psf_models.html

Deil Christoph, 02/14/2013 07:29 PM

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<html>
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<head>
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<style type="text/css">
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/**
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 * HTML5 ✰ Boilerplate
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 *
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 * style.css contains a reset, font normalization and some base styles.
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 *
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 * Credit is left where credit is due.
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 * Much inspiration was taken from these projects:
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 * - yui.yahooapis.com/2.8.1/build/base/base.css
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 * - camendesign.com/design/
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 * - praegnanz.de/weblog/htmlcssjs-kickstart
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 */
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/**
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 * html5doctor.com Reset Stylesheet (Eric Meyer's Reset Reloaded + HTML5 baseline)
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 * v1.6.1 2010-09-17 | Authors: Eric Meyer & Richard Clark
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 * html5doctor.com/html-5-reset-stylesheet/
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 */
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html, body, div, span, object, iframe,
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h1, h2, h3, h4, h5, h6, p, blockquote, pre,
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abbr, address, cite, code, del, dfn, em, img, ins, kbd, q, samp,
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small, strong, sub, sup, var, b, i, dl, dt, dd, ol, ul, li,
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fieldset, form, label, legend,
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table, caption, tbody, tfoot, thead, tr, th, td,
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article, aside, canvas, details, figcaption, figure,
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footer, header, hgroup, menu, nav, section, summary,
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time, mark, audio, video {
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  margin: 0;
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  padding: 0;
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  border: 0;
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  font-size: 100%;
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  font: inherit;
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  vertical-align: baseline;
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}
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sup { vertical-align: super; }
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sub { vertical-align: sub; }
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article, aside, details, figcaption, figure,
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footer, header, hgroup, menu, nav, section {
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  display: block;
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}
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blockquote, q { quotes: none; }
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blockquote:before, blockquote:after,
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q:before, q:after { content: ""; content: none; }
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ins { background-color: #ff9; color: #000; text-decoration: none; }
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mark { background-color: #ff9; color: #000; font-style: italic; font-weight: bold; }
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del { text-decoration: line-through; }
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abbr[title], dfn[title] { border-bottom: 1px dotted; cursor: help; }
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table { border-collapse: collapse; border-spacing: 0; }
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hr { display: block; height: 1px; border: 0; border-top: 1px solid #ccc; margin: 1em 0; padding: 0; }
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input, select { vertical-align: middle; }
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/**
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 * Font normalization inspired by YUI Library's fonts.css: developer.yahoo.com/yui/
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 */
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body { font:13px/1.231 sans-serif; *font-size:small; } /* Hack retained to preserve specificity */
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select, input, textarea, button { font:99% sans-serif; }
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/* Normalize monospace sizing:
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   en.wikipedia.org/wiki/MediaWiki_talk:Common.css/Archive_11#Teletype_style_fix_for_Chrome */
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pre, code, kbd, samp { font-family: monospace, sans-serif; }
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em,i { font-style: italic; }
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b,strong { font-weight: bold; }
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</style>
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<style type="text/css">
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/* Flexible box model classes */
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/* Taken from Alex Russell http://infrequently.org/2009/08/css-3-progress/ */
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.hbox {
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        display: -webkit-box;
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        -webkit-box-orient: horizontal;
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        -webkit-box-align: stretch;
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        display: -moz-box;
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        -moz-box-orient: horizontal;
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        -moz-box-align: stretch;
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        display: box;
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        box-orient: horizontal;
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        box-align: stretch;
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}
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.hbox > * {
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        -webkit-box-flex: 0;
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        -moz-box-flex: 0;
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        box-flex: 0;
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}
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.vbox {
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        display: -webkit-box;
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        -webkit-box-orient: vertical;
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        -webkit-box-align: stretch;
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        display: -moz-box;
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        -moz-box-orient: vertical;
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        -moz-box-align: stretch;
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        display: box;
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        box-orient: vertical;
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        box-align: stretch;
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}
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.vbox > * {
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        -webkit-box-flex: 0;
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        -moz-box-flex: 0;
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        box-flex: 0;
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}
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.reverse {
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        -webkit-box-direction: reverse;
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        -moz-box-direction: reverse;
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        box-direction: reverse;
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}
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.box-flex0 {
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        -webkit-box-flex: 0;
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        -moz-box-flex: 0;
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        box-flex: 0;
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}
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.box-flex1, .box-flex {
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        -webkit-box-flex: 1;
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        -moz-box-flex: 1;
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        box-flex: 1;
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}
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.box-flex2 {
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        -webkit-box-flex: 2;
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        -moz-box-flex: 2;
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        box-flex: 2;
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}
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.box-group1 {
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        -webkit-box-flex-group: 1;
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        -moz-box-flex-group: 1;
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        box-flex-group: 1;
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}
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.box-group2 {
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        -webkit-box-flex-group: 2;
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        -moz-box-flex-group: 2;
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        box-flex-group: 2;
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}
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.start {
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        -webkit-box-pack: start;
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        -moz-box-pack: start;
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        box-pack: start;
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}
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.end {
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        -webkit-box-pack: end;
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        -moz-box-pack: end;
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        box-pack: end;
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}
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.center {
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        -webkit-box-pack: center;
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        -moz-box-pack: center;
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        box-pack: center;
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}
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</style>
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<style type="text/css">
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/**
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 * Primary styles
