<?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>euler(Flat) -- Euler characteristic</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Flat.html">next</a> | <a href="___E__P__Y.html">previous</a> | <a href="___Flat.html">forward</a> | <a href="___E__P__Y.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>euler(Flat) -- Euler characteristic</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>euler(F) or euler(A)</tt></div> </dd></dl> </div> </li> <li><span>Function: <a href="../../Macaulay2Doc/html/_euler.html" title="Euler characteristic">euler</a></span></li> <li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>an <a href="___Flat.html">intersection of hyperplane(s)</a></span>, a flat</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><tt>k</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the beta invariant of the flat <tt>F</tt> or arrangement <tt>A</tt></span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div>The beta invariant of an arrangement <tt>A</tt> is, by definition, the Euler characteristic of complement of <tt>A</tt> in complex projective space.<p/> The beta invariant of a flat <tt>F</tt> is the beta invariant of the <a href="_restriction.html">restriction</a> of <tt>A</tt> to <tt>F</tt>.<table class="examples"><tr><td><pre>i1 : A = typeA(3) o1 = {x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 2 3 2 4 3 4 o1 : Hyperplane Arrangement </pre> </td></tr> <tr><td><pre>i2 : euler A -- for a real arrangement, equals number of bounded chambers o2 = 2</pre> </td></tr> </table> </div> </div> </div> </body> </html>