e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
388 lines
17 KiB
HTML
388 lines
17 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/gamma-function.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:45:38 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Gamma Function</title>
|
||
|
||
<style>
|
||
|
||
.intgl {display:inline-block; margin: -4% 0 4% -1%; transform: translateX(20%) translateY(35%);}
|
||
.intgl .to {text-align:center; width:2em; font: 0.8em Verdana; margin: 0 0 -5px 8px;}
|
||
.intgl .symb {font: 180% Georgia;}
|
||
.intgl .symb:before { content: "\222B";}
|
||
.intgl .from {text-align:center; width:2em; font: 0.8em Verdana; overflow:visible; }
|
||
|
||
.sigma {display:inline-block; margin: -4% 0 4% -1%; transform: translateX(20%) translateY(35%);}
|
||
.sigma .to {text-align:center; width:2em; font: 0.8em Verdana; margin: 0 0 -12px 0;}
|
||
.sigma .symb {font: 200% Georgia; transform: translateY(16%); }
|
||
.sigma .symb:before { content: "\03A3";}
|
||
.sigma .from {text-align:center; width:2em; font: 0.8em Verdana; overflow:visible; }
|
||
|
||
.limit { display: inline-table;
|
||
text-align: center;
|
||
vertical-align: middle;
|
||
margin: 0 4px 0 2px;
|
||
border-collapse: collapse;
|
||
}
|
||
.limit em {
|
||
display: table-row;
|
||
text-align: center;
|
||
font-style: inherit;
|
||
}
|
||
.limit strong {
|
||
display: table-row;
|
||
text-align: center;
|
||
font-weight: inherit;
|
||
font-size: 80%;
|
||
line-height: 9px;
|
||
}
|
||
|
||
.boxa {
|
||
text-align: center;
|
||
display: inline-block;
|
||
vertical-align:bottom;
|
||
margin: 0 25px 35px 0;
|
||
}
|
||
|
||
|
||
</style>
|
||
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun Advanced"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Gamma Function</h1>
|
||
|
||
<p>The Gamma Function serves as a super powerful version of the <a href="factorial.html">factorial function</a>.</p>
|
||
<p>Let us first look at the factorial function:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="width:60px; text-align:center;"><img src="images/factorial.svg" alt="Factorial Symbol" height="117" width="26"></td>
|
||
<td>
|
||
<p>The <b>factorial function</b> (symbol: <b><font size="+1">!</font></b>) says to <b>multiply all whole numbers</b> from our chosen number down to 1.</p>
|
||
<p>Examples:</p>
|
||
<ul>
|
||
<div class="bigul">
|
||
<li><b>4!</b> = 4 × 3 × 2 × 1 = 24</li>
|
||
<li><b>7!</b> = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040</li>
|
||
<li><b>1!</b> = 1</li>
|
||
</div>
|
||
</ul> </td>
|
||
</tr>
|
||
</tbody></table></div><br>
|
||
<p>We can easily calculate a factorial from the previous one:</p>
|
||
<p class="center"><img src="images/factorial-how.svg" alt="factorial multiply" height="61" width="279"></p>
|
||
<p>As a table:</p>
|
||
<div class="beach">
|
||
|
||
<table align="center" width="400" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th>n</th>
|
||
<th>n!</th>
|
||
<th> </th>
|
||
<th> </th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>1</td>
|
||
<td><b>1</b></td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>2</td>
|
||
<td>2 × <b>1</b></td>
|
||
<td>= 2 × <b>1!</b></td>
|
||
<td>= 2</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>3</td>
|
||
<td>3 × <b>2 × 1</b></td>
|
||
<td>= 3 × <b>2!</b></td>
|
||
<td>= 6</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>4</td>
|
||
<td>4 × <b>3 × 2 × 1</b></td>
|
||
<td>= 4 × <b>3!</b></td>
|
||
<td>= 24</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>5</td>
|
||
<td>5 × <b>4 × 3 × 2 × 1</b></td>
|
||
<td>= 5 × <b>4!</b></td>
|
||
<td>= 120</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td>6</td>
|
||
<td>etc</td>
|
||
<td>etc</td>
|
||
<td> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div><p>In fact we have this general rule:</p>
|
||
<p class="center large">n! = n × (n−1)!</p>
|
||
<p>Which says</p>
|
||
<p class="center">"the factorial of any number is <b>that number</b> times<br>
|
||
the <b>factorial of (that number minus 1)</b>"</p>
|
||
<p>So 10! = 10 × 9!, ... and 125! = 125 × 124!, etc.</p>
|
||
<p>Note: this is a "recurrence relation"</p>
|
||
|
||
|
||
<h2>Beyond Whole Numbers</h2>
|
||
|
||
<p>Can we have a function that works more generally? If so, what properties do we want?</p>
|
||
<p>Firstly, it should "hit the mark" at each whole number:</p>
|
||
<p class="center larger"><b>f(x) = x!</b> for whole numbers</p>
|
||
<p>Here are some examples that do that:</p>
|
||
<div class="center">