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 *
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 * Author: IPython Development Team
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 */
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body {
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    overflow: hidden;
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}
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blockquote {
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    border-left: 4px solid #DDD;
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    padding: 0 15px;
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    color: #777;
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}
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span#save_widget {
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    padding: 5px;
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    margin: 0px 0px 0px 300px;
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    display:inline-block;
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}
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span#notebook_name {
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    height: 1em;
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    line-height: 1em;
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    padding: 3px;
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    border: none;
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    font-size: 146.5%;
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}
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.ui-menubar-item .ui-button .ui-button-text {
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    padding: 0.4em 1.0em;
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    font-size: 100%;
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}
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.ui-menu {
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  -moz-box-shadow:    0px 6px 10px -1px #adadad;
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  -webkit-box-shadow: 0px 6px 10px -1px #adadad;
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  box-shadow:         0px 6px 10px -1px #adadad;
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}
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.ui-menu .ui-menu-item a {
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    border: 1px solid transparent;
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    padding: 2px 1.6em;
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}
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.ui-menu .ui-menu-item a.ui-state-focus {
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    margin: 0;
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}
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.ui-menu hr {
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    margin: 0.3em 0;
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}
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#menubar_container {
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    position: relative;
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}
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#notification_area {
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    position: absolute;
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    right: 0px;
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    top: 0px;
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    height: 25px;
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    padding: 3px 0px;
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    padding-right: 3px;
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    z-index: 10;
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}
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.notification_widget{
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    float : right;
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    right: 0px;
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    top: 1px;
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    height: 25px;
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    padding: 3px 6px;
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    z-index: 10;
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}
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.toolbar {
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    padding: 3px 15px;
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}
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#cell_type {
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    font-size: 85%;
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}
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div#main_app {
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    width: 100%;
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    position: relative;
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}
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span#quick_help_area {
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    position: static;
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    padding: 5px 0px;
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    margin: 0px 0px 0px 0px;
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}
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.help_string {
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    float: right;
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    width: 170px;
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    padding: 0px 5px;
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    text-align: left;
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    font-size: 85%;
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}
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.help_string_label {
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    float: right;
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    font-size: 85%;
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}
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div#notebook_panel {
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    margin: 0px 0px 0px 0px;
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    padding: 0px;
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}
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div#notebook {
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    overflow-y: scroll;
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    overflow-x: auto;
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    width: 100%;
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    /* This spaces the cell away from the edge of the notebook area */
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    padding: 5px 5px 15px 5px;
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    margin: 0px;
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    background-color: white;
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}
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div#pager_splitter {
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    height: 8px;
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}
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#pager_container {
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    position : relative;
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}
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div#pager {
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    padding: 15px;
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    overflow: auto;
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    display: none;
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}
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div.ui-widget-content {
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    border: 1px solid #aaa;
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    outline: none;
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}
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.cell {
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    border: 1px solid transparent;
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}
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div.cell {
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    width: 100%;
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    padding: 5px 5px 5px 0px;
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    /* This acts as a spacer between cells, that is outside the border */
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    margin: 2px 0px 2px 0px;
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}
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div.code_cell {
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    background-color: white;
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}
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/* any special styling for code cells that are currently running goes here */
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div.code_cell.running {
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}
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div.prompt {
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    /* This needs to be wide enough for 3 digit prompt numbers: In[100]: */
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    width: 11ex;
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    /* This 0.4em is tuned to match the padding on the CodeMirror editor. */
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    padding: 0.4em;
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    margin: 0px;
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    font-family: monospace;
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    text-align:right;
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}
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div.input {
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    page-break-inside: avoid;
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}
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/* input_area and input_prompt must match in top border and margin for alignment */
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div.input_area {
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    color: black;
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    border: 1px solid #ddd;
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    border-radius: 3px;
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    background: #f7f7f7;
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}
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div.input_prompt {
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    color: navy;
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    border-top: 1px solid transparent;
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}
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div.output_wrapper {
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    /* This is a spacer between the input and output of each cell */
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    margin-top: 5px;
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    margin-left: 5px;
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    /* FF needs explicit width to stretch */
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    width: 100%;
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    /* this position must be relative to enable descendents to be absolute within it */
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    position: relative;
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}
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/* class for the output area when it should be height-limited */
