<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>mPosHullFacesAndDuals -- Faces and their duals of positive hull.</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_read__Convex__Hull__Faces.html">next</a> | <a href="_m__Pos__Hull__Faces.html">previous</a> | <a href="_read__Convex__Hull__Faces.html">forward</a> | <a href="_m__Pos__Hull__Faces.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>mPosHullFacesAndDuals -- Faces and their duals of positive hull.</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>mPosHullFacesAndDuals(L)</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of <a href="../../Macaulay2Doc/html/___Vector.html" title="the class of all elements of free modules that are handled by the engine">Vector</a>s</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li> </ul> </div> </li> <li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_to__File.html">toFile => ...</a>, -- Store result in a file.</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><p>Returns a list of</p> <p>- a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> A with the rays of the positive hull of L in its rows</p> <p>- a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> Adual with the rays of the dual in its rows</p> <p>- a list of lists with the faces of the positive hull sorted by increasing dimension.</p> <p></p> <p>Each face is a list with two elements. The first is a list of the vertices of the face, the second a list of the vertices of the dual face.</p> <p>The rays of the faces are represented by the indices of the rows of A.</p> <p>This uses the Convex functions posHull and faces.</p> <p>The positive hull of L is required to be strictly convex and of full dimension.</p> <div/> <table class="examples"><tr><td><pre>i1 : L= {{0,1,1,0,0},{0,1,0,1,0},{0,1,0,0,0},{1,0,0,0,1},{1,0,-1,-1,-1},{1,0,0,0,0}};</pre> </td></tr> <tr><td><pre>i2 : L=apply(L,vector) o2 = {| 0 |, | 0 |, | 0 |, | 1 |, | 1 |, | 1 |} | 1 | | 1 | | 1 | | 0 | | 0 | | 0 | | 1 | | 0 | | 0 | | 0 | | -1 | | 0 | | 0 | | 1 | | 0 | | 0 | | -1 | | 0 | | 0 | | 0 | | 0 | | 1 | | -1 | | 0 | o2 : List</pre> </td></tr> <tr><td><pre>i3 : P=mPosHullFacesAndDuals(L) o3 = {| 0 1 1 0 0 |, | 1 0 2 0 -1 |, {{{{}, {1, 2, 3, 4, 5, 6, 7, 8}}}, | 0 1 0 1 0 | | 1 0 0 2 -1 | | 0 1 0 0 0 | | 1 0 0 0 1 | | 1 0 0 0 1 | | 1 0 0 0 -1 | | 1 0 -1 -1 -1 | | 0 1 1 -1 0 | | 1 0 0 0 0 | | 0 1 -1 1 0 | | 0 1 -1 -1 2 | | 0 1 -1 -1 0 | ------------------------------------------------------------------------ {{{1}, {2, 3, 4, 6, 7, 8}}, {{2}, {1, 3, 4, 5, 7, 8}}, {{3}, {1, 2, 3, ------------------------------------------------------------------------ 4}}, {{4}, {1, 2, 4, 5, 6, 8}}, {{5}, {1, 2, 3, 5, 6, 7}}, {{6}, {5, 6, ------------------------------------------------------------------------ 7, 8}}}, {{{1, 3}, {2, 3, 4}}, {{2, 3}, {1, 3, 4}}, {{3, 4}, {1, 2, 4}}, ------------------------------------------------------------------------ {{3, 5}, {1, 2, 3}}, {{4, 5}, {1, 2, 5, 6}}, {{5, 6}, {5, 6, 7}}, {{1, ------------------------------------------------------------------------ 2}, {3, 4, 7, 8}}, {{1, 4}, {2, 4, 6, 8}}, {{1, 5}, {2, 3, 6, 7}}, {{1, ------------------------------------------------------------------------ 6}, {6, 7, 8}}, {{2, 4}, {1, 4, 5, 8}}, {{2, 5}, {1, 3, 5, 7}}, {{2, 6}, ------------------------------------------------------------------------ {5, 7, 8}}, {{4, 6}, {5, 6, 8}}}, {{{1, 3, 5}, {2, 3}}, {{2, 3, 5}, {1, ------------------------------------------------------------------------ 3}}, {{3, 4, 5}, {1, 2}}, {{4, 5, 6}, {5, 6}}, {{2, 4, 5}, {1, 5}}, {{1, ------------------------------------------------------------------------ 4, 5}, {2, 6}}, {{1, 2, 5}, {3, 7}}, {{1, 2, 3}, {3, 4}}, {{2, 3, 4}, ------------------------------------------------------------------------ {1, 4}}, {{1, 3, 4}, {2, 4}}, {{1, 5, 6}, {6, 7}}, {{2, 5, 6}, {5, 7}}, ------------------------------------------------------------------------ {{1, 2, 4}, {4, 8}}, {{1, 2, 6}, {7, 8}}, {{1, 4, 6}, {6, 8}}, {{2, 4, ------------------------------------------------------------------------ 6}, {5, 8}}}, {{{2, 3, 4, 5}, {1}}, {{1, 3, 4, 5}, {2}}, {{1, 2, 3, 5}, ------------------------------------------------------------------------ {3}}, {{1, 2, 3, 4}, {4}}, {{2, 4, 5, 6}, {5}}, {{1, 4, 5, 6}, {6}}, ------------------------------------------------------------------------ {{1, 2, 5, 6}, {7}}, {{1, 2, 4, 6}, {8}}}, {{{1, 2, 3, 4, 5, 6}, {}}}}} o3 : List</pre> </td></tr> </table> </div> </div> <div class="waystouse"><h2>Ways to use <tt>mPosHullFacesAndDuals</tt> :</h2> <ul><li>mPosHullFacesAndDuals(List)</li> </ul> </div> </div> </body> </html>