjNzMzE9evXlaamPn/+HI0bN1YaLeQHvKlR8zGgnh77X+PpU+DxY8DQEHBxwfARIyASibA0z6b+byArK6uQz+Tt/adPnyIlJQV6enrQ0NBAeno6DAwM4OjoCF1dXTx8+BA6Ojro378/pt++Db1Dh4ACo66SKGrqLABIIB+xTQGQrq+PmLVrcS4nRyEK0dHRcHNzUxIFFxeX9x4hqVFTkaiF4j9OSkoKateujYiICNSpU6eyu/NeSCQSXLlyRTFqePDgAVq0aAEPDw/k5ubizJkzuHXrFhwcHGBpaYns7GwkJibC5OlTXMzNhb4Kbbw9dfY8oPD9mEA+O0oGYC6A6QB2ikQ40rUrmjZtCk9PT3h4eBTr3Fej5mOlrEKhfv35RLCwsMA333yD8ePH498o/omJidi8eTP69OkDS0tLjB49GlKpFD/99BOioqLg7e2NXbt24fjx4xg4cCASEhIQHR2N8+fP48qVK3j06BG6TZ+OcebmkGhrl9jeKgBvIPdV9IDcDJXPSwCvIJ+C7AH5zClHTU1cvnwZaWlpcHBwUIuEmv8UaqH4hBgxYgRSUlKwb9++yu5KiUgkEly4cAHBwcHw8PCAq6srjh49Cj8/P9y+fRvXrl2Dr68vli9fjtq1ayM6Ohpbt27FzZs3MWbMGKWYguzsbAwYMAAnT57E/Nu3ofnrr4C+fonpJvKnzj6GfOpsQfQBjIDcuZ0MoFmzZti7dy/i4uJQu3ZtdOvWDUePHoX0rdgNNWo+RdRC8QmhqamJ5cuXY9KkScjIyKjs7hQiISEBGzduRK9evWBhYYHx48dDIBBg+fLlSElJwZ49e9CxY0ds2bIFzs7OmDJlCnx9fREbG4uNGzeiadOmhabdPnv2DG3atEF2djYOHz6MuLg4hKakYN6gQdhlb4+XAF4DyIF82baikOCfmU4FkUHu6I4HkGVri4YNG2Lt2rWIj49H586dMXfuXFSvXh2zZ89GfHx8+d0oNWo+MtQ+ik+Qvn37wsXFBXPmzKnUfuTm5uLSpUsKX0NcXBzatm2LDh06wM/PTzEjSCaTISwsDOvWrUNERAR69uyJYcOGoVGjRkXGY6SmpuLu3bs4c+YMFi9eDDMzM0ilUqSkpMDFxQWurq5wc3ODi4sLxgwfjmvffQfbzEzg2DEknjuHs1Kp0tTZ/HTaegDMIU+4mA5gJoC9AO4IhegiEkGvZUv07NkT3bp1U0Rl37x5E+vXr8eOHTvQrFkzDB06FJ06dVLHRaj5KCmrj+K9VrirrO1TWuGuIoiLi6NYLC5xGdGKID4+nuvXr2ePHj1oYmLChg0bcsaMGfztt9+Ym5urVPbJkyf89ttv6eDgwIYNG3LNmjV89eoVSVImk/Hp06c8ffo0V6xYwVGjRrFVq1a0srKioaEhXV1dqaurS39/fx4+fJgPHjygRCJRqv/ChQt0d3cnSWZmZnLpzJmMA+gF0BigIcA6ANflrd62G6ALQH2A5gA7ALwJkG5ufPPmDXfv3s3evXvTyMiIrVu35urVq5mYmEiSTE9P5+bNm9m8eXPa2NhwxowZlXL/1ah5F6iMFe4qC/WIomTmzZuHq1ev4sCBAxXaTk5ODi5evKgYNTx58gTt2rVTjBoKpt4G/kkNvm7dOpw7dw69e/dGly5doKGhgTt37uDu3buKn0KhEG5ubooRQv7P06dPY8qUKdixY8c713+YPHky9PT04OrqiunTp6Nhw4b4xcgIJrt3Ayqa5jIALGnZEkFnzihGCRkZGThx4gT27t2LY8eOwcPDAz179kT37t1hY2ODqKgorF+/Htu2bUPDhg0xdOhQdOnSBdoqONnVqKlI1CMKNUpkZmbSycmJJ0+eLPe64+LiuHbtWnbr1o3GxsZs3LgxZ82axYsXLxZ6q88nJiaGY8eOpbm5Oe3s7Ni0aVM2aNCA+vr6tLGxYevWrTl27FiuXr2aZ8+eZXJycqE6ZDIZZ86cyerVqzMqKuqdfZTJZLSxsWHdunXZsGFDnjt3Tv6FVEr26EGJri6LXEu74CYS8WpgILW0tFinTh2+fPmyUDuZmZk8ePAgBwwYQFNTU7Zs2ZJLly5lXFwcMzMzuW3bNnp7e9PKyorTpk3jX3/9VfobrkZNOYEyjigq/aFflk0tFCUTEBBAU1NTCoVCOjs5cf348eTeveStWwwPD6eLiwtFIhF9fHz46NGjd9aVnZ3N06dPc/LkyaxduzbFYjH79evHrVu3MikpqVDZ27dvc/fu3fzmm2/42Wef0dDQkABoaGjIli1bcvLkyfzll1946dIlvnjxQqXryczMZN++fdm0adNCbb7NgwcP2K5dO2poaDA0NJRSqVTpe0lODtfZ2DBHR4dpAoGSOEgByvT1SVtb8tAhkuSff/5JY2NjmpmZ8f79+8W2m5WVxaNHj/LLL7+kmZkZmzZtysWLF/PRo0eMjo7m5MmTaWFhQV9fX+7YsYNZWVkqXbsaNeVFpQgFADPIV5L8K++naTHlpJCvs3MDwKECx6sDuJJ3/i4A2qq0qxaKkrl9+zYz09MZ6uzMm1patBIIeFVfnykiEY2EQu5etIiZmZmcPHkymzRpUuj8R48ecc2aNezatSuNjIzo6enJkJAQXr58mRKJhOnp6fzjjz+4detWBgcHs1u3bnRxcaGOjg6dnJxYq1YtGhgY0NnZmSEhISU+3N9FcnIymzdvzj59+jAjI6PYci9evODkyZMpFovp6+vL0aNHF1lu1apVbNGiBaXp6Tzcpw/vWluTrq5knTo8a2bGqz/8QMpkSuc8f/6cNWvWpLa2No8dO1Zin7Ozs3nixAkGBgbS3NycjRs35qJFixgVFcVdu3axdevWtLCw4MSJE3n37t3S3RA1aspIZQnF9wCC8j4HAVhUTLm0Yo7vBtA37/MaACNVaVctFCoyahSlIhGjAVoD3AVwLcBmAGlgQP75J9PS0qirq8ubN28yLCyMEydOpKurKy0sLNi/f3+uX7+ex44d4y+//MJJkyaxY8eOrFatGkUiEd3d3dmnTx/OmTOH27dv57Jly9iuXTuamZlxzJgxvHnz5ntfwt27d+nk5MTg4OBCI4N8cnJyuGLFClpaWjIwMJAJCQn08PDg2bNnC5VNSEigubk5b9++TZJMTEykiYmJwok+c+ZMzpgxo9h2unTpQg0NDS5cuFDla8jNzWVYWBiHDx9OS0tLenh4cN68eQwLC2NQUBCtra352WefcevWre8UQjVq3pfKEop7AKrkfa4C4F4x5QoJBQABgGcANPP2mwE4qUq7aqFQgfv3OVJDgyJ5+AA9AL4BOA7giDwzS0bTply1ahUNDQ2pq6tLNzc3du7cmb1796avry9tbGyor6/PRo0aceDAgVywYAEPHjzI+/fvK2YwxcTEcMaMGaxSpQpbtGjBLVu2MD09vVwuISIigpaWlty4cWOR38tkMh4+fJguLi5s06YNb9y4QVI+GjI3Ny80y4ok+/Tpw+nTpysd69GjB9esWUOSDA8PZ4sWLd7Zr2+++YZCoZB9+vQpVryKQyKR8MyZMxw9ejStra1Zt25dhoSEcNmyZWzfvj3FYjHHjRvHW7dulapeNWpUobKE4uVb+y+KKScBcBXAZQDd8o6ZA/i7QJmqAG6r0q5aKFRg8mRSS4sSgBcAfgswB+BggNPyhCITYB2xmEKhkCKRiC1atGBgYCCXLFnC48ePMzY2tsgHYU5ODvfu3Us/Pz+KxWKOHz9e8YZeXmzatImWlpaMiIgo8vsbN26wdevWrFWrFo8ePUpZAVPR0qVL+dVXXxU65/jx43R0dCwkZMePH2f+31R6ejr19fWZlpb2zv79+uuv1NLSoru7O9+8eVPayyMpF40LFy5w/PjxtLW1pZubG8eNG8dhw4bRxsaGzZo148aNG0vsixo1qlJhQgF5TNLtIraupRAKm7yfjgAeQZ7R2aIIobj1jn4MyxObq/b29hV5Lz8NPv+cBZ20wwEuyxtRjMw7lqWry99Xr2atWrW4Z8+eEqv8+++/GRQURCsrK3p5eVWIqUQqlTI4OJiOjo5F2u4TEhI4ePBgWllZceXKlczJySlUxsfHh4fyHNH5pKens3r16jxx4kSh8hKJhPb29rx27RpJsmXLljx16lSJfb1z5w5NTEzKJWZFKpXy4sWLnDhxIu3t7VmjRg326tWLn332Gc3MzDhy5Ehev379vdpQo+ajNj29dc5myJcLUJueKpJhw0ihUCEUQ/JEYi3A5gWmf6bdukWRSFSsQzU7O5u7d+9mmzZtaG5uzokTJ/LOnTsV0uWMjAz26dOHzZs3LzQ9Nj09nXPnzqWZmRmnTp1a5FRVkkxJSaGRkVEhAZs+fTr79OlTbNuzZ8/mqFGjSMr9FMHBwSr1+eXLl6xVqxa1tbVVEhdVkMlkjIyM5NSpU1m9enU6ODiwZcuWtLKyYqNGjbhu3Tq+fv26XNpS89+isoTih7ec2d8XUcYUgE7eZ/O8GU5ueft73nJmj1KlXbVQvJukpCTumDuXL3V0KAF4AqAewF8BJgM0ArgXYKarK6dOnVrkrKf79+9z6tSptLS0pI+PD7dv387MzMwK7XOzZs3Yt29fpXakUim3bNlCOzs79unTp8Q3902bNtHf31/p2K1bt2hubs6EhIRiz4uLi6OZmRnT09MZHh7O5s2bq9x3iUTC7t27UygUcvHixSqfpwoymYx//PEHp0+fTicnJ1paWtLZ2ZmGhoYMDAxkZGSkktlt+fLlbNiwIbW1tTlo0CClutavX08nJyfq6+vTz8+PT548Kde+qvn4qSyhEAM4nffwPw3ALO94IwAb8j43B3AL8gXHbgEYUuB8RwCRAP7OEw0dVdpVC8W7SUpKoqOjIw2KSFNBgGEAXQQC6mpr09vbmzExMSTlcQA7duxgq1ataGFhwSlTpvDevXsV3t87d+6wevXqnDVrltJD7+zZs2zQoAGbNm3KixcvqlRXly5duHXrVsW+VCplixYtuGrVqhLP7dixIzdv3qyyn+JtvvvuOwqFQgYEBChdR3khk8l48+ZNzpo1i87OzjQyMqKRkRFr1KjBFStW8OXLl9y3bx8PHDjAESNGcFDfvuS335ING/Js/fq0MDLi7T//ZHZ2NkeMGEEvL69y76Oaj5tKEYrK2tRCUTypqans0qULGzduzJjbt8mOHUmRiNTUlP+6DQxIfX1yxw7FOdHR0Zw0aRItLCzYunVr7tq164MFg4WHh9PS0pJbtmxRHLt//z67detGBwcH7ty5U+WHblpaGg0NDZmamqo4tn79ejZp0kSl2UkHDhxgy5YtSarup3ibI0eOUEtLi/Xr16/wqa63b9/mN998w2rVqlFXV5fa2tps3749L1y4wOBx4zhIV5fMi0CfBHCUpibZrRspk/HJkycEwL///rtC+6jm40ItFGp45coVVqtWjePHj2d2dvY/X0RHk/Pnk9OmkVu2kBkZivQSXl5elZZeYsOGDbS0tFTEOzx//pzjx4+nWCzmwoULS23q2rdvH9u0aaPYT0pKooWFhcrxHDk5ObS2tuadO3c4a9Yslf0Ub3P//n2amJjQ3Ny8xKj38uLu3bsMCgqijY0NNTQ02FUo5IACo8iJ+ZMY9PXJEyf4+PFjAuCvv/76Qfqn5uNALRT/YWQyGX/66SdaWFhw//797ywbFRXFCRMmUCwWs127dty7d6+yqHwApFIpg4KC6OzszOjoaGZnZyv6P3LkyDJHcQ8YMIArV65U7Pfv359TpkwpVR1BQUGcOHEiT58+XSo/xdukpaXRzc2N2traDA8PL3M9ZeH+/fucKhRyUAGhCAcozsuGm/HFFxw2bBgFAgG3b9/+QfumpnJRC8V/lNTUVHbr1o2NGjUq1tGbkZHB0NBQtmzZktbW1gwODuaDBw8+cE//6UuvXr3YsmVLJicnc//+/XR2dmaHDh3eKxYjJyeHZmZmfPz4MUkyLCyMDg4OpfYz/P3337SwsGBqamqZ/BQFkclk7NmzJ4VCIZcsWVLmesrCDC0tJaEgwBUAnQFa6Ohw/vz5NDIy4vnz5z9ov9RULmqh+A9S0NRUlE/h1q1bHDt2LMViMTt06MD9+/cXGXfwoUhMTGSTJk3Yr18/Xrx4kV5eXqxTp065ZLgNDw+np6cnSXkCwRo1avDw4cNlqsvX15e7du3iZ599Vi59mz9/PoVCIQcMGFAhTu6imOHqWkgoFD6qX3/lvXv3qKenp+TPUfPpU1ahUC+F+i+EJJYtW4ZfDoUBAAAOyklEQVTOnTvjxx9/xNKlS6GTtz50RkYGNm/ejObNm8PPzw8mJib4448/cOzYMXTv3h1aWlrl2pfs7GwMGTIEDg4OMDQ0hIeHB44fOwYkJCAnMRE9e/ZEtWrVIBAIUK9ePTRv3hxCoRD+/v7o378/bty4gXbt2r13Pw4cOIBu3boBABYsWAB3d3d07ty5THUFBgZi/fr18PHxwdmzZ9+7b9OnT8eRI0ewc+dONGzYEFlZWe9dZ3FIJBJkZWVB6u0NqbY2sjQ0IAGQBeC2ri7YoAHi3N0xbNgwjB8/XmntcTVqiqUs6lLZ2395RFHQ1FTQfHTjxg2OHj2aZmZm7NSpEw8ePFhkrqPyJi0tjSEhIYyJiaFUKuXhWbNoIBAwRlub2Vpa/Kl6da6cMYMCgYAtWrSgmZkZZ8yYUa4BYzKZjLa2trx79y7v3r1Lc3NzhQmqLGRmZtLc3Jxbt25ls2bNyq2ff//9N01NTWlubs64uLhyq7cgISEhRF5+r/wtxNiYL2rVYl0bG+rp6dHKyopBQUHFrh2i5tMFatPTv5OsrCwOHjyY9vb2NDAwYP369RVprLdt20Z9fX3FpqOjQwDs27cvs7Ky+ObNG27YsIGenp60s7Pj7NmzK+wBpBIREaRIxLp5AX0EKBUImCIQUEMgYJs2bRgbG1vuzUZGRtLFxYUymYze3t78+eef37vOCRMmcMqUKdTX1y9zLqeiSE9Pp5ubG4VCIS0tLf/5nR89SmZnMyoqig0bNqSJiQlNTEzYunXrEhdpUqNGVdRC8S+l0Bv5mjU00NJijKMj6e1NHjhAmVTKZcuW0dDQkFZWVrx69SpHjBhBU1NTdunShUeOHPk43g6bNmUiQB2AdwvYxTOEQlr/f3v3HhxVdQdw/PtLYnY3CUUkBaKYmCIOUF/BHaWMii98jWMZHwjDI1GpQ/iz0xorjI9IhcoUxk7qCLVGI75qH0q1ImqhrVKsZEAdBDQaBHxAK4aHeRH21z/uDW5C2OfN7ib5fWZ2stl7cs7vl7vJuffcu+cUFOjatWt7pdm7775bq6qqtLa2VoPBoCe/iy1btmhRUZFeeOGFnq8SePDgQR03bpyKiP5m0SL967XXagFoo4h+U1KijY88