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div.output_scroll {
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  /* ideally, this would be max-height, but FF barfs all over that */
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  height: 24em;
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  /* FF needs this *and the wrapper* to specify full width, or it will shrinkwrap */
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  width: 100%;
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  overflow: auto;
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  border-radius: 3px;
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  box-shadow: inset 0 2px 8px rgba(0, 0, 0, .8);
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}
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/* output div while it is collapsed */
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div.output_collapsed {
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  margin-right: 5px;
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}
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div.out_prompt_overlay {
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  height: 100%;
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  padding: 0px;
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  position: absolute;
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  border-radius: 3px;
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}
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div.out_prompt_overlay:hover {
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  /* use inner shadow to get border that is computed the same on WebKit/FF */
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  box-shadow: inset 0 0 1px #000;
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  background: rgba(240, 240, 240, 0.5);
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}
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div.output_prompt {
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    color: darkred;
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    /* 5px right shift to account for margin in parent container */
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    margin: 0 5px 0 -5px;
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}
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/* This class is the outer container of all output sections. */
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div.output_area {
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    padding: 0px;
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    page-break-inside: avoid;
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}
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/* This class is for the output subarea inside the output_area and after
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   the prompt div. */
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div.output_subarea {
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    padding: 0.44em 0.4em 0.4em 1px;
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}
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/* The rest of the output_* classes are for special styling of the different
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   output types */
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/* all text output has this class: */
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div.output_text {
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    text-align: left;
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    color: black;
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    font-family: monospace;
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}
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/* stdout/stderr are 'text' as well as 'stream', but pyout/pyerr are *not* streams */
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div.output_stream {
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    padding-top: 0.0em;
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    padding-bottom: 0.0em;
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}
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div.output_stdout {
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}
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div.output_stderr {
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    background: #fdd; /* very light red background for stderr */
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}
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div.output_latex {
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    text-align: left;
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    color: black;
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}
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div.output_html {
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}
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div.output_png {
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}
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div.output_jpeg {
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}
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div.text_cell {
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    background-color: white;
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    padding: 5px 5px 5px 5px;
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}
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div.text_cell_input {
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    color: black;
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    border: 1px solid #ddd;
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    border-radius: 3px;
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    background: #f7f7f7;
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}
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div.text_cell_render {
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    font-family: "Helvetica Neue", Arial, Helvetica, Geneva, sans-serif;
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    outline: none;
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    resize: none;
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    width:  inherit;
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    border-style: none;
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    padding: 5px;
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    color: black;
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}
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/* The following gets added to the <head> if it is detected that the user has a
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 * monospace font with inconsistent normal/bold/italic height.  See
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 * notebookmain.js.  Such fonts will have keywords vertically offset with
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 * respect to the rest of the text.  The user should select a better font. 
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 * See: https://github.com/ipython/ipython/issues/1503
495
 *
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 * .CodeMirror span {
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 *      vertical-align: bottom;
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 * }
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 */
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501
.CodeMirror {
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    line-height: 1.231;  /* Changed from 1em to our global default */
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}
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.CodeMirror-scroll {
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    height: auto;     /* Changed to auto to autogrow */
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    /*  The CodeMirror docs are a bit fuzzy on if overflow-y should be hidden or visible.*/
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    /*  We have found that if it is visible, vertical scrollbars appear with font size changes.*/
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    overflow-y: hidden;
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    overflow-x: auto; /* Changed from auto to remove scrollbar */
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}
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/* CSS font colors for translated ANSI colors. */
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.ansiblack {color: black;}
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.ansired {color: darkred;}
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.ansigreen {color: darkgreen;}
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.ansiyellow {color: brown;}
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.ansiblue {color: darkblue;}
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.ansipurple {color: darkviolet;}
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.ansicyan {color: steelblue;}
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.ansigrey {color: grey;}
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.ansibold {font-weight: bold;}
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.completions {
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    position: absolute;
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    z-index: 10;
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    overflow: hidden;
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    border: 1px solid grey;
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}
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.completions select {
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    background: white;
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    outline: none;
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    border: none;
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    padding: 0px;
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    margin: 0px;
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    overflow: auto;
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    font-family: monospace;
541
}
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option.context {
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  background-color: #DEF7FF;
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}
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option.introspection {
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  background-color: #EBF4EB;
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}
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/*fixed part of the completion*/
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.completions p b {
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    font-weight:bold;
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}
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.completions p {
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    background: #DDF;
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    /*outline: none;
558
    padding: 0px;*/
559
    border-bottom: black solid 1px;
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    padding: 1px;
561
    font-family: monospace;
562
}
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pre.dialog {
565
    background-color: #f7f7f7;
566
    border: 1px solid #ddd;
567
    border-radius: 3px;
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    padding: 0.4em;
569
    padding-left: 2em;
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}
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p.dialog {
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    padding : 0.2em;
574
}
575