|
||
<div class="boxa"><img src="images/graph-factorial-a.svg" alt="factorial graph a" height="374" width="182"><br>
|
||
Straight Lines
|
||
</div>
|
||
<div class="boxa"><img src="images/graph-factorial-b.svg" alt="factorial graph b" height="374" width="182"><br>
|
||
Anything Goes</div>
|
||
<div class="boxa"><img src="images/graph-factorial-c.svg" alt="factorial graph c" height="374" width="182"><br>
|
||
Smooth</div>
|
||
</div>
|
||
<p>Next, we want this to be true:</p>
|
||
<p class="center larger"><b>f(</b><b><b>z</b>) =</b> <b><b>z</b></b> <b>f(</b><b><b>z</b></b><b>−1)</b> in between the whole numbers</p>
|
||
<p class="center"><img src="images/graph-gamma-gap1.svg" alt="gamma graph gap=1" height="242" width="279"><br>
|
||
<b>f(b) = b f(b−1)</b><br>
|
||
Example: <b>f(2.62) =</b> <b><b>2.62</b> × f(</b><b><b>1.62</b>)</b></p>
|
||
<p>We also need a special condition to keep it smooth called "logarithmically convex": the <a href="../sets/functions-composition.html">composition</a> of the <a href="../sets/function-logarithmic.html">logarithm</a> with our function should be <a href="../calculus/concave-up-down-convex.html">convex</a>.</p>
|
||
<p>Many functions have been discovered with those properties. They each have good and bad points.</p>
|
||
<p>The one most liked is called the Gamma Function (<b>Γ</b> is the Greek capital letter Gamma):</p>
|
||
<div class="center large">Γ(z) =
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>z−1</sup> e<sup>−x</sup> dx</div>
|
||
<!-- (z) = INT{0,INF} x^z-1 e^-x dx -->
|
||
<p>It is a <a href="../calculus/integration-definite.html">definite integral</a> with limits from 0 to infinity.</p>
|
||
<p>It matches the factorial function for whole numbers (but sadly we must subtract 1):</p>
|
||
<div class="center large"><b>Γ(n) = (n−1)!</b> for whole numbers</div>
|
||
<p>So:</p>
|
||
<ul>
|
||
<li>Γ(1) = 0!</li>
|
||
<li>Γ(2) = 1!</li>
|
||
<li>Γ(3) = 2!</li>
|
||
<li>etc</li></ul>
|
||
<p>Let's see how to use it.</p>
|
||
|
||
<div class="dotpoint">
|
||
<p>How about n=1</p><br>
|
||
<div class="center large">Γ(1) =
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>1−1</sup> e<sup>−x</sup> dx</div>
|
||
<div class="center large"> =
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>0</sup> e<sup>−x</sup> dx</div>
|
||
<div class="center large">=
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div>e<sup>−x</sup> dx</div>
|
||
<div class="center large">= <span class="limit"><em>lim</em><strong>x→∞</strong></span> (−e<sup>−x</sup>) − (−e<sup>−0</sup>)</div>
|
||
<div class="center large">= 0 − (−1)</div>
|
||
<div class="center large">= 1</div>
|
||
</div> <p>Good so far, but does it work generally (<b>z</b> not restricted to integers)?</p>
|
||
<div class="center large"><b>Γ(z+1) = z Γ(z)</b> </div>
|
||
|
||
<div class="example">
|
||
<p>Let us try:</p>
|
||
<div class="center large">Γ(z+1) =
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>z+1−1</sup> e<sup>−x</sup> dx</div>
|
||
<div class="center large"> =
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>z</sup> e<sup>−x</sup> dx</div>
|
||
<p>We can use <a href="../calculus/integration-by-parts.html">Integration by Parts</a> with u=x<sup>z</sup> and v=e<sup>−x</sup>. There are many steps, but the key points are:</p>
|
||
<div class="center large">Γ(z+1) = <span style="display:inline-block; transform: scaleY(2);">[</span> −x<sup>z</sup> e<sup>−x</sup> <span style="display:inline-block; transform: scaleY(2);">]</span>
|
||
<div style="display:inline-block; font: 0.8em Verdana; text-align:center;transform: translateY(30%) translateX(-30%);">
|
||
<div>∞</div><br>
|
||
<div>0</div>
|
||
</div>+<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div>zx<sup>z-1</sup> e<sup>−x</sup> dx</div>
|
||
<div class="center large">Γ(z+1) = <span class="limit"><em>lim</em><strong>x→∞</strong></span>(−x<sup>z</sup> e<sup>−x</sup>) − (−0<sup>z</sup> e<sup>−0</sup>) + z
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div>x<sup>z-1</sup> e<sup>−x</sup> dx</div>
|
||
<p>And −x<sup>z</sup> e<sup>−x</sup> goes to 0 as z goes to infinity, so it all simplifies to:</p>
|
||
<div class="center large">Γ(z+1) = z
|
||
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div>x<sup>z-1</sup> e<sup>−x</sup> dx</div>
|
||
<p>And the remaining integral is actually the Gamma Function for z, so:</p>
|
||
<div class="center large">Γ(z+1) = z Γ(z)</div>
|
||
<p>So it works generally.</p>
|
||
</div>
|
||