oqFQSDs6OvThhx/Ws846y9P2zcBlHUV/sH69an5+lyPyUF6evlJaqmVlZVpaWqpFRUVaXFys1dXVumvXrnRH3EW736+Xg97Rw0XUU/z+Xusoxo4dq6tXr9bhw4drfX29Z/VOnDhRp06desy05F5ZvGiRrgdty8rqss81EFB95hk9fPiw1tTUaCAQ6JX2zcCTaEeRk7aLI6YrVaioYM+33/IR8EP3ZWlu5tIdO5BAgMbmZmpra5k1axbZ2dnpjPYYoVCIWSLkAjXdN2ZnO49esH37dpqamnjuueeYPn0648eP96zuOXPmsGLFCnbt2uVZneGqJkygw+/n69bWLvuclhZOnDGDQ7NmEQqFqK6u7pX2jYmV3fWUKXbu5PDOncwAyoExYZtygWvOOYdJkyZRUVGRcZ2EqnL77bezp6iIPwUCHHNfVW4uFBQkXH9NTQ3BYBCfz0dFRYXzYigEn3/Oi3V1+Hw+6urqePzxx7n44osTbqe7qVOnsn37djZv3syhQ4c8q/eot95C29p63OdNubnsb2igpqaGsrIy79s2Jg7WUWSIUHs7sw4f7vGIPBt4fts2ysvL0xBZdJWVlWzdupW/1tcTuOYayMtzOoe8PNp8PlqXLIGcHNrb22ltbXXGPONw8skns2DBAm677TbnhdpaGDECRo/mvQcf5LSdO3l66VL27dvHsmXLPMsrPz+fW265hWHDhrF+/XrP6u0UGjyYWVlZPZ+FqZI/bBhz585l9uzZ7N271/P2jYlZIuNV6X70t2sUoVBIKyoq9JLcXG3uYXz/Lb9f8/z+jFyDYMeOHQqoz+f77g6tQEBXTp+u+sgjWjJy5DG3a3bOVhuv+fPna/lFF6nm5amCbnNnx/0aVEtLVT36MGH4VN3XXXedDh48WKuqqrSxsVGBLneiVVdXJ9RGKBTSimnT9JKsrGP3eU6O6vXXq6qz3rbf7z+6qJIxycCuUfRdnUfkb6xcSaC8HFpavtuYl8eThYXcOGkSgwYNSl+Qx1FSUhLxDGFHZaV3jalCfT00NwPwDlACPAA81dhI0ahR3LdsGTfeeGNSzXSewbz22mu0tLRwyaBBnLl8OWzdCkDT3r3k5OUl1UZlZSVbGxt5o7aWwNy5zlBaWxuv+/0UDh3K2Y8+yrcHDrBgwQKGDBnC2LFjk2rPmGTY0FOaffbZZyxfvpzNmzcz4tZbKQAKsrJ4OhCA0lJa772XP+zfn7HDTinV0dGlE92NsybvYOALoGbCBMrLy9nq/kNP1A033MCUKVMYOnQobNrEC3v2MK2pCVatcgqcdx40NSVcf5d9Pm8eBSIUqPL0lVfSVFnJ9Px8Bo8ezahRo2hoaGD16tX4/f6kcjImGXZGkWbRjsj9QNOdd6YuoEyWnQ1ZWXDkCAAB4ARgAZDj9zPpggu4tKWFNWvWeHME/tVXsGULJ7jtdSrZtg0ZOZLJN9/MkiVLKCwsjKvaaPv85qVLEwrXmN5iZxSm78jKgtNPB3deq7O7b5s509v23nvvaKcEzjq+7wKfAfXAwYMHmTFjhrdtGpOBrKMwGe/oRHdHjnCkrIzW8ePpyMvj4uxsikVYlJNDx8qVvN3QwLp167jqqqu8abjb5H0FOGv85gDD29qoqalhzZo1HDhwwJv2jMlQNvRkMt7ChQu5//77j36/Erj31lu5b8wYXmppYc7LL7N45kxKSkqoq6tjzJgxx68sHqefDtu2OddGups4EREBiPt2X2P6GumLb/JgMKgbN25Mdximn+ro6KCjo4P777mH3StW8LuWFnLa26kHTgRGBwJ889JLzHvsMfbu3cvatWvTHbIxMRGRelUNxvtzSQ09ichJIvK6iHzsfj1mcnsRuVRENoc9WkVkirvtCRFpDNt2bjLxGOOFhQsXEggEWLxkCSv37yfQ3s5Cn49Ps7O52u9nkCpnzp6Nz+fj2WefTXe4xvS6pM4oROQhYJ+qLhaRu4AhqloVofxJQAMwUlWbReQJ4GVV/WM87doZhTHGxC8tZxTAj4En3edPAlOilL8JeFVVm5Ns1xhjTIok21EMV9UvAdyvw6KUnwZ0P1f/pYi8LyLLRMSXZDzGGGM8FvWuJxF5AxjRw6b58TQkIkXAWcBrYS//AvgKZ4LUFUAV0OOcyiJyB3AHQHFxcTxNG2OMSULUjkJVrzjeNhHZIyJFqvql2xFEmuJyKvAXVT0cVveX7tM2EakFfhYhjhU4nQnBYLDv3apljDF9VLJDT6twptLH/fpShLLT6Tbs5HYuiHND+hScqXuMMcZkkGQ7isXAZBH5GJjsfo+IBEXksc5CInIacCrwj24//7SIfAB8gDNDwsIk4zHGGOOxpD6ZrapfA5f38PpGYE7Y9zuAU3ood1ky7RtjjOl9ffKT2SLyX5y52ZJRCPzPg3AyieXUN/S3nPpbPtB/c8pX1e/H+4N9sqPwgohsTOSDJ5nMcuob+ltO/S0fsJy6s9ljjTHGRGQdhTHGmIgGckexIt0B9ALLqW/obzn1t3zAcupiwF6jMMYYE5uBfEZhjDEmBgOmo4hl7Qy3XLGIrBGRrSLyofthwYwUa05u2e+JyOciUpPKGOMV4xon54rIv0Vkizuh5C3piDUSEblaRLaLSIM7BX/37T4Red7d/k4mv886xZDTT92/mfdF5E0RKUlHnPGIllNYuZtEREUk4++EiiUnEZnq7qstIvJM1EpVdUA8gIeAu9zndwG/Ok65dcBk93kBkJfu2JPNyd3+MPAMUJPuuJPNCTgDGO0+Pxn4Ejgx3bGHxZcNfAL8AGfCy/eAcd3KzAMedZ9PA55Pd9we5HRp598LUNkfcnLLDQL+CWwAgumO24P9NBrYhLN+EMCwaPUOmDMKYlg7Q0TGATmq+jqAqh7SzF47I6b1QETkPGA4sCZFcSUjak6q+pGqfuw+/wJnMsq4P0TUi84HGlT1U1VtB57DyStceJ5/BC6XzkW4M1PUnFR1bdjfywZgZIpjjFcs+wngAZwDmNZUBpegWHL6CfBbVf0GQFUjTeYKDKChJ2JbO+MMoElE/iwim0RkiYhkpzTK+ETNSUSygF8DP09xbImKa40TETkf58jpkxTEFqtTgF1h3+/m2ClsjpZR1Q5gPzA0JdElJpacwt0OvNqrESUvak4iUgacqqovpzKwJMSyn84AzhCRt0Vkg4hcHa3SpOZ6yjQerJ2RA1wElAE7geeBCuD3XsSXCA9ymgf8TVV3ZcoBq8drnDwFlKtqyIvYPNLTL7r77YWxlMkkMccrIjOBIDCpVyNKXsSc3IOsZTj/A/qKWPZTDs7w0yU4Z33/EpEzVbXpeJX2q45Ck187YzewSVU/dX/mRWACaewoPMjpR8BFIjIP55pLrogcUtXjXrjrbR7khIh8D3gFWKCqG3op1ETtxpktudNI4IvjlNktIjnAYGBfasJLSCw5ISJX4HT4k1S1LUWxJSpaToOAM4F17kHWCGCViFyvzsSnmSjW994GddYGahSR7Tgdx7vHq3QgDT3FsnbGu8AQEekc774M+DAFsSUqak6qOkNVi1X1NJyFoerS2UnEIGpOIpIL/AUnlxdSGFus3gVGi0ipG+s0nLzChed5E/B3da8sZqioObnDNMuB62MZ984AEXNS1f2qWqiqp7l/PxtwcsvUTgJie++9iHPjASJSiDMU9WnEWtN9lT6FdwMMBd4EPna/nuS+HgQeCys3GXgfZ42MJ4DcdMeebE5h5SvI/LueouYEzAQOA5vDHuemO/ZueVwLfIRz7WS++1o1zj8aAD/wAtAA/Af4Qbpj9iCnN4A9YftkVbpjTjanbmXXkeF3PcW4nwRYinMQ/AEwLVqd9slsY4wxEQ2koSdjjDEJsI7CGGNMRNZRGGOMicg6CmOMMRFZR2GMMSYi6yiMMcZEZB2FMcaYiKyjMMYYE9H/AbpoFS+mvZAyAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pr = nx.algorithms.link_analysis.pagerank(kn)\n",
    "pr_vector = list(pr.values())\n",
    "pr_vector = np.array(pr_vector) * 3000\n",
    "nx.draw_networkx(kn, node_size=pr_vector, labels=None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление собственных значений\n",
    "\n",
    "- Как вычислить собственные значения и собственные векторы? \n",
    "\n",
    "Все задачи на собственные значения делятся на два класса:\n",
    "\n",
    "- полная задача на собственные значения (нужны все собственные значения и собственные векторы)\n",
    "- частичная задача на собственные значения (требуются минимальное/максимальное собственное значение, или собственные значения из заданного промежутка)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вычисление собственых значений с помощью характеристического многочлена\n",
    "\n",
    "Задача на собственные значения имеет вид\n",
    "\n",
    "$$ Ax = \\lambda x, $$\n",
    "\n",
    "или\n",
    "\n",
    "$$ (A - \\lambda I) x = 0,$$\n",
    "\n",
    "поэтому матрица $A - \\lambda I$ имеет нетривиальное ядро и должна быть вырожденной. Это значит, что её **детерминант** равен нулю: \n",
    "\n",
    "$$ p(\\lambda) = \\det(A - \\lambda I) = 0. $$\n",
    "\n",
    "- Уравнение называется **характеристическим** и является полиномом степени $n$. \n",
    "- Многочлен степени $n$ имеет $n$ комплексных корней!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вспомним, что такое детерминант\n",
    "\n",
    "Детерминант квадратной матрицы определён как\n",
    "\n",
    "$$\\det A = \\sum_{\\sigma \\in S_n} \\mathrm{sgn}({\\sigma})\\prod^n_{i=1} a_{i, \\sigma_i},$$\n",
    "\n",
    "где \n",
    "- $S_n$ множество всех **перестановок** чисел $1, \\ldots, n$\n",
    "- $\\mathrm{sgn}$ обозначает **знак** перестановки ( $(-1)^p$, где $p$ – число требуемых транспозиций)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Свойства детерминанта\n",
    "\n",
    "Детерминант обладает многими полезными свойствами:\n",
    "\n",
    "1. $\\det(AB) = \\det(A) \\det(B)$\n",
    "2. Разложение по минорам: мы можем вычислить детерминант с помощью процедуры разложения по выбранной строке или столбцу. В этом случае сложность вычисления будет экспоненциальна по $n$.\n",
    "\n",
    "Можно ли вычислить детерминант за $\\mathcal{O}(n^3)$? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Собственные значения и характеристическое уравнение\n",
    "\n",
    "Вернёмся к собственным значениям.\n",
    "\n",
    "Характеристическое уравнение можно использовать для вычисления собственных значений, что приводит нас к наивному алгоритму:\n",
    "\n",
    "$$p(\\lambda) = \\det(A - \\lambda I)$$\n",
    "\n",
    "1. Вычислить коэффициенты многочлена\n",
    "2. Найти его корни\n",
    "\n",
    "**Это хорошая идея?**  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Посмотрим на небольшой пример"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Imaginary part')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "n = 40\n",
    "a = [[1.0 / (i - j + 0.5) for i in range(n)] for j in range(n)]\n",
    "a = np.array(a)\n",
    "ev = np.linalg.eigvals(a)\n",
    "#There is a special numpy function for chacteristic polynomial\n",
    "cf = np.poly(a)\n",
    "ev_roots = np.roots(cf)\n",
    "#print('Coefficients of the polynomial:', cf)\n",
    "#print('Polynomial roots:', ev_roots)\n",