576
.shortcut_key {
577
    display: inline-block;
578
    width: 15ex;
579
    text-align: right;
580
    font-family: monospace;
581
}
582

583
.shortcut_descr {
584
}
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586
/* Word-wrap output correctly.  This is the CSS3 spelling, though Firefox seems
587
   to not honor it correctly.  Webkit browsers (Chrome, rekonq, Safari) do.
588
 */
589
pre, code, kbd, samp { white-space: pre-wrap; }
590

591
#fonttest {
592
    font-family: monospace;
593
}
594

595
.js-error {
596
    color: darkred;
597
}
598

    
599
</style>
600
<style type="text/css">
601
.rendered_html {color: black;}
602
.rendered_html em {font-style: italic;}
603
.rendered_html strong {font-weight: bold;}
604
.rendered_html u {text-decoration: underline;}
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.rendered_html :link { text-decoration: underline }
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.rendered_html :visited { text-decoration: underline }
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.rendered_html h1 {font-size: 197%; margin: .65em 0; font-weight: bold;}
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.rendered_html h2 {font-size: 153.9%; margin: .75em 0; font-weight: bold;}
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.rendered_html h3 {font-size: 123.1%; margin: .85em 0; font-weight: bold;}
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.rendered_html h4 {font-size: 100% margin: 0.95em 0; font-weight: bold;}
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.rendered_html h5 {font-size: 85%; margin: 1.5em 0; font-weight: bold;}
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.rendered_html h6 {font-size: 77%; margin: 1.65em 0; font-weight: bold;}
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.rendered_html ul {list-style:disc; margin: 1em 2em;}
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.rendered_html ul ul {list-style:square; margin: 0em 2em;}
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.rendered_html ul ul ul {list-style:circle; margin-left: 0em 2em;}
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.rendered_html ol {list-style:upper-roman; margin: 1em 2em;}
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.rendered_html ol ol {list-style:upper-alpha; margin: 0em 2em;}
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.rendered_html ol ol ol {list-style:decimal; margin: 0em 2em;}
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.rendered_html ol ol ol ol {list-style:lower-alpha; margin: 0em 2em;}
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.rendered_html ol ol ol ol ol {list-style:lower-roman; margin: 0em 2em;}
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622
.rendered_html hr {
623
    color: black;
624
    background-color: black;
625
}
626

627
.rendered_html pre {
628
    margin: 1em 2em;
629
}
630

631
.rendered_html blockquote {
632
    margin: 1em 2em;
633
}
634

635
.rendered_html table {
636
    border: 1px solid black;
637
    border-collapse: collapse;
638
    margin: 1em 2em;
639
}
640

641
.rendered_html td {
642
    border: 1px solid black;
643
    text-align: left;
644
    vertical-align: middle;
645
    padding: 4px;
646
}
647

648
.rendered_html th {
649
    border: 1px solid black;
650
    text-align: left;
651
    vertical-align: middle;
652
    padding: 4px;
653
    font-weight: bold;
654
}
655

656
.rendered_html tr {
657
    border: 1px solid black;
658
}    
659

660
.rendered_html p + p {
661
    margin-top: 1em;
662
}
663

    
664

    
665
</style>
666
<style type="text/css">
667
/* Overrides of notebook CSS for static HTML export
668

669
*/
670
body {
671
  overflow: visible;
672
  padding: 8px;
673
}
674
.input_area {
675
  padding: 0.4em;
676
}
677

    
678
</style>
679
<meta charset="UTF-8">
680
<style type="text/css">
681
.highlight .hll { background-color: #ffffcc }
682
.highlight  { background: #f8f8f8; }
683
.highlight .c { color: #408080; font-style: italic } /* Comment */
684
.highlight .err { border: 1px solid #FF0000 } /* Error */
685
.highlight .k { color: #008000; font-weight: bold } /* Keyword */
686
.highlight .o { color: #666666 } /* Operator */
687
.highlight .cm { color: #408080; font-style: italic } /* Comment.Multiline */
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.highlight .cp { color: #BC7A00 } /* Comment.Preproc */
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.highlight .c1 { color: #408080; font-style: italic } /* Comment.Single */
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.highlight .cs { color: #408080; font-style: italic } /* Comment.Special */
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.highlight .gd { color: #A00000 } /* Generic.Deleted */
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.highlight .ge { font-style: italic } /* Generic.Emph */
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.highlight .gr { color: #FF0000 } /* Generic.Error */
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.highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */
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.highlight .gi { color: #00A000 } /* Generic.Inserted */
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.highlight .go { color: #808080 } /* Generic.Output */
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.highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */
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.highlight .gs { font-weight: bold } /* Generic.Strong */
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.highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */
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.highlight .gt { color: #0040D0 } /* Generic.Traceback */
701
.highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */
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<body>
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<div class="text_cell_render border-box-sizing rendered_html">
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<h1>
770
  PSF Models
771
</h1>
772
</div>
773
<div class="text_cell_render border-box-sizing rendered_html">
774
<p>Here I prototype some PSF models we are considering in Python to make a few plots for the internal note.</p>
775
<p>Some things are already implemented in gammalib:
776
http://gammalib.sourceforge.net/doxygen/classGCTAPsf2D.html
777
http://gammalib.sourceforge.net/doxygen/classGLATPsf.html</p>
778
<p>Everything we actually end up using should eventually be implemented properly in gammalib.</p>
779
<p>TODO:</p>
780
<ul>
781
<li>Plot cumulative distributions, i.e. containment fraction</li>
782
<li>Plot containment radius, which is like containment fraction, but with x- and y-axis exchanged.</li>
783
</ul>
784
</div>
785
<div class="cell border-box-sizing code_cell vbox">
786
<div class="input hbox">
787
<div class="prompt input_prompt">In&nbsp;[1]:</div>
788
<div class="input_area box-flex1">
789
<div class="highlight"><pre><span class="c"># Execute this if you didn&#39;t start this notebook with --pylab=inline</span>
790
<span class="o">%</span><span class="k">pylab</span> <span class="n">inline</span>
791
</pre></div>
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<pre>
801
Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
802
For more information, type &apos;help(pylab)&apos;.
803
</pre>
804
</div>
805
</div>
806
</div>
807
</div>
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</div>
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811
<div class="prompt input_prompt">In&nbsp;[2]:</div>
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813
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
814
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sort</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span>
815
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
816
</pre></div>
817