<p>And here is a plot of the Gamma Function:</p>
|
||
<p class="center"><img src="images/gamma-plot.svg" alt="gamma function " height="400" width="600"></p>
|
||
<p>But at x = 0 or less it works everywhere <b>except at integer values</b> because</p>
|
||
<ul>
|
||
<li><b style="white-space:nowrap;">Γ(0) = Γ(1)/0</b> is undefined (<a href="dividing-by-zero.html">dividing by zero</a>),</li>
|
||
<li>and so Γ(−1) = Γ(0)/−1 is also undefined,</li>
|
||
<li>etc <span style="font-family:Verdana,Arial,Tahoma,sans-serif"><span style="font-family:Verdana,Arial,Tahoma,sans-serif"><span style="font-family:Verdana,Arial,Tahoma,sans-serif"><b style="font-family:Verdana,Arial,Tahoma,sans-serif;text-align:center"><b style="font-family:Verdana,Arial,Tahoma,sans-serif;text-align:center"><font color="#2664b5"><span style="font-size:17.6px"><span></span></span></font></b></b></span></span></span></li></ul>
|
||
<p><b>Try comparing</b> two values on the graph that are 1 apart on the x axis and see if it is true that Γ(z+1) = z Γ(z)</p>
|
||
|
||
<h3>Complex</h3>
|
||
<p>The Gamma Function also works for <a href="complex-numbers.html">Complex Numbers</a> so long as the real part is greater than 0.</p>
|
||
|
||
<h3>Half</h3>
|
||
<p>We can calculate the gamma function at <b>a half</b> (quite a few steps!) to get a surprising result:</p>
|
||
<p class="center large">Γ(<span class="intbl"><em>1</em><strong>2</strong></span>) = <span class="times">√π</span></p>
|
||
<p>Knowing that <b>Γ(z+1) = z Γ(z)</b> we get these "half-integer" factorials:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="center">Gamma</td>
|
||
<td>Γ(z+1) = z Γ(z)</td>
|
||
<td><br>
|
||
</td>
|
||
<td>Factorial</td></tr>
|
||
<tr>
|
||
<td style="text-align:center; width:100px;">Γ(<span class="intbl"><em>1</em><strong>2</strong></span>)</td>
|
||
<td style="text-align:center;"><br>
|
||
</td>
|
||
<td style="text-align:center; width:100px;"><span class="times">√π</span></td>
|
||
<td style="text-align:center;">(−<span class="intbl"><em>1</em><strong>2</strong></span>)!</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">Γ(<span class="intbl"><em>3</em><strong>2</strong></span>)</td>
|
||
<td style="text-align:center;">= <span class="intbl"><em>1</em><strong>2</strong></span>Γ(<span class="intbl"><em>1</em><strong>2</strong></span>) =</td>
|
||
<td style="text-align:center;"><span class="intbl"><em>1</em><strong>2</strong></span><span class="times">√π</span></td>
|
||
<td style="text-align:center;">(<span class="intbl"><em>1</em><strong>2</strong></span>)!</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">Γ(<span class="intbl"><em>5</em><strong>2</strong></span>)</td>
|
||
<td style="text-align:center;">= <span class="intbl"><em>3</em><strong>2</strong></span>Γ(<span class="intbl"><em>3</em><strong>2</strong></span>) =</td>
|
||
<td style="text-align:center;"><span class="intbl"><em>3</em><strong>4</strong></span><span class="times">√π</span></td>
|
||
<td style="text-align:center;">(<span class="intbl"><em>3</em><strong>2</strong></span>)!</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">Γ(<span class="intbl"><em>7</em><strong>2</strong></span>)</td>
|
||
<td style="text-align:center;">= <span class="intbl"><em>5</em><strong>2</strong></span>Γ(<span class="intbl"><em>5</em><strong>2</strong></span>) =</td>
|
||
<td style="text-align:center;"><span class="intbl"><em>15</em><strong>8</strong></span><span class="times">√π</span></td>
|
||
<td style="text-align:center;">(<span class="intbl"><em>5</em><strong>2</strong></span>)!</td>
|
||
</tr><tr>
|
||
<td class="center">...</td>
|
||
<td class="center">...</td>
|
||
<td class="center">...</td>
|
||
<td class="center">...</td></tr>
|
||
</tbody></table>
|
||
</div><br>
|
||
<p>Also check if the graph above gets them right.</p>
|
||
|
||
|
||
<h2>Applications</h2>
|
||
|
||
<p>Just like the Factorial function there are many uses for the Gamma function in Combinatorics, Probability and Statistics. It is also very useful in Calculus and Physics.</p>
|
||
<p><br></p>
|
||
<p>So there you have it: the Gamma Function may be a little hard to
|
||
calculate but it neatly extends the factorial function beyond whole
|
||
numbers.</p>
|
||
|
||
<div class="related">
|
||
<a href="../combinatorics/combinations-permutations.html">Combinations and Permutations</a>
|
||
<a href="index.html">Numbers Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/gamma-function.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:45:39 GMT -->
|
||
</html> |