    "plt.scatter(ev_roots.real, ev_roots.imag, marker='x', label='roots')\n",
    "b = a + 0 * np.random.randn(n, n)\n",
    "ev_b = np.linalg.eigvals(b)\n",
    "plt.scatter(ev_b.real, ev_b.imag, marker='o', label='Lapack')\n",
    "#plt.scatter(ev_roots.real, ev_roots.imag, marker='o', label='Brute force')\n",
    "plt.legend(loc='best')\n",
    "plt.xlabel('Real part')\n",
    "plt.ylabel('Imaginary part')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Мораль**: \n",
    "\n",
    "- Не делайте так, если только у вас нет серьёзной причины\n",
    "- Поиск корней полинома – очень **плохо обусловленная** задача (задача может быть обусловлена не так плохо, но с использованием другого базиса в пространстве многочленов). Заметим, что матрицей Грама для мономов\n",
    "\n",
    "$$h_{ij} = \\int_0^1 x^i x^j\\, dx = \\frac{1}{i+j+1},$$\n",
    "\n",
    "является матрица Гильберта, у которой сингулярные числа экспоненциально убывают. Таким образом, мономы почти линейно зависимы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Круги Гершгорина\n",
    "\n",
    "- Есть интересная теорема, которая часто помогает локализовать собственные значения. \n",
    "\n",
    "- Она называется **теоремой Гершгорина**.  \n",
    "\n",
    "- Она утверждает, что все собственные значения $\\lambda_i,  i = 1, \\ldots, n$ находятся внутри объединения **кругов Гершгорина** $C_i$, где $C_i$ – окружность на комплексной плоскости с центром в $a_{ii}$ и радиусом\n",
    "\n",
    "$$r_i = \\sum_{j \\ne i} |a_{ij}|.$$\n",
    "\n",
    "Более того, если круги не пересекаются, то они содержат по одному собственному значению внутри каждого круга."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Доказательство \n",
    "\n",
    "Сначала покажем, что если матрица $A$ обладает строгим диагональным преобладанием, то есть \n",
    "\n",
    "$$ |a_{ii}| > \\sum_{j \\ne i} |a_{ij}|, $$\n",
    "\n",
    "тогда такая матрица невырождена."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Разделим диагональную и недиагональную часть и получим\n",
    "\n",
    "$$\n",
    "    A = D + S = D( I + D^{-1}S),\n",
    "$$\n",
    "\n",
    "где $\\Vert D^{-1} S\\Vert_1 < 1$. \n",
    "Поэтому, в силу теоремы о ряде Неймана, матрица $I + D^{-1}S$ обратима и, следовательно, $A$ также обратима."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Теперь докажем утверждение теоремы от противного: \n",
    "\n",
    "- если любое из собственных чисел лежит вне всех кругов, то матрица $(A - \\lambda I)$ обладает свойством строгого диагонального преобладания\n",
    "- поэтому она обратима\n",
    "- это означает, что если $(A - \\lambda I) x = 0$, то $x = 0$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Демо"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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swd/f/77L69mzJ4MGDeL8+fNs3boVgCtXrlCpUiU8PDzYvHkzx+/yOmvWrMnBgwe5desWKSkp/Oc//6FFixbUr1+fc+fOsXPnTpo1a0ZaWhpHjhyhQYMGnDx5kjZt2tCiRQu+++47kpOT8fT0tPIa0pzdnYUmJiaGPXv2UKFCBYKCgggODubtt9+mTp06VKlS5a4/tmwhOTmZxMTErGzz589n5MiRpKamEhgYSHBwsC5ITibfxUdEflVKLQfiABOwG8vwms28/z4MHnz70Fvx4ubH8+HmzZu3FYxOnTpl7W4NUK1aNcaOHUvTpk2pWrUqDRs2pEyZMoB5g/7w4cPx8/PDZDLx2GOP8eWXX963vUceeYRr165RrVq1rOGxXr16ERYWRnBwMP7+/jz88MN/e16NGjV49tln8fPzo27dugQEBABQuHBhli9fzssvv8yVK1cwmUyMGjWKevXq0bt3b65cuYKI8Oqrr+rCo2U5ceIEUVFRREZGsnPnztsKzbhx4wgMDMwaajZKyZIlqVevHvXq1bttZ5ukpKSsQjlv3rysgtSqVSvCwsLo2rUrFSvm6Npmmp2p+w0l2UpwcLDceTG53377jQYNGuR8IYsWmbfxnDhh7vG8/36+tvfkVHJyMiVLlsRkMvHUU0/Rv39/nnrqKZu3aw+5fg80p5SRkUFcXByRkZFERkZy6tQpunTpQnh4OG3atDG80ORXYmIiGzduJDIykk2bNuHr60t4eDjh4eHUr19f94psSCkVKyLBOZrXaYuPQUaPHs2mTZtISUmhQ4cOTJs2zWU+zM7yHmi5l5KSwk8//URkZCRRUVGUKlUq6wu5WbNmWdtiXE1KSgpbtmzJKrTFixfPet3NmzfX2z2tTBcfLU/0e+B64uLiiIiIYOnSpTRu3Jjw8HDCwsKoX7++0dHsTkTYvXt3ViE6efIkL774IkOHDuWhhx4yOp5LyE3xcahzuxlRCDUzve5dR0pKCt9++y3NmjWjW7du1KpVi8OHD7Nt2zZGjx5dIAsPmE+eGxgYyIQJE4iLi2PXrl0opQgNDaVz585ERUWRnp5udMwCw2F6PseOHaNUqVL6sgoGEMv1fK5du6av5+PE/vzzT7766ivmzp1LYGAgw4YNo0uXLnpo6QFu3rzJsmXLmDlzJqdPn2bIkCEMGDCASpUqGR3N6eSm5+Mwn8rq1auTkJDAuXPnjI5SIGVeyVRzLunp6axbt46ZM2cSHR1N37592bFjB3Xq1DE6mtMoVqwYffr0oU+fPsTGxhIREUH9+vXp0qULw4YNo3nz5voHsQ04TM9H07ScExHWrFnD2LFjKVy4MCNGjKBHjx52O+7G1V26dIn58+czY8YMqlevzpQpUwgNDTU6lsNzyh0ONE3Lme3bt/PWW29x6dIlpkyZwhNPPKF/mduIyWRiwYIFTJgwgeDgYN5//329U859OO0OB5qm3du+ffsIDw/n+eefZ+DAgcTHxxMWFqYLjw25u7vTv39/Dh8+TPPmzWnVqhUDBgzg5MmTRkdzerr4aJqDO378OH379qV9+/a0bduWI0eO0LdvX5c9NscRFStWjNGjR3PkyBGqVKmCv78/o0eP5sKFC0ZHc1q6+Giag7p8+TKjRo0iMDCQmjVr8vvvvzNq1KjbTlSr2Zenpyfvv/8++/fv5/r169SvX5/Jkydz69Yto6M5HV18NM0B/fjjj/j6+nLz5k0OHjzIu+++S+nSpY2OpVl4eXkRERHBzp07iY6OJigoCL0dO3ccZldrTdPMvZ1XX32VrVu3smDBAtq0aWN0JO0+6taty8qVK1myZAldu3Zl4MCBjBs3TvdOc0D3fDTNQWT2dooXL87evXt14XESSimee+454uPjOXjwoO4F5ZDu+WiawTJ7O1u2bNG9HSdWpUoVfvjhBxYvXqx7QTmgez6aZqDsvZ19+/bpwuPklFI8//zzxMfHc+DAAd0Lug/d89E0A6SmpvLKK6+wbt065s+fT9u2bY2OpFlRlSpVWLlyZVYvaNSoUbz11lv6mKxsdM9H0+zs7NmztGvXjsTEROLj43XhcVGZvaC4uDhWrlzJc889x43sV18u4KxSfJRSnkqp5UqpQ0qp35RSzayxXE1zNXv27CEkJITWrVuzcuVKvft0AVCtWjW2bt2Kh4cHLVu21GdHsLBWz2casE5EHgYaA79Zabma5jKWL1/O448/zgcffMCkSZMoVEgPPBQUxYoVY8GCBfTs2ZOmTZuyY8cOoyMZLt/bfJRSpYHHgBcBRCQVSM3vcrX8S05OJjk5mbS0NEwmEyaTCaUUHh4euLu7U7hwYcqWLauv92JjGRkZTJw4kXnz5rF+/XoCAwONjqQZQCnFmDFjeOSRR+jWrRtTp06lf//+RscyjDW+dWoD54C5SqnGQCzwiohczz6TUmowMBjA29vbCs0WbGlpaRw8eJADBw6QmJhIYmIiSUlJJCUlZf2dnp5OmTJlcHd3x93dHTc3N0QkqxDdunWLK1euUKFCBby8vPDy8qJq1apZ/9asWZPAwEAqV65s9Mt1WsnJyfTp04czZ84QHR2t16VGly5d2Lp1K08++ST79u3jww8/LJA/APN9SQWlVDCwC3hURH5VSk0DrorI/93rOfqSCrmTWWhiYmKIjY0lNjaW/fv34+3tjZ+fH9WqVbutaGT+W6pUqQfuXWMymTh79mxWwcpexI4ePUpcXBzFixcnKCjotluVKlXs9Oqd1/HjxwkLCyMkJIQZM2bo4z2021y6dIkePXoAsHTpUjw9PQ1OlH92vZ6PUqoKsEtEfCz3WwJviUjXez1HF5/7ExEOHz5MZGQkUVFRxMXF4e3tTXBwcNaXv7+/P6VKlbJLlmPHjmUVvcxb8eLFefzxxwkLC6NDhw6ULFnS5lmcyR9//EG7du145ZVXePXVV/UuttpdmUwmXnnlFXbt2sWGDRsoX7680ZHyxe4Xk1NK/QwMFJHDSqkJQAkRGXOv+XXx+TuTycSOHTuIjIwkMjKSGzduEB4eTlhYGC1atLBLockpEeHo0aOsW7eOyMhIdu3aRYsWLbLyVqtWzeiIhvrtt994/PHHGT9+PIMGDTI6jubgRIQ333yTdevWsWnTJipVqmR0pDwzovj4A3OAwsCfQD8RuXSv+XXxMRMRfvnlF+bMmcOaNWuoWbMm4eHhhIeH4+/v7zS/lq9cucL69euJjIxk7dq11KpVix49etCvXz8qVKhgdDy72rt3L506deKDDz7ghRdeMDqO5iREhAkTJrB06VL+85//ULVqVaMj5Uluig8iYvdbUFCQFGRXr16ViIgI8fX1lXr16sknn3wiJ06cMDqWVaSlpcnmzZulb9++4unpKX369JFdu3ZJRkaG0dFsbs+ePVK5cmX5/vvvjY6iOanJkydL3bp15dSpU0ZHyRMgRnJYB3TxsaP9+/fL8OHDpWzZsvKPf/xDNm3a5NJfyufPn5cPP/xQateuLYGBgTJnzhy5fv260bFs4sCBA1KlShVZunSp0VE0Jzd58mRp0KCBnDlzxugouaaLj4PZtm2btG7dWry8vGTcuHFy8uRJoyPZVXp6uqxdu1bCwsKkfPnyMnbsWLl06ZLRsazmyJEjUq1aNfn222+NjqK5iHHjxomvr6+cP3/e6Ci5oouPg4iPj5euXbuKj4+PzJ8/X1JTU42OZLg///xT+vfvLxUrVpQPP/xQbty4YXSkfDl27Jh4e3vL7NmzjY6iuZCMjAx54403JCgoyKl+qOWm+Ojze9jAsWPHeOGFF+jQoQMdOnTg0KFD9OnTBw8PD6OjGa5WrVp8/fXXbNmyhR07dlCvXj2+/vprTCaT0dFy7cqVK3Tu3JnXX3+dgQMHGh1HcyFKKaZOnUpoaCjPPvusU/7/eBBdfKzo7NmzvPzyywQHB1OnTh1+//13Xn75ZX1w4V00bNiQH374gWXLlvHtt9/i6+vLDz/8YO6OO4H09HSef/552rVrx8svv2x0HM0FKaX47LPPKFSoEGPG3PPIFaeli48ViAgLFy7E19cXpRS//fYb48ePd6hjcxxVaGgomzdv5tNPP2XChAl07dqVhIQEo2M90NixY7l58yaffvqp0VE0F+bu7s6SJUv48ccf+eabb4yOY1W6+ORTUlIS3bp141//+hdr165l2rRpTn2QmBGUUnTq1InY2FhCQ0MJCAhg7ty5DtsLWrhwIcuWLWPZsmV6KFWzOU9PTyIjI3nrrbfYvn270XGsRhefPMrs7fj7++Pn50dMTIw+W3E+eXh4MG7cODZt2sTnn3/ukL2g6OhoXn31VSIjI53+VCia86hfvz7z58/nmWee4cSJE0bHsQpdfPLgzt7OpEmTKFy4sNGxXEbjxo2Jjo4mNDSUwMBAh+kFJSYm8vTTTzNnzhwaNWpkdBytgOncuTOvvfYa3bp1c4krourik0sbN27UvR07yOwFbdy4kS+++ILu3buTnJxsWJ6UlBSeeuophg4dypNPPmlYDq1ge/3112nUqBH9+vVziB9k+aGLTw6JCNOmTaNPnz4sXbpU93bspHHjxuzcuZOyZcvSvHlzjh07ZvcMIsKQIUPw8fFh7Nixdm9f0zIppZg1axZ//fUXkydPNjpOvhS8Kxjlwa1btxg2bBgxMTHs3LkTHx8foyMVKEWKFGH27NlMnz6dZs2asWTJElq3bm239pcsWUJMTAzR0dFOc7JXzXUVLVqUlStXEhAQwOOPP05ISIjRkfJE93we4MyZM7Rt25bLly+zfft2XXgMopRi5MiRLFq0iB49ehAREWGXdk+fPs2oUaOYN28eJUqUsEubmvYgVatWZdq0afTr14+UlBSj4+SJLj73ERcXR0hICB06dGDZsmX6gmkOoF27duzYsYPp06fz0ksvkZaWZrO2RISXXnqJAQMG0KRJE5u1o2l50aNHD+rXr8/EiRONjpInuvjcw3/+8x86derEJ598wvjx4ylUSK8qR/HQQw+xc+dOEhISCA8P5+bNmzZpZ8mSJRw5coTx48fbZPmalh9KKSIiIvjmm2+Ijo42Ok6u6W/Uu1i7di3PPfccK1as4OmnnzY6jnYXpUuXZuXKlZQvX54nnniC69evW3X52Yfb9OmRNEdVuXJlpx1+08XnDqtWreLFF18kMjKSli1bGh1Huw93d3fmz5+Pj48PHTt25OrVq1ZZrh5u05yJsw6/Wa34KKXclFK7lVKrrbVMe1u9ejWDBw/mxx9/JDQ01Og4Wg64ubkxe/ZsfH196dq1q1V6QHq4TXMmzjr8Zs2ezyvAb1Zcnl1t2LCB/v37ExUVRVBQkNFxtFwoVKgQM2bMoG7duvneBqSH2zRn5IzDb1YpPkqp6kBXYI41lmdvv/zyC7169eKHH35w2n3mC7pChQoxe/ZsqlSpQvfu3fN8/ZPRo0fTr18/PdymOZ0ePXpQr149PvzwQ6Oj5IiyxikalFLLgSlAKWC0iDxxl3kGA4MBvL29g44fP57vdq3h+PHjhIaGMm/ePDp27Gh0HKckIly4cIHExETOnj3LrVu3MJlMiAgeHh4ULlyYSpUqUbVqVcqXL2/TPQdNJhNhYWHUr1+fzz77LFfP3bNnD506deL333/Xl8PQnNIff/xBaGgohw4dokKFCnZvXykVKyLBOZo3v8VHKfUE0EVEhimlWnOP4pNdcHCwxMTE5Ktda7h+/TrNmzenX79+jBo1yug4Di81NZUDBw4QGxtLbGwse/fuJSEhgdOnT1OyZEm8vLyoXLkyRYsWxd3dfPIMk8lESkoKZ8+eJSkpiatXr1K5cmWqV6+On58fQUFBBAUF0ahRI6sNc12+fJmmTZvyxhtvMGDAgBw/r0uXLnTu3JmRI0daJYemGWH48OEUKVKETz75xO5t27v4TAFeAExAUaA08IOI9L7Xcxyh+GRkZPDss89SqlQpvvnmG33alLsQEfbu3UtkZCQ//vgj8fHx1KpVK6tgBAQE4O3tTZUqVShatGiOlnnr1i1Onz7NiRMn2LNnT1YhO3r0KI0aNaJz586Eh4cTGBiYr/fk8OHDtGzZkpUrV/Loo48+cP6tW7fSr18/Dh06pM/Zpzm106dP88gjjxAXF0fNmjXt2rZdi88dDbfGSXpxfRkxAAAgAElEQVQ+7777LuvWrWPz5s16w3I2IsK2bdtYtmwZkZGReHh4EB4ezhNPPEHTpk1tdpaHGzduEB0dzerVq4mMjOTGjRuEhYXRvXt32rRpk6ehurVr1zJgwAB27dqFt7f3PecTEZo3b87w4cPp3fuev5k0zWn83//9HydPnmTevHl2bTc3xQcRsdoNaA2sftB8QUFBYqQVK1ZIjRo1JCkpydAcjuTy5cvyxRdfSIMGDaRhw4YyefJkOXDggGRkZBiS59ChQ/Kvf/1L/Pz8pG7duvLJJ5/IxYsXc72cjz76SAICAuT69ev3nGflypXi5+cn6enp+YmsaQ7j8uXLUrFiRdm3b59d2wViJKf1IqczWvNmZPE5cOCAVKhQQWJiYgzL4Ej++OMPGTJkiHh6esqzzz4rW7duNazg3E1GRoZs375devXqJZ6entK/f385dOhQrp7/wgsvyPPPP3/X6SaTSRo0aCCrV6+2VmRNcwgff/yxhIeH27XN3BSfAnWGA5PJRN++fXnvvfcK/LE8p0+fZvjw4TRt2pTKlStz8OBBvv/+ex577DGH2v6llKJ58+YsXLiQw4cPU7t2bVq0aMGgQYNydIltpRRfffUVcXFxLFu27G/TFyxYQIUKFejSpYst4muaYYYNG8aePXvYvn270VHuLqdVypo3o3o+U6ZMkfbt2zvUL3t7u3z5srz99ttSrlw5efXVV+XcuXNGR8q1ixcvyptvvinlypWTMWPGyIULFx74nJ07d0rlypXlzJkzWY/dvHlTatSoIdu3b7dlXE0zzDfffCMtWrSw23ceuufzdwcOHODjjz9mzpw5DvXL3p6ioqJo2LAhCQkJ7N69m08++cSQYwHyq2zZskydOpW9e/dy5coVGjZseNdeTXahoaH06dOH4cOHZz327bff4uvrS/PmzW0dWdMM0adPH86ePcu2bduMjvJ3Oa1S1rzZu+eTlpYmwcHB8uWXX9q1XUdx8eJFeeGFF6R27dqyZcsWo+NY3Y4dO6RevXryzDPPyNmzZ+85382bN+Xhhx+W77//XjIyMsTf31/Wr19vx6SaZn+ff/659OjRwy5toXs+t/voo4/w9PRk8ODBRkexux9//BFfX1/KlCnD3r17adWqldGRrK5Zs2bs2bOHmjVr4uvryw8//HDX+YoWLcrcuXN5+eWX+fHHH0lOTqZ9+/Z2Tqtp9tWnTx/Wr1/P6dOnjY5yu5xWKWve7Nnzydy77a+//rJbm44gPT1dJk6cKNWrV5fNmzcbHcduduzYIT4+PvLWW2+JyWS66zxjxowRb29v+fjjj+2cTtOMMXjwYJk0aZLN20H3fP7n1VdfZdy4cXY/0tdI169f59lnn2Xt2rVER0fTunVroyPZTbNmzYiOjmbnzp08+eSTd73Gz8iRI0lISKB27doGJNQ0+3vppZf46quv8nzCXVtw6eLz008/cfToUYYOHWp0FLs5fvw4jz76KCVLlmTz5s14eXkZHcnuKlasyMaNG6lRowahoaH88ccft03/7rvvaNmyJVOmTMH8Y03TXJu/vz/e3t6sXu04l1tz2eIjIrz11lu89957eHh4GB3HLg4dOsSjjz7KCy+8wNy5c3N8vjVX5OHhQUREBCNHjqRly5bEx8cDkJ6ezpdffsm//vUvUlNTWblypcFJNc0+hg0bxsyZM42O8T85HZ+z5s0e23yWLVsmAQEBBeaUKfv27RMvLy+ZN2+e0VEczvfffy+VKlWSmJgYWb16tTRp0kRERNauXSv169eXtLQ0gxNqmu2lpKRIpUqV5PDhwzZrg4K+zcdkMvH2228zZcoUm147xlEcOnSIDh068PHHH9O3b1+j4zicZ599lq+++oouXbowdepUhg0bBkDHjh3x8vJi/vz5BifUNNsrUqQI/fv358svvzQ6illOq5Q1b7bu+cyaNUvatGlTIM5kcOzYMalevbrMnz/f6CgOLyIiQpRSsmfPnqzHdu3aJdWrV5cbN24YmEzT7OPYsWNSrlw5SU1NtcnyKcg9n5SUFCZOnMjUqVNd/kwGycnJhIeHM3r0aPr06WN0HIcnIjRt2pRnn32Wy5cvA9C0aVNCQkKYMWOGwek0zfZ8fHyoU6cOP//8s9FRXK/4LF++nIYNGxISEmJ0FJvKyMigT58+hISE8PLLLxsdxylERkby+uuv06lTJ3r27El6ejoA77zzDp9//nnWfU1zZWFhYURGRhodw/WKz8yZM287f5ermjhxImfPnmXGjBku38OzhmvXrrF9+/asbWPp6em8+eabAAQEBFCtWjXWrFljcEpNs73w8HAiIyMNP8zApYrP7t27SUhIoGvXrkZHsalVq1Yxb948VqxYoa/CmkMbNmygWbNmlC5dGnd3d77//ntWrVrFd999BzjgbqiaZiO+vr5kZGRw4MABQ3O4VPGJiIhgyJAhuLu7Gx3FZs6fP8/QoUNZvHgxlStXNjqO04iMjCQ8PDzrfrly5Vi6dCmjRo0iMTGRZ555hri4uL8dkKpprkYpRXh4OFFRUYbmyHfxUUrVUEptVkr9ppQ6oJR6xRrBcuvy5cssW7aMAQMGGNG83YwYMYJevXrpywDkgslkYs2aNYSFhd32eEBAAC+99BKDBw+mSJEi9OvXz3F2Q9U0G3KE7T7W6PmYgNdFpAEQCgxXSjW0wnJzZcGCBXTq1IkqVarYu2m7WbFiBbt372bSpElGR3EqO3fupEaNGnh7e/9t2ttvv01CQgLffvstQ4YMYf78+dy8edOAlJpmP61ateK3334z9EzX+S4+IpIkInGWv68BvwHV8rvcXGYgIiIi6+BBV3ThwgVGjBjB3LlzKVasmNFxnEpUVNRtQ27ZFS5cmHnz5jF69GiKFStGSEgIS5cutXNCTbOvwoUL07FjR0N3srHqNh+llA8QAPx6l2mDlVIxSqmYc+fOWbNZDh48yM2bN2nRooVVl+tIJk+eTLdu3fRwWx7cub3nTv7+/vTt25eJEyfywgsvsHz5cjum0zRjZO71ZhRlrd3tlFIlga3A+yJy96t5WQQHB0tMTIxV2gWYMmUKSUlJfP7551ZbpiM5ceIEAQEB7N+/v0CepTo/kpKSaNSoEefPn7/vLukXL16kXr16rFu3jrZt25KUlESJEiXsmFTT7Ovs2bPUq1ePS5cuWe1wDaVUrIgE52Req/R8lFIewApg0YMKjy086Jets5swYQJDhw7VhScPYmNjCQoKeuB/rnLlyvHaa6/x4YcfEhISwqZNm+yUUNOMUalSJUqXLs3Ro0cNad8ae7sp4GvgNxH5JP+Rcuf06dMcOnSIxx57zN5N28XBgwdZvXo1b7zxhtFRnFJm8cmJV155hZ9//pmAgADD9wTSNHsICgoiNjbWkLat0fN5FHgBaKuU2mO5dbHCcnNkzZo1dOjQgcKFC9urSbuaMmUKr7/+OmXKlDE6ilPKTfEpUaIEY8eOJT4+ntWrV+vT7WguLzg42HmLj4j8IiJKRPxExN9y+9Ea4XLClYfczp07x+rVqxk0aJDRUZxWTExMjosPwIsvvkhMTAxly5YlOjrahsk0zXhBQUFYc/t7bjj1GQ5u3LjB5s2b6dy5s9FRbOKbb77hqaeeoly5ckZHcUpJSUncunULHx+fHD+nZMmS9OrViwoVKrBq1SrbhdM0BxAUFERcXJwh53lz6vPQREdH88gjj7jkl3N6ejoRERF6t998yOnOBnd66aWXaNmyJWlpaTZKpmmOoWLFilk7HdSpU8eubTt1zyc2Npbg4Bzt1ed01q1bR6VKlVz29dlDbrb3ZNewYUMaNmxIfHy8LkCayzNqpwOnLj4xMTEu++W8dOlSfYG4fMpr8QEYMGAAhQsX5uDBg1ZOpWmOxaidDpy6+OTny8WRZZ4I01V3pLCX/Hw+unbtSkpKCrt27bJyKk1zLEbtdOC0xefKlSskJiby8MMPGx3F6u53IkwtZ27evMn58+dztbNBdhUrVqRatWqsXr3ausE0zcHUr1/fkANNnbb47N69Gz8/P5e8ds/9ToSp5UxSUhJeXl75Om1Ix44d+e9//2vFVJrmeLy8vDh9+rTd93hz2uLjqkNuAKtXr/7btWe03ElMTMz36YgGDhzI2bNnMZlMVkqlaY6naNGilChRggsXLti1XactPnv27CEwMNDoGFZ35coVTpw4gb+/v9FRnFpSUhJVq1bN1zIyd9P++eefrZRK0xyTl5cXSUlJdm3TaYvPiRMn8jye78ji4uJo3LixSw4n2lPmsFt+KKUoW7YsW7dutVIqTXNMVatWJTEx0a5tOm3xscaXiyNy5eFEe0pMTMx3zwfMvwh3795thUSa5rh0zycXrDGs4oh08bEOa/04qVOnDocPH7ZCIk1zXF5eXrrnkxPXrl0jIyODUqVKGR3F6vbs2UNAQIDRMZyetYqPn58fp06dskIiTXNcVatW1T2fnMjs9Vjr6nuO5OTJky65LcverDXs1qBBA65fv64vr6C5ND3slkPW2I3WEV27dg0Rcckenb1Zq+dTo0YN3NzcOHfunBVSaZpj0sNuOeSq23tcuUdnb1euXLHKBfgy3w97/yrUNHvy9PTk6tWrdm3TKYvPhQsXKF++vNExrM5Ve3T2JiKkp6fj4eGR72VVqFCBjIwMu/8q1DR78vDwsPvB1FYpPkqpTkqpw0qpP5RSb1ljmfeTlpbmepfNXrSIM888Q+WffwYfH1i0yOhEjmXRIvN6KVTogesnPT0dNzc3q/QgPTw8EBFOnz6d72VpFrl4LzX7cHd3t3vxyfeRjEopN2AG8DiQAPxXKRUpIjY7F73JZHKtgzAXLYLBg7l14wbFAI4fh8GDzdN69TIymWOwrB9u3DDff8D6sebnw93dHREhNTXVKssr8HL5Xmr24e7ubvdrV1mj5xMC/CEif4pIKrAEeNIKy70nlys+b78NN25gItuvgRs3zI9rWevnNvdZP9b8fLi5ueniY025fC81+3DWYbdqwMls9xMsj91GKTVYKRWjlIrJ755DRlxv3KZOnADgb6/K8niBd6/1cI/Hrfn5yBy6y8jIsNoyC7RcvpeafSil7P4Zt0bxudvA+t/+94vILBEJFpHgihUr5qtBd3d31zruwnLdHncg/S6PF3j3Wg/3eNyan4/M5RQpUsQqyyvwcvleavaRlpZmlR10csMaxScBqJHtfnXAprsGGTE+aVPvvw/Fi1MYuJX5WPHi5se1rPVzm/usH2t+PkwmE4UKFbL7f0yXlcv3UrMPk8nklMXnv0BdpVQtpVRhoCcQaYXl3pPLFZ9evWDWLCpUqsQ5gJo1YdYsvQE2k2X9ULMmKPXA9ZPZ87HG8FtaWhpKKfLbW9cscvleavZhxHb0fLcmIial1AhgPeAGfCMiB/Kd7D48PT25cuWKLZuwv169qOrvT9Izz8BBm+0o6Lx69crxF5RSCjc3N9LT0/P9H+ry5csUKlTIJQ9qNkwu3kvNPtLS0pyv+ACIyI/Aj9ZYVk4YcSoIe3DV12WE4sWLk5ycjKenZ76Wk/l+6IN/NVeWnJxM8TuHQ23MKc9wYMQZWO2hbNmypKSkcOPOXVG1XLPWiRJPnTqFyWSicuXKVkilaY7JiOujOWXxcdUeglKKqlWrkpCQYHQUp2etHyiHDx+mSJEirndGDU3LxojzZTpl8XHlHoKvry/x8fFGx3B61vqBEh8fr4fcNJdnxHklnbL4KKUMuf6EPQQFBREbG2t0DKdnrc/H77//Tt26da2QSNMcl+755IIuPtr9VK1a1So9n4SEBBo3bmyFRJrmuPQ2n1yoVq0aJ0+efPCMTiaz+LjcKYTszFo/Ti5evMijjz5qhUSa5risdeXf3HDa4uPn5+eS20aqVKlCiRIlOHLkiNFRnJo1tvn8+eefpKWl8fjjj1splaY5Jt3zyYWgoCBiYmKMjmETnTt3Zs2aNUbHcGrW2Nttzpw5eHp62v34B02zp7S0NC5dukSlSpXs2q5TF5+4uDiXHJ4KDw8nMtKmZyhyeZk9n/x8PqKiovT2Hs3lnTlzhgoVKuDm5mbXdp22+FSuXJnixYtz7Ngxo6NYXbt27di9ezcXL140OorTKlWqFEWKFOHMmTN5ev7169c5fPgwXbt2tXIyTXMsf/75JzVr1rR7u05bfMB19wwrVqwYbdq0Ye3atUZHcVpKKQICAoiLi8vT8zdu3EjRokVp0aKFlZNpmmOJjY0lMDDQ7u3q4uOg/vGPf/Ddd98ZHcOpBQcH5/nz8e2335KSkqKH3TSXFxsbS3BwsN3bderiExwc7LI7HXTv3p1ff/3VJYcV7SWvP06SkpLYsGEDdevW1TsbaC4vJiaGoKAgu7fr1MUnNDSU//73v9y8edPoKFZXvHhx+vbty5dffml0FKeV1+Ize/ZsHnnkEVq1amWDVJrmOK5du8bJkydp2LCh3dt26uJTrlw5/P39+emnn4yOYhNDhw5l7ty5pKSkGB3FKdWuXZvk5ORc7XSQlpbGrFmzSE1NJSwszIbpNM14u3fvxtfX1+7X8gEnLz7g2rsl161bl8DAQL7//nujozglpRSBgYG56v2sWrWK6tWrc/ToUdq0aWPDdJpmvNjYWEOG3MBFik9UVBQZGRlGR7GJN998k0mTJpGammp0FKeUm50O0tPTmTBhAqGhobRv356iRYvaOJ2mGcuonQ0gn8VHKfWhUuqQUmqvUmqlUip/l43Mg7p161KmTBmX3eutTZs2PPTQQ8yZM8foKE4pN9t9Fi5ciKenJydPniQ8PNzGyTTNeEbtbAD57/lsBBqJiB9wBPhn/iPlnisPvQFMmTKF9957j+vXrxsdxenktPjcunWL8ePHM3HiRDZt2kSXLl3skE7TjGPkzgaQz+IjIhtExGS5uwuonv9IuZc59OaqAgMDeeyxx/jss8+MjuJ0ateujclk4vfff7/vfBEREfj6+pKWloafnx8VK1a0U0JNM8bWrVtp0qSJITsbgHW3+fQH7nlIvlJqsFIqRikVc+7cOSs2a97l+syZMxw8eNCqy3Uk7733Hp9++qk+7ieXlFKEhYXd98fJqVOnmDx5MlOmTGHx4sU89dRTdkyoacaIjIw0dHj5gcVHKbVJKbX/Lrcns83zNmACFt1rOSIyS0SCRSTY2r8q3dzcGDhwoEsfE1OnTh3efPNN+vfv77I7V9jK/YZlRYQhQ4YwbNgwvLy8iIyMpE+fPnZOqGn2lZGRwerVqw09nOCBxUdE2otIo7vcVgEopfoCTwC9xMBTTA8ePJiFCxeSnJxsVASbe+2110hJSXHpImsL9ztR64IFC0hISGDs2LHMnTuX8PBwKlSoYEBKTbOf2NhYypQpY+gl4vO7t1sn4E0gXERuWCdS3tSoUYNWrVq59PnQ3NzcmDt3LuPGjdPDb7lwrxO1njp1ijFjxjBv3jzc3d2JiIhg2LBhBqXUNPsxesgN8r/NZzpQCtiolNqjlDL0J/mwYcOYMWOGS17jJ9PDDz/MP//5T5577jl95oNcuHPoLS0tjd69ezN8+HD8/f3ZsGEDnp6ehISEGJhS0+zD6YuPiNQRkRoi4m+5DbVWsLxo164dN2/eZOfOnUbGsLnXXnsNb29vhg4d6tKF1pq6du3K+vXrsw7WHTVqFCVKlOCdd94BYObMmQwbNgyllJExNc3mjh8/TlJSEqGhoYbmcPozHGRXqFAhXnrpJWbOnGl0FJtSSjF37lzi4+P59NNPjY7jFCpXrkyDBg3YunUrX375JZs3b+a7777Dzc2Nv/76i+3bt/Pcc88ZHVPTbC4qKoquXbva/cqld3Kp4gPw4osvsnbtWpffJlKiRAlWrVrFRx99xLp164yO4xTCw8OJiIhg/PjxrFq1itKlSwPwySef0K9fP335BK1AiIyMdIiT5rpc8SlbtiwjRoxg/PjxRkexOW9vb5YtW0afPn3YtWuX0XEcXt26dVm1ahXz58/P2svn2LFjfPfdd7zxxhsGp9M027t48SK7du2iQ4cORkdxveID8Prrr7Nhwwb27t1rdBSbe/TRR5k/fz7h4eEue347a9i/fz8jRoygVq1apKenZz0+btw4Ro4cSaVKlQxMp2n2MW/ePMLDwylZsqTRUcwH2dn7FhQUJLb22WefSdeuXW3ejqP497//LRUrVpSdO3caHcXhxMXFSeXKlWXx4sXyzTffSJcuXUREJD4+XipXrixXr141OKGm2V56errUqVNHtm/fbrM2gBjJYR1w2eKTkpIiNWvWlG3bttm8LUexZs0aqVixomzcuNHoKA7j559/lkqVKsny5ctFROTGjRtSoUIFOXr0qHTp0kWmTZtmcEJNs4/169eLv7+/ZGRk2KyN3BQflxx2AyhSpAjvvvsub731VoHZHblLly4sW7aM3r1788UXXxSY130vc+bM4R//+AcLFizg6aefBswHnPbt25d33nmHgwcPMmTIEINTapp9ONzhBDmtUta82aPnIyJiMpmkUaNG8u9//9su7TmKo0ePSqNGjWTgwIFy69Yto+PYXWpqqowcOVLq1asnhw4d+tv0I0eOiLu7u8yZM8eAdJpmf8ePH5dy5cpJcnKyTdtB93zM3Nzc+PTTT3n55Ze5evWq0XHspnbt2uzYsYNz587Rtm1bTp06ZXQkuzlz5gydO3fmyJEj/Prrr9SvX/9v8/z0008UL17csFPJa5q9zZo1i969e1OiRAmjo2Rx6eID0L59ezp27Mjo0aONjmJXpUqV4ocffqBTp04EBAQwb948lx6GExEWL16Mn58fTZs2Zc2aNXh6/v3Cun/99RfvvPMO77//vj5Bq1YgpKamMmfOHF566SWjo9wup10ka97sNeyW6cqVK+Lt7S3r16+3a7uOYs+ePeLv7y9dunSRhIQEo+NY3enTp+Wpp56Shg0bSnR09D3ny8jIkHbt2smUKVPEZDKJt7e3xMbG2jGpptnf4sWLpW3btnZpCz3sdrvSpUsze/ZsBg0aVKCG3zI1btyYX3/9lZCQEAICAvjiiy+yznHmzNLS0vjqq6/w8/Ojfv36xMbG0qRJk3vOP2vWLK5du8bo0aNxc3NjyJAhzJgxw46JNc2+RITp06c75tnac1qlrHmzd88n06BBg2TQoEGGtO0o9u7dK507d5ZatWrJwoULJT093ehIuZaRkSFLly6VunXrStu2bSUmJuaBzzl27JhUqFBBDhw4kPXY+fPnpXz58vL777/bMq6mGWbDhg1St25dSUtLs0t76ON87q6gD79lt2XLFmnatKn4+fnJqlWrxGQyGR3pgdLT0+XHH3+UoKAgCQwMlA0bNuToeZnDbVOnTv3btEmTJknPnj2tHVXTDJeeni6BgYGydOlSu7Wpi899bNiwQapVq+aS2z5yKyMjQ1auXCnBwcFSq1Yt+eCDD+TcuXNGx/qbCxcuyMcffyx16tSRxo0by5IlS3LVY5syZYqEhobe9dfftWvXpEqVKhIXF2fNyJpmuCVLlkhQUJBNDyq9ky4+DzB58mRp0qSJ3Lhxw9AcjiQ6OlpefPFFKVOmjPTu3VvWr18vKSkphuVJTU2VTZs2Sb9+/cTT01N69+4tO3bsyPV/pMjIyAf+2Jg+fbp07Ngxv5E1zWGkpqZKnTp17H62E118HiAjI0N69uwpvXv3tuuvAmdw/vx5+fjjj6VZs2ZSpkwZ6d69uyxYsEDOnz9v87YvXrwoixYtkp49e4qnp6eEhITI1KlT5cyZM3la3oEDB6RixYqya9eu+85369YtqV27tvz00095akfTHE1ERIS0b9/e7u3mpvgo8/z5o5QaDXwIVBSR8w+aPzg4WGJiYvLdbn7cuHGDli1b0rNnT8aMGWNoFkd19uxZ1qxZQ2RkJJs2baJGjRoEBQVl3QICAvJ8dtzr168THx9PTEwMsbGxxMbGcvz4cdq0aUNYWBhPPPEEXl5eec5+8eJFQkJCGDduHH369Hng/N999x3Tpk1j165djnP6EU3Lg+vXr1O3bl0iIyMJDg62a9tKqVgRyVGj+S4+SqkawBzgYSDIWYoPwMmTJwkNDWX27Nl06dLF6DgOLS0tjQMHDmQVipiYGPbt20exYsXw8vKiatWqeHl5UblyZYoWLZp19gCTycStW7c4c+YMSUlJJCYmkpSUxPXr12nUqBFBQUEEBwcTFBREo0aNKFy4cL6zmkwmOnfuTOPGjfnoo49y9JyMjAwCAwMZP348Tz31VL4zaJpRpkyZwu7du1m6dKnd27Z38VkOTAJWAcHOVHwAduzYQbdu3di2bRsPP/yw0XGciohw8eLFrIKSmJjImTNnSE1NxWQyISJ4eHhQuHBhKlWqlFWgvLy8KF++PIUK2eYws1GjRnH48GFWr16dq0sFr127ltdee419+/bpU+9oTunixYvUr1+f7du3U69ePbu3b7fio5QKB9qJyCtKqb+4T/FRSg0GBgN4e3sHHT9+PM/tWtu8efOYMGECW7dupWbNmkbH0fLhgw8+YP78+ezYseOup9e5HxGhTZs29OzZk6FDh9oooabZzquvvsqNGzf46quvDGk/N8XngT/vlFKbgCp3mfQ2MBbI0fVYRWQWMAvMPZ+cPMdeXnzxRa5cuUK7du3YunUr1apVMzqSlgfTpk1j9uzZbNu2LdeFB0ApxfTp02nTpg1du3alRo0aNkipabaxc+dOlixZ4jRXcH5g8RGR9nd7XCnlC9QC4i0baKsDcUqpEBE5bdWUdvDKK69w69Yt2rVrx5YtW6hS5W71VnNUX375JZ999hlbt26latWqeV5Oo0aNGDVqFIMGDWLt2rV65wPNKdy8eZN+/frxxRdfULFiRaPj5EieB91FZJ+IVBIRHxHxARKAQGcsPJneeOMNevfuTevWrQvUZQic3fTp05kyZQqbNm3C29s738t78803OX/+PF9//bUV0mma7Y0bN47GjRvTvT6wlgoAABInSURBVHt3o6PkmN6qeod33nmHIkWK8Nhjj/HTTz/pbUAO7qOPPiIiIoKtW7fi4+NjlWW6u7szd+5c2rZtS4cOHaxS0DTNVnbu3MnChQudZrgtk9V2N7L0gB64p5szGDNmDKNGjaJVq1bs37/f6DjaXWRkZPDOO+8we/ZsqxaeTL6+vlnDb9Y4Fk7TbMEZh9syFYhLKuTFyJEjef/992nTpg2rVq0yOo6WTXJyMt27d2fLli1s27aN6tWr26SdN954Qw+/aQ7NGYfbMulht/vo1asX9erV46mnnmLfvn28/fbbegO0wY4dO8aTTz5JSEgIixcvpkiRIjZry8PDg3nz5unhN80hOetwWybd83mAJk2aEB0dTVRUFD179uT69etGRyqwtmzZQrNmzRg0aBCzZ8+2aeHJlDn8NnDgQDIyMmzenqblxI0bN5x2uC2TLj45ULVqVbZu3UqRIkVo2bIlJ06cMDpSgRMREUGPHj1YtGgRI0eOtGsP9I033uD69eu8++67dmtT0+5FRBgwYAAhISFOOdyWSRefHCpatCjz58+nV69eNG3alJUrVxodqUC4dOkSffr0Yfr06ezYsYN27drZPYOHhwcrVqzgm2++YcWKFXZvX9Oymzp1Kn/88YdhZzGwFl18ckEpxeuvv86yZct44403eP7557lw4YLRsVzWmjVr8PX1pXTp0vz666889NBDhmWpUqUK//73vxk6dCjx8fGG5dAKtqioKGbMmMG///1vihUrZnScfNHFJw9atGhBfHw8lStXxtfXV/eCrOzSpUv07duXkSNH8u233zJ9+vQ8X7rBmgIDA5k+fTpPPvkkZ8+eNTqOVsAcOHCAAQMGsGLFCpc4BZguPnlUvHhxPv30U5YuXap7QVaU2dspVaoUe/fupU2bNkZHuk2PHj3o1asX3bt3JzU11eg4WgFx4cIFnnzyST766COaNm1qdByr0MUnn+7sBc2fP5/09HSjYzmdEydO0Lt3b4fr7dzNpEmT8PT0ZOTIkfoAVM3mTCYTPXr0oFu3bjm6MKKz0MXHCjJ7QStWrGDWrFn4+/sTFRWlv5hy4Pz587z22msEBATg4+PjkL2dOxUqVIiFCxfyyy+/EBERYXQczcW9/vrreHh48MEHHxgdxap08bGiZs2a8csvvzB58mT++c9/0rJlS3755RejYzmk5ORk3nvvPR5++GFSUlLYv38/7733nsP2du5UunRpIiMjmThxIhs3bjQ6juaivvrqK9atW8fixYtzdWFEZ6CLj5UppQgLCyM+Pp5BgwbRq1cvwsLC2Ldvn9HRHEJqaiozZsygbt26HDhwgF27djFz5ky8vLyMjpZrDz30EMuXL+f555/n559/NjqO5mIWLVrEu+++y+rVq/N0fSpHp4uPjbi5udG3b1+OHDlCu3btaN++PZ06dSIyMrJAbhM6deoU48ePx8fHh6ioKNasWcPixYupU6eO0dHypWXLlixevJinn36aX3/91eg4motYvnw5o0ePZuPGjdStW9foODahi4+NFSlShFGjRnH8+HF69+7N1KlTqV27Nu+//z5nzpwxOp5NiQg//fQT3bt3x9fXl/Pnz7NhwwbWrVtHYGCg0fGspn379sydO5fw8HB2795tdBzNyUVFRTF8+HDWrl1Lw4YNjY5jOyJi91tQUJAUZHFxcTJo0CDx9PSUnj17ytatWyU9Pd3oWFZz4cIFmTZtmtSvX18aNWokM2fOlKtXrxody+ZWrFghlStXlri4OKOjaE4qKipKKlasKNHR0UZHyRMgRnJYB3TxMdClS5dk2rRp0qhRI6ny/+3dcWyUdZ7H8fe3ncKFoz2vbFtaruXUgHFTq2WqMYAVKLE9kEGMXlBriMEr2t3TPb1s8EzVqMRoTveaKJ6ckN07rb0tZZdBwFLgpKcJZTt17aCgXch6x80Iu0qsYBdm6Pf+6NBALZQy7fNMZ76vZMIzD9OZT38dng+/55k+z9Sp+uCDD6rf79eTJ0+6HW3Euru79eWXX9Zbb71Vs7KydPny5drW1qZ9fX1uR3NUc3Oz5ubm6t69e92OYsaZjRs3am5urra3t7sd5bKNpHxEXfg4cFlZmXZ0dDj+uons0KFDbNmyBb/fT0dHB/PmzcPn83H77bczdepUt+N9z5kzZ2hvb8fv9+P3+zl+/DhLlizB5/NRUVEx7k/9EY+tW7fywAMPsGnTJubOnet2HDMONDQ08Pjjj7Nt2zZKS0vdjnPZRCSgqmWX9Nh4y0dE/h74MRAFtqrqT4f7Giufizt+/Djbt2/H7/fT0tJCXl4eZWVleL1evF4vpaWlZGZmOpZHVTl8+DCBQOC8W1FR0UDhlJWVkZZmhxDPam1t5b777qOhoYGFCxe6HccksPXr1/PUU0/R0tJCcXGx23Hi4lj5iMh84ElgsaqeEpFcVR32pFdWPpcuEolw4MABAoEAHR0dBAIBgsEghYWFeL1eSkpKmDZtGgUFBeTn51NQUEBWVtaILzkQjUY5evQooVCIcDhMOBw+r3AmT548UH5nb3l5eWP0XSeHtrY27r77bp555hkefvhht+OYBHPmzBmeeOIJmpub2bZtG9dcc43bkeLmZPn8ElinqjtH8nVWPvE5t5A++eQTQqHQeaURjUYHyigrKwuPxzNw6+vrIxqNEo1GOX36NMeOHSMcDvPVV1+Rk5NDfn7+QIlNnz6dWbNm4fV6yc3NdfvbHpcOHTqEz+ejvLyc+vp6JkyY4HYkkwC++eYb7r33Xnp7e2lqamLKlCluRxoVTpbPb4HNQBXwJ+AfVfU3F3hsDVADUFRU5P3iiy8u+3XNxZ04cYJwOEwoFOLEiRMDZROJREhLSyMjIwOPx0NGRgY5OTkUFBSQk5ODx2NXVR8LPT09VFdX09PTQ1NT07i98qQZHZ9//jlLly5l4cKFvPLKK2RkZLgdadSMavmIyE5gqCPeTwJrgN3Ao8CNwH8CV+kwT2ozH5Nq+vr6qKuro6Ghgc2bN1NSUuJ2JOOCHTt2cP/99/Pcc89RU1PjdpxRN5LyGfa/uqp6waOlIvIwsClWNvtEpA/4AfCHSw1rTCpIS0tjzZo1FBcXU1FRwRtvvMGdd97pdizjEFWlvr6eF198kaamJsrLy92O5Lp497P8GlgAvC8iM4EJwB/jTmVMkrrnnnuYOXMmy5YtIxgMUldXZ58STHKnTp3ioYceorOzk7179zJ9+nS3IyWEeN/1G4CrRGQ/0AisGG6XmzGpzuv1sm/fPlpaWli0aBFHjhxxO5IZI8FgkNmzZ/Ptt9/y4YcfWvGcI67yUdXTqlqtqsWqOktVd49WMGOS2dSpU9mzZw9z5syhtLSUDRs22PWfkkgkEuH5559nwYIF1NbW0tTUNG4uF+IUm+8b45KMjAzq6urYtWsXr776qs2CkkQwGOTmm2/mgw8+oLOzk5UrV4749+5SgZWPMS4rKSmhvb2d2bNn2yxoHBs829m+fTuFhYVux0pYVj7GJACbBY1vNtsZOSsfYxLI4FnQunXrUvLig+NFb28vzz77rM12LoOVjzEJ5txZ0FtvvUVJSQmbN2+2XXEJJBqN8uabbzJz5kw+/vhjm+1cBisfYxJUSUkJe/bs4aWXXqKuro45c+bQ1tbmdqyUpqo0NzdTXFzM22+/zcaNG2lubrbZzmWwk3kZk8BEhMWLF1NVVcU777zDihUruPbaa3nhhRe4/vrr3Y6XUnbv3s3q1auJRCLU19dz22232UwnDjbzMWYcSE9Pp7q6moMHD1JVVUVlZSXV1dUcPnzY7WhJr7Ozk8rKSmpqanjssccIBAJUVlZa8cTJyseYcWTixIk88sgjdHd3M2PGDG688UZqamro6upyO1pSUVXa2tq46667WLx4MUuXLuXTTz9l+fLldjqkUWKjaMw4lJmZydNPP83BgwcpLCxk0aJFzJ07l4aGBk6dOuV2vHGrp6eHtWvXct1117Fq1SrKy8vp7u6mtrbWrsU0yuK+jPblsEsqGDO6otEoW7ZsYe3atXR1dbFy5UpWrVpl5xK7RMFgkNdff53GxkYqKiqora1l3rx5tmtthEZySQWb+RiTBDweD8uWLaO1tZW2tjZ6e3uZNWsWPp+P9957j76+PrcjJpzTp0/T2NhIeXk5VVVV5OXlEQwGaWpqYv78+VY8Y8xmPsYkqZMnT9LY2Mhrr73G119/zR133IHP5+OWW25JqqtnjsR3333Hzp078fv9bNmyheLiYmpra/H5fCk7JqPJsctoXy4rH2Oco6p0dXXh9/vx+/0cOnSIqqoqfD4fVVVVXHHFFW5HHFNffvkl7777Ln6/n/fffx+v14vP58Pn83H11Ve7HS+pWPkYYy4oFAoNbIzb2tq46aab8Pl8LFmyhCuvvNLteHFTVfbv3z8wu/nss8+orKwcKNvs7Gy3IyYtKx9jzCU5efIkra2t+P1+tm7dyqRJk/B6vXi9XsrKyvB6vQm/sQ6FQgQCAQKBAB0dHQQCASZOnDgwuykvL7dPqjnEyscYM2J9fX10d3eftyH/6KOPmDJlynll5GYhDVU0kUjke/mKiorsAwMucKx8ROQG4F+BPwOiQK2q7hvu66x8jBkfBhdSIBCgs7OT9PR08vPzyc/Pp6Cg4ILLkyZNGrYEVJWenh7C4TChUIhwOHze8rnrJk6caEWTwJwsnx3Az1R1u4gsAn6qqvOG+zorH2PGL1Xl+PHjQxbF4OXe3l48Hg8ZGRl4PB48nv7TSUYiEaLR6MBt8uTJQ5bY4HWZmZlWNAlsJOUT74lFFciKLf8FEIrz+YwxCU5EyM7OJjs7m+Li4os+tq+vb6BgIpEIkUiEtLS0gSLKyMggPT3dTlmTguItn58ALSLyz/T/wursCz1QRGqAGoCioqI4X9YYMx6kpaUxYcIEO+BvvmfY8hGRncDUIf7qSaAC+AdVbRaRvwXWAwuHeh5VXQesg/7dbped2BhjzLg3bPmo6pBlAiAi/w48GrvbBLw5SrmMMcYksXh3tIaAW2PLC4DuOJ/PGGNMCoj3mM/fAfUi4gH+ROyYjjHGGHMxcZWPqn4AeEcpizHGmBRhn280xhjjOCsfY4wxjrPyMcYY4zgrH2OMMY6z8jHGGOM4Kx9jjDGOs/IxxhjjOFcuJicifwC+cPyFh/cD4I9uh3CZjYGNAdgYgI0BjHwMpqtqzqU80JXySVQi0nGp16JIVjYGNgZgYwA2BjC2Y2C73YwxxjjOyscYY4zjrHzOt87tAAnAxsDGAGwMwMYAxnAM7JiPMcYYx9nMxxhjjOOsfIwxxjjOymcQEXlGRP5PRH4buy1yO5NTRKRKRD4Tkd+JyGq387hBRH4vIsHYz77D7TxOEJENInJMRPafsy5bRFpFpDv251+6mXGsXWAMUmZbICKFIvJfInJARD4RkUdj68fsfWDlM7SfqeoNsds2t8M4QUTSgdeAvwF+CNwjIj90N5Vr5sd+9qnyOx4/B6oGrVsN7FLVGcCu2P1k9nO+PwaQOtuCKPC4ql4L3Az8KPbvf8zeB1Y+5qybgN+p6mFVPQ00AktdzmQcoKptwNeDVi8FfhFb/gVwh6OhHHaBMUgZqhpW1c7Y8rfAAWAaY/g+sPIZ2o9FpCs2FU/q3Q3nmAb87zn3j8TWpRoFdohIQERq3A7jojxVDUP/hgnIdTmPW1JuWyAifw2UAu2M4fsgJctHRHaKyP4hbkuB14GrgRuAMPCyq2GdI0OsS8XP4c9R1Vn07378kYiUux3IuCbltgUiMhloBn6iqj1j+VqesXzyRKWqCy/lcSLyb8C7YxwnURwBCs+5/1dAyKUsrlHVUOzPYyLyK/p3R7a5m8oVR0UkX1XDIpIPHHM7kNNU9ejZ5VTYFohIBv3F87aqboqtHrP3QUrOfC4mNsBnLQP2X+ixSeY3wAwRuVJEJgDLAb/LmRwlIn8uIplnl4HbSJ2f/2B+YEVseQWw2cUsrkilbYGICLAeOKCqr5zzV2P2PrAzHAwiIv9B/zRbgd8Dq87u80x2sY+S/guQDmxQ1TUuR3KUiFwF/Cp21wM0pMIYiMg7wDz6T59/FHga+DXwS6AI+B/gblVN2gPyFxiDeaTItkBE5gL/DQSBvtjqf6L/uM+YvA+sfIwxxjjOdrsZY4xxnJWPMcYYx1n5GGOMcZyVjzHGGMdZ+RhjjHGclY8xxhjHWfkYY4xx3P8DulTE0tr/CX8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "n = 3\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "a = [[5, 1, 1], [1, 0, 0.5], [2, 0, 10]]\n",
    "#a = [[1.0 / (i - j + 0.5) for i in xrange(n)] for j in xrange(n)]\n",
    "a = np.array(a)\n",
    "#a = np.diag(np.arange(n))\n",
    "a = a + 2 * np.random.randn(n, n)\n",
    "#u = np.random.randn(n, n)\n",
    "#a = np.linalg.inv(u).dot(a).dot(u)\n",
    "xg = np.diag(a).real\n",
    "yg = np.diag(a).imag\n",
    "rg = np.zeros(n)\n",
    "ev = np.linalg.eigvals(a)\n",
    "for i in range(n):\n",
    "    rg[i] = np.sum(np.abs(a[i, :])) - np.abs(a[i, i])\n",
    "    crc = plt.Circle((xg[i], yg[i]), radius=rg[i], fill=False)\n",
    "    ax.add_patch(crc)\n",
    "plt.scatter(ev.real, ev.imag, color='r', label=\"Eigenvalues\")\n",
    "plt.axis('equal')\n",
    "plt.legend()\n",
    "ax.set_title('Eigenvalues and Gershgorin circles')\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Замечание**: Существуют более сложные фигуры, под названием **[овалы Cassini](https://en.wikipedia.org/wiki/Cassini_oval)**, которые содержат спектр\n",
    "\n",
    "$$\n",
    "    M_{ij} = \\{z\\in\\mathbb{C}: |a_{ii} - z|\\cdot |a_{jj} - z|\\leq r_i r_j\\}, \\quad r_i = \\sum_{l\\not= i} |a_{il}|.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Степенной метод\n",
    "\n",
    "- Часто в вычислительной практике требуется найти не весь спектр, а только некоторую его часть, например самое большое или самое маленькое собственые значения. \n",
    "- Также отметим, что для Эрмитовых матриц $(A = A^*)$ собственные значения всегда действительны (докажите!).\n",
    "- Степенной метод – простейший метод вычисления **максимального по модулю** собственного значения. Это также первый пример **итерационного метода** и **Крыловского метода**.\n",
    "-  Другие примеры будут далее в курсе"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Степенной метод: детали\n",
    "\n",
    "Задача на собственые значения\n",
    "\n",
    "$$Ax = \\lambda x, \\quad \\Vert x \\Vert_2 = 1 \\ \\text{для устойчивости}.$$ \n",
    "\n",
    "может быть записана как итерации с неподвижной точкой, которые называются **степенным методом** и дают максимальное по модулю собственное значение матрицы $A$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Степенной метод имеет вид\n",
    "\n",
    "$$ x_{k+1} = A x_k, \\quad x_{k+1} := \\frac{x_{k+1}}{\\Vert x_{k+1} \\Vert_2}.$$\n",
    "\n",
    "и $x_{k+1}\\to v_1$, где $Av_1 = \\lambda_1 v_1$ и $\\lambda_1$ максимальное по модулю собственное значение, и $v_1$ – соответствующий собственный вектор.\n",
    "\n",
    "На $(k+1)$-ой итерации приближение для $\\lambda_1$ может быть найдено следующим образом\n",
    "\n",
    "$$ \\lambda^{(k+1)} = (Ax_{k+1}, x_{k+1}), $$\n",
    "\n",
    "Заметим, что $\\lambda^{(k+1)}$ не требуется для $(k+2)$-ой итерации, но может быть полезно для оценки ошибки на каждой итерации: $\\|Ax_{k+1} - \\lambda^{(k+1)}x_{k+1}\\|$. \n",
    "\n",
    "Метод сходится со скоростью геометричекой прогрессии, с константой $q = \\left|\\frac{\\lambda_{2}}{\\lambda_{1}}\\right| < 1$, где $\\lambda_1>\\lambda_2\\geq\\dots\\geq \\lambda_n$. \n",
    "Это означает, что сходимость может быть сколь угодно медленной при близких значениях у $\\lambda_1$ и $\\lambda_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Анализ сходимости для $A=A^*$\n",
    "\n",