    
818
</div>
819
</div>
820
</div>
821
<div class="text_cell_render border-box-sizing rendered_html">
822
<h2>
823
  Definition of models
824
</h2>
825
</div>
826
<div class="text_cell_render border-box-sizing rendered_html">
827
<p>We define PSF models as probability density functions (PDFs) in 2D, i.e. as $\frac{dP}{dx dy}$. This is explained in more detail below.</p>
828
<p>TODO:</p>
829
<ul>
830
<li>Implement as classes? Compute normalization integral only once!</li>
831
<li>Implement multi-Gauss and multi-King with correct normalisations.</li>
832
<li>Implement models in ROOT so that we can fit and draw random numbers easily!</li>
833
</ul>
834
</div>
835
<div class="cell border-box-sizing code_cell vbox">
836
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837
<div class="prompt input_prompt">In&nbsp;[3]:</div>
838
<div class="input_area box-flex1">
839
<div class="highlight"><pre><span class="k">def</span> <span class="nf">gauss</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
840
    <span class="sd">&quot;&quot;&quot;2D Gauss PDF dP / (dx dy) at r = sqrt(x ** 2 + y ** 2)</span>
841
<span class="sd">    Reference: TODO</span>
842
<span class="sd">    &quot;&quot;&quot;</span>
843
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
844
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">r</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
845
</pre></div>
846

    
847
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848
</div>
849
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<div class="input hbox">
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<div class="prompt input_prompt">In&nbsp;[4]:</div>
853
<div class="input_area box-flex1">
854
<div class="highlight"><pre><span class="k">def</span> <span class="nf">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
855
    <span class="sd">&quot;&quot;&quot;2D King PDF dP / (dx dy) at  r = sqrt(x ** 2 + y ** 2)</span>
856
<span class="sd">    Reference: This is the Fermi LAT PSF, see Equation (36) in [1]</span>
857
<span class="sd">    &quot;&quot;&quot;</span>
858
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mf">1.</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">)</span>
859
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mf">1.</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">gamma</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">gamma</span><span class="p">)</span>
860
</pre></div>
861

