EPAL-v1/ex.I.6.26.ipynb

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    "<table style=\"width:100%\">\n",
    "    <tr>\n",
    "        <td>\n",
    "            <a href=\"http://v-v-kisil.scienceontheweb.net/MoebInv-notebooks/EPAL-v1/ex.I.6.26.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.6.26.ipynb\">\n",
    "  <img src=\"../svg/open-github.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "            <a href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.6.26.ipynb\">\n",
    "  <img src=\"../svg/exec-colab.svg\" alt=\"Open In Colab\"/>\n",
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    "        <td>\n",
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    "\n",
    "<a  align=\"center\" href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/Introduction/Software_installation_GUI_integration.ipynb\">Google CoLab software installation</a>\n"
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    "# Check properties of reflection with respect to orthogonality\n",
    "**Exercise I.6.26** [ [1] ](*[1])\n",
    "$\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
    "  \\newcommand{\\rs}{\\sigma_r}\n",
    "  \\newcommand{\\lvec}[1]{\\overrightarrow{#1}}\n",
    "$\n",
    "  Let $\\det \\cycle[\\tilde]{s}{\\bs}\\neq 0$. Then, for\n",
    "  cycle similarity with $\\cycle[\\tilde]{s}{\\bs}$:  \n",
    "+ Verify the identities\n",
    "$-\\cycle[\\tilde]{s_1}{\\bs}    \\overline   {\\cycle[\\tilde]{s}{\\bs}}\\cycle[\\tilde]{s_1}{\\bs}\n",
    "    ={ \\det_{\\bs}}(\\cycle[\\tilde]{s_1}{\\bs})\\cdot\n",
    "    \\cycle[\\tilde]{s_1}{\\bs}\n",
    "$ and $    -\\cycle[\\tilde]{}{\\bs}\\overline\n",
    "    {\\cycle{}{\\bs}}\\cycle[\\tilde]{}{\\bs}\n",
    "    =-{\\det_{\\bs}}(\\cycle[\\tilde]{}{\\bs})\\cdot\\cycle{}{\\bs},\n",
    "$\n",
    "    where $\\cycle{s}{\\bs}$  is  a cycle $\\bs$-orthogonal to\n",
    "    $\\strut \\cycle[\\tilde]{}{\\bs}$.  Note the difference in the signs in\n",
    "    the right-hand sides of both identities.\n",
    "+ Describe all cycles which are fixed (as points in the\n",
    "    projective space $\\Space{P}{3}$) by the similarity with the given\n",
    "    cycle $\\cycle[\\tilde]{s}{\\bs}$.\n",
    "\n",
    "**Hint.**\n",
    "      Use a decomposition of a generic cycle into a sum\n",
    "      $\\cycle[\\tilde]{s}{\\bs}$ and a cycle orthogonal to\n",
    "      $\\cycle[\\tilde]{s}{\\bs}$."
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    "**Solution.** The direct check:"
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    "from init_cycle import *\n",
    "print(\"C conjugated with itself is C multiplied by det(C): %s\" % \\\n",
    "C.cycle_similarity(C,e,sign_mat1,sign_mat1,pow(sign_mat1,-1).evalm())\\\n",
    ".subs(pow(s1,2)==1,subs_options.algebraic)\\\n",
    ".is_equal(C.exmul(C.hdet(e,sign_mat1)).subs(pow(s1,2)==1,subs_options.algebraic),False))"
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    "A similar check for orthogonal cycles"
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    "Cn=C1.subject_to(C1.is_orthogonal(C,e))\n",
    "print(\"A cycle orthogonal to C conjugated with C is itself times -det(C): %s \" % \\\n",
    "Cn.cycle_similarity(C,e).is_equal(Cn.exmul(-C.hdet(e)),False))"
   ]
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    "\n",
    "This notebook is a part of the *MoebInv notebooks* project [ [2] ](#[2]).\n",
    "\n",
    "### References\n",
    "<a id=\"references\"></a>\n",
    "<a id=\"[1]\"></a>\n",
    "1. Vladimir V. Kisil. *Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\\mathbb{R})$*. [Imperial College Press](https://www.worldscientific.com/worldscibooks/10.1142/p835), London, 2012. Includes a live DVD.\n",
    "\n",
    "   <a id=\"[2]\"></a>\n",
    "2.  Vladimir V. Kisil, [*MoebInv notebooks*](https://github.com/vvkisil/MoebInv-notebooks), 2019.\n",
    "\n",
    "[Back to Folder](https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/)\n"
   ]
  }
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