    "- Рассмотрим степенной метод более подробно для случая эрмитовой матрицы \n",
    "- Через несколько слайдов вы увидите, что любая эрмитова матрица диагонализуема, поэтому существует ортонормированный базис из собственных векторов $v_1,\\dots,v_n$ такой что $Av_i = \\lambda_i v_i$. \n",
    "- Разложим $x_0$ в этом базисе с коэффициентами $c_i$:\n",
    "\n",
    "$$ x_0 = c_1 v_1 + \\dots + c_n v_n. $$\n",
    "\n",
    "- Поскольку $v_i$ – собственные векторы, выполнены следующие равенства\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "x_1 &= \\frac{Ax_0}{\\|Ax_0\\|} = \\frac{c_1 \\lambda_1 v_1 + \\dots + c_n \\lambda_n v_n}{\\|c_1 \\lambda_1 v_1 + \\dots + c_n \\lambda_n v_n \\|}  \\\\\n",
    "&\\vdots\\\\\n",
    "x_k &= \\frac{Ax_{k-1}}{\\|Ax_{k-1}\\|} = \\frac{c_1 \\lambda_1^k v_1 + \\dots + c_n \\lambda_n^k v_n}{\\|c_1 \\lambda_1^k v_1 + \\dots + c_n \\lambda_n^k v_n \\|}\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "- Получаем следующее выражение \n",
    "\n",
    "$$\n",
    "x_k = \\frac{c_1}{|c_1|}\\left(\\frac{\\lambda_1}{|\\lambda_1|}\\right)^k\\frac{ v_1 + \\frac{c_2}{c_1}\\frac{\\lambda_2^k}{\\lambda_1^k}v_2 + \\dots + \\frac{c_n}{c_1}\\frac{\\lambda_n^k}{\\lambda_1^k}v_n}{\\left\\|v_1 + \\frac{c_2}{c_1}\\frac{\\lambda_2^k}{\\lambda_1^k}v_2 + \\dots + \\frac{c_n}{c_1}\\frac{\\lambda_n^k}{\\lambda_1^k}v_n\\right\\|},\n",
    "$$\n",
    "\n",
    "которое сходится к $v_1$ при $\\left| \\frac{c_1}{|c_1|}\\left(\\frac{\\lambda_1}{|\\lambda_1|}\\right)^k\\right| = 1$ и $\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^k \\to 0$ если $|\\lambda_2|<|\\lambda_1|$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Что необходимо помнить о степенном методе\n",
    "\n",
    "- Степенной метод даёт оценку для максимального по модулю собственного числа или спектрального радиуса матрицы\n",
    "- Одна итерация требует одного умножения матрицы на вектор. Если можно умножить вектор на матрицу за $\\mathcal{O}(n)$ (напрмиер, она разреженная), тогда степенной метод можно использовать для больших $n$\n",
    "- Сходимость может быть медленной\n",
    "- Для грубой оценки максимального по модулю собственного значения и соответствующего вектора достаточно небольшого числа итераций\n",
    "- Вектор решения лежит в **Крыловском подпространстве** $\\{x_0, Ax_0,\\dots,A^{k}x_0\\}$ и имеет вид $\\mu A^k x_0$, где $\\mu$ нормировочная постоянная. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## А как найти весь спектр?\n",
    "\n",
    "**Используем матричное разложение!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##  Для какой матрицы легко найти весь спектр?\n",
    "\n",
    "Существует класс матриц, для которого собственные числа можно найти очень легко, – это **треугольные матрицы**\n",
    "\n",
    "$$\n",
    "  A = \\begin{pmatrix}\n",
    "   \\lambda_1 & * & * \\\\\n",
    "   0 & \\lambda_2 & * \\\\\n",
    "   0 & 0 & \\lambda_3 \\\\\n",
    "  \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "Собственные числа матрицы $A$ – $\\lambda_1, \\lambda_2, \\lambda_3$. Почему? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Потому что детерминант имеет вид \n",
    "\n",
    "$$ \\det(A - \\lambda I) = (\\lambda - \\lambda_1) (\\lambda - \\lambda_2) (\\lambda - \\lambda_3). $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Таким образом, вычисление собственных значений для треугольной матрицы – простая задача. Теперь на помощь приходят унитарные матрицы. Пусть $U$ унитарная матрица, то есть $U^* U = I$.  Тогда выполнены следующие равенства\n",
    "\n",
    "$$ \\det(A - \\lambda I) = \\det(U (U^* A U - \\lambda I) U^*) = \\det(UU^*) \\det(U^* A U - \\lambda I) = \\det(U^* A U - \\lambda I), $$\n",
    "\n",
    "где мы используем свойства детерминанта от произведения матриц, $\\det(AB) = \\det(A) \\det(B)$. \n",
    "Это означает, что матрицы $U^* A U$ и $A$ имеют одинаковые характеристические многочлены, и, следовательно, одинаковые собственные значения. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Если мы приведём матрицу $A$ к верхнетреугольному виду $T$ с помощью унитарной матрицы $U$: $U^* A U = T$, мы решили задачу! \n",
    "Умножим слева и справа на $U$ и $U^*$ соответственно, получим нужное нам разложение:  \n",
    "\n",
    "$$ A = U T U^*. $$\n",
    "\n",
    "- Это знаменитое **разложение Шура**. \n",
    "- Напомним, что использование унитарных матриц приводит к устойчивым алгоритмам, таким образом обственные значения вычисляются очень точно. \n",
    "- Разложение Шура показывает, почему нам нужны матричные разложения: они представляют матрицу в виде произведения трёх матриц подходящей структуры.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Теорема Шура\n",
    "\n",
    "**Теорема:** Каждая матрица $A \\in \\mathbb{C}^{n \\times n}$ может быть представлена в виде формы Шура $A = UTU^*$, где $U$ унитарная, а $T$ верхнетреугольная.\n",
    "\n",
    "**Набросок доказательства**.\n",
    "1. Каждая матрица имеет как минимум один ненулевой собственный вектор (для корня характеристического многочлена матрица $(A-\\lambda I)$ вырождена и имеет нетривиальное ядро). Пусть \n",
    "\n",
    "$$Av_1 = \\lambda_1 v_1, \\quad \\Vert v_1 \\Vert_2 = 1.$$\n",
    "\n",
    "2. Пусть $U_1 = [v_1,v_2,\\dots,v_n]$, где $v_2,\\dots, v_n$ любые векторы ортогональные $v_1$. Тогда \n",
    "  \n",
    "  $$\n",
    "      U^*_1 A U_1 = \\begin{pmatrix}\n",
    "      \\lambda_1 & *  \\\\\n",
    "      0 & A_2 \n",
    "      \\end{pmatrix},\n",
    "  $$\n",
    "  \n",
    "  где $A_2$ матрица размера $(n-1) \\times (n-1)$. Это называется **блочнотреугольной формой**. Теперь мы можем проделать аналогичную процедуру для матрицы $A_2$ и так далее.  \n",
    "  \n",
    "  \n",
    "**Замечание**: Поскольку в доказательстве необходимы собственные векторы, оно не является практичным алгоритмом."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Приложение теоремы Шура\n",
    "\n",
    "Важное приложение теоремы Шура связано с так называемыми **нормальными матрицами**.  \n",
    "\n",
    "**Определение.** Матрица $A$ называется **нормальной матрицей**, если  \n",
    "\n",
    "$$ AA^* = A^* A. $$\n",
    "\n",
    "**Q:** Какие примеры нормальных матриц вы можете привести?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Примеры: эрмитовы матрицы, унитарные матрицы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Нормальные матрицы\n",
    "\n",
    "**Теорема**: $A$ – **нормальная матрица**, тогда и только тогда, когда $A = U \\Lambda U^*$, где $U$ унитарна и $\\Lambda$ диагональна. \n",
    "\n",
    "**Набросок доказательства:** В одну сторону доказательство очевидно (если разложение существует, необходимо проверить выполнение определения нормальной матрицы).  \n",
    "В другую сторону доказательство более сложное. Рассмотрим форму Шура для матрицы $A$. Тогда $AA^* = A^*A$ означает, что $TT^* = T^* T$.  \n",
    "Сравнив поэлементно матрицы в левой и правой части, сразу становится видно, что единственная верхнетреугольная матрица, для которой такое равенство будет выполнено – это диагональная матрица!\n",
    "\n",
    "#### Важное следствие\n",
    "\n",
    "Любая нормальная матрица – **унитарно диагонализуема**. Это означает, что она может быть приведена к диагональному виду с помощью унитарной матрицы $U$. Другими словами, каждая нормальная матрица имеет ортогональный базис из собственных векторов."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как вычислить разложение Шура?\n",
    "\n",
    "- Узнаем это через несколько слайдов :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вариационный принцип для собственных значений\n",
    "\n",
    "- Во многих задачах необходимо найти максимальный или минимальный собственный вектор и соответствующее ему значение\n",
    "- Тогда, если $A$ эрмитова матрица, **отношение Релея** определяется как\n",
    "\n",
    "$$R_A(x) = \\frac{(Ax, x)}{(x, x)},$$\n",
    "\n",
    "и максимальное собственное значение равно максимальному значению $R_A(x)$, аналогично для минимального собственного значения. \n",
    "\n",
    "- Таким образом, мы можем использовать методы оптимизации для поиска этих экстремальных собственных значений.\n",
    "\n",
    "А теперь приведём понятие, которое является обобщением собственных чисел"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Спектр и псевдоспектр\n",
    "\n",
    "- Для динамических систем с матрицей $A$, спектр может много сообщить о поведении системы (например, о её устойчивости)\n",
    "\n",
    "- Однако для **не нормальных матриц**, спектр может быть неустойчивым относительно малых возмущений матрицы\n",
    "\n",
    "- Для измерения подобных возмущений было разработана концепция **псевдоспектра**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Псевдоспектр\n",
    "\n",
    "Рассмотрим объединение всех возможных собственных значений для всевозможных возмущений матрицы $A$.\n",
    "\n",
    "$$\\Lambda_{\\epsilon}(A) = \\{ \\lambda \\in \\mathbb{C}: \\exists E, x \\ne 0: (A + E) x = \\lambda x, \\quad \\Vert E \\Vert_2 \\leq \\epsilon. \\}$$\n",
    "\n",
    "Для малых $E$ и нормальных $A$ это круги вокруг собственных значений, для не нормальных матриц, структура может сильно отличаться. Подробности можно найти тут: http://www.cs.ox.ac.uk/pseudospectra/\n",
    "\n",
    "<img src='./pseudospectrum.gif'>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Возвращаемся к вычислению разложения Шура\n",
    "\n",
    "- Нужно найти унитарную матрицу $U$ и верхнетреугольную матрицу $T$, такие что для данной матрице $A$ выполнено\n",
    "\n",
    "$$ A = U T U^*. $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## QR алгоритм\n",
    "\n",
    "- QR алгоритм был предложен в 1961 г. независимо В. Н. Кублановской и J. Francis'ом. Статью про историю этого алгоритма и его авторов можно прочитать [тут](http://www.dm.unibo.it/~guerrini/html/an_09_10/QR_50_years_later.pdf).   \n",
    "\n",
    "- <font color='red'> **Не путайте** QR алгоритм и QR разложение! </font>\n",
    "\n",
    "- QR разложение – это представление матрицы в виде произведения двух матриц, а QR алгоритм использует QR разложение для вычисления разложения Шура.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##  Путь к QR алгоритму\n",
    "\n",
    "Рассмотрим выражение\n",
    "\n",
    "$$A = Q T Q^*,$$\n",
    "\n",
    "и перепишем его в виде\n",
    "\n",
    "$$\n",
    "   Q T = A Q.\n",
    "$$\n",
    "\n",
    "Слева замечаем QR разложение матрицы $AQ$.\n",
    "\n",
    "Используем его чтобы записать одну итерацию метода неподвижной точки для разложения Шура."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вывод QR алгоритма из уравнения неподвижной точки\n",
    "\n",
    "Запишем следующий итерационный процесс \n",
    "\n",
    "$$\n",
    "    Q_{k+1} R_{k+1} = A Q_k, \\quad Q_{k+1}^* A = R_{k+1} Q^*_k\n",
    "$$\n",
    "\n",
    "Введём новую матрицу\n",
    "\n",
    "$$A_k = Q^* _k A Q_k = Q^*_k Q_{k+1} R_{k+1} = \\widehat{Q}_k R_{k+1}$$\n",
    "\n",
    "тогда аппроксимация для $A_{k+1}$ имеет вид\n",
    "\n",
    "$$A_{k+1} = Q^*_{k+1} A Q_{k+1} = ( Q_{k+1}^* A = R_{k+1} Q^*_k)  = R_{k+1} \\widehat{Q}_k.$$\n",
    "\n",
    "Итак, мы получили стандартную форму записи QR алгоритма."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Финальные формулы обычно записывают в **QRRQ**-форме:\n",
    "\n",
    "1. Инициализируем $A_0 = A$.\n",
    "2. Вычислим QR разложение матрицы $A_k$: $A_k = Q_k R_k$.\n",
    "3. Обновим аппроксимацию $A_{k+1} = R_k Q_k$.\n",
    "\n",
    "Продолжаем итерации пока $A_k$ не станет достаточно треугольной (например, норма подматрицы под главной диагональю не станет достаточно мала)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Что известно о сходимости и сложности\n",
    "\n",
    "**Утверждение**\n",
    "\n",
    "Матрицы $A_k$ унитарно подобны матрице $A$\n",
    "\n",
    "$$A_k = Q^*_{k-1} A_{k-1} Q_{k-1} = (Q_{k-1} \\ldots Q_1)^* A (Q_{k-1} \\ldots Q_1)$$\n",
    "\n",
    "а произведение унитарных матриц – унитарная матрица.\n",
    "\n",
    "Сложность одной итерации $\\mathcal{O}(n^3)$, если используется QR разложение для общего случая.\n",
    "\n",
    "Мы ожидаем, что $A_k$ будет **очень близка к треугольной матрице** для достаточно большого $k$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Leading 3x3 block of a:\n",
      "[[ 2.41052440e+000 -5.41127562e-017 -4.10963400e-017]\n",
      " [ 2.42500623e-168  3.49984625e-001  5.13648922e-017]\n",
      " [ 0.00000000e+000  6.56745067e-273  1.53236733e-002]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "n = 4\n",
    "a = [[1.0/(i + j + 0.5) for i in range(n)] for j in range(n)]\n",
    "niters = 200\n",
    "for k in range(niters):\n",
    "    q, rmat = np.linalg.qr(a)\n",
    "    a = rmat.dot(q)\n",
    "print('Leading 3x3 block of a:')\n",
    "print(a[:3, :3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сходимость и сложность QR алгоритма\n",
    "\n",
    "- QR алгоритм сходится от первого диагонального элемента к последнему.\n",
    "\n",
    "- По крайней мере 2-3 итерации необходимо для определения каждого диагонального элемента матрицы $T$. \n",
    "\n",
    "- Каждый шаг состоит в вычислении QR разложения и одного произведения двух матриц, в результате имеем сложность $\\mathcal{O}(n^3)$.\n",
    "\n",
    "**Q**: означает ли это итоговую сложность $\\mathcal{O}(n^4)$? \n",
    "\n",
    "**A**: к счастью, нет! \n",
    "\n",
    "- Мы можем ускорить QR алгоритм, используя сдвиги, поскольку матрица $A_k - \\lambda I$ имеет те же векторы Шура (столбцы матрицы $U$).\n",
    "\n",
    "- Подробности в следующей лекции"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Выводы по способам вычисления разложений\n",
    "\n",
    "- LU и QR разложения можно вычислить с помощью **прямых** методов за конечное число операций.\n",
    "\n",
    "- SVD и разложение Шура не могут быть вычислены с помощью прямых методов (почему?). Они могут быть получены с помощью **итерационных методов**\n",
    "\n",
    "- Однако итерационные методы также имеют сложность $\\mathcal{O}(n^3)$ операций благодаря быстрой сходимости."
   ]
  },
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    "## Резюме по сегодняшней лекции\n",
    "\n",
    "- Собственные значения и векторы\n",
    "- Теорема Гершгорина\n",
    "- Степенной метод\n",
    "- Теорема Шура\n",
    "- Нормальные матрицы\n",
    "- QR алгоритм"
   ]
  },
  {
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   "metadata": {
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   "source": [
    "## Анонс следующей лекции\n",
    "\n",
    "- Ускорение сходимости QR алгоритма\n",
    "- Методы вычисления SVD"
   ]
  }
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