    
862
</div>
863
</div>
864
</div>
865
<div class="text_cell_render border-box-sizing rendered_html">
866
<h2>
867
  Transformation of probability densities
868
</h2>
869
</div>
870
<div class="text_cell_render border-box-sizing rendered_html">
871
<h3>Definitions</h3>
872
<p>There are three equivalent useful ways to describe a two-dimensional, radially symmetric PDF like a PSF.</p>
873
<ul>
874
<li>$A(r) = \frac{dP}{dx dy}$ is the 2D probability density (unit: deg$^{-2}$) in $r = \sqrt{x^2 + y^2}$ (unit: deg). This is the distribution we used above to define our models, but it is rarely shown. A distribution with this shape will be found if histogramming events in $x$ for a small $y$-width stripe along the $x$-axis, or by evaluating the 2D PDF $P(x,y)$ at $P(x, y=0)$.</li>
875
<li>$B(t) = \frac{dP}{dt}$ is the 1D probability density (unit: deg$^2$) in $t = r^2$ (unit: deg$^2$). This is the theta square distribution that is often shown or fitted. A distribution with this shape will be found if histogramming events in $t$.</li>
876
<li>$C(r) = \frac{dP}{dr}$ is the 1D probability density (unit: deg) in $r= \sqrt{x^2 + y^2}$ (unit: deg). This distribution is not often shown or used, but is better suited to visualize differences between different distributions, because the peak is not at the edge. A distribution with this shape will be found if histogramming events in $r$.</li>
877
</ul>
878
<p>Here we derive the relationship between $A$, $B$ and $C$ and illustrate them for the Gauss and King models.
879
When implementing the fit function $B(t)$ for the theta square-distribution it might be better to re-code the formula for $B(t)$ directly.</p>
880
<h3>Relations</h3>
881
<p>Given $A(r)$, how to obtain / evaluate $B(t)$?</p>
882
<ul>
883
<li>$t = r^2$, so $dt = 2 r dr$.</li>
884
<li>$dP = 2\pi r dr A(r)$ follows from the definition of $A$ as the probability $dP$ in a ring at radius $r$ of width $dr$.</li>
885
<li>$B(t) = \frac{dP}{dt} = \frac{2\pi r dr A(r)}{2 r dr} = \pi A(r)$, i.e. using the formula for $A(r)$ above we obtain the formula for $B(t)$ by substituting $r \to \sqrt{t}$ and multiplying by $\pi$.</li>
886
<li>$C(r) = \frac{dP}{dr} = \frac{2 \pi r dr A(r)}{dr} = 2 \pi r A(r)$, i.e. using the formula for $A(r)$ above we obtain the formula for $C(r)$ by multiplying with $2 \pi r$.</li>
887
<li>The relation between $B$ and $C$ is also simple, should we need it: $C(r) = 2 r B(t)$.</li>
888
</ul>
889
</div>
890
<div class="cell border-box-sizing code_cell vbox">
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<div class="prompt input_prompt">In&nbsp;[5]:</div>
893
<div class="input_area box-flex1">
894
<div class="highlight"><pre><span class="k">def</span> <span class="nf">compute_B</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">r2</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
895
    <span class="sd">&quot;&quot;&quot;B(r2) for a given A(r)</span>
896
<span class="sd">    See derivation above.&quot;&quot;&quot;</span>
897
    <span class="n">r</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">r2</span><span class="p">)</span>
898
    <span class="k">return</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">A</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
899
</pre></div>
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904
<div class="cell border-box-sizing code_cell vbox">
905
<div class="input hbox">
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<div class="prompt input_prompt">In&nbsp;[6]:</div>
907
<div class="input_area box-flex1">
908
<div class="highlight"><pre><span class="k">def</span> <span class="nf">compute_C</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
909
    <span class="sd">&quot;&quot;&quot;C(r) for a given A(r)</span>
910
<span class="sd">    See derivation above.&quot;&quot;&quot;</span>
911
    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span><span class="n">pi</span> <span class="o">*</span> <span class="n">r</span> <span class="o">*</span> <span class="n">A</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
912
</pre></div>
913

    
914
</div>
915
</div>
916
</div>
917
<div class="text_cell_render border-box-sizing rendered_html">
918
<p>For the Gauss and King models we implement the $B$ and $C$ PDFs here explicitly in addition to the $A$ PDF given above.</p>
919
<p>This is useful for fitting and for comparing with other code that might use e.g. $B$ directly.</p>
920
</div>
921
<div class="cell border-box-sizing code_cell vbox">
922
<div class="input hbox">
923
<div class="prompt input_prompt">In&nbsp;[7]:</div>
924
<div class="input_area box-flex1">
925
<div class="highlight"><pre><span class="k">def</span> <span class="nf">gauss_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
926
    <span class="sd">&quot;&quot;&quot;2D Gauss PDF dP(r2) / dr2 at r2 = x ** 2 + y ** 2&quot;&quot;&quot;</span>
927
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
928
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">r2</span> <span class="o">/</span> <span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
929
</pre></div>
930

    
931
</div>
932
</div>
933
</div>
934
<div class="cell border-box-sizing code_cell vbox">
935
<div class="input hbox">
936
<div class="prompt input_prompt">In&nbsp;[8]:</div>
937
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938
<div class="highlight"><pre><span class="k">def</span> <span class="nf">gauss_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
939
    <span class="sd">&quot;&quot;&quot;2D Gauss PDF dP(r) / dr at r = sqrt(x ** 2 + y ** 2)&quot;&quot;&quot;</span>
940
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">*</span> <span class="n">r</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span>
941
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">r</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
942
</pre></div>
943

    
944
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945
</div>
946
</div>
947
<div class="cell border-box-sizing code_cell vbox">
948
<div class="input hbox">
949
<div class="prompt input_prompt">In&nbsp;[9]:</div>
950
<div class="input_area box-flex1">
951
<div class="highlight"><pre><span class="k">def</span> <span class="nf">king_B</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
952
    <span class="sd">&quot;&quot;&quot;2D King PDF dP(r2) / dr2 at r2 = x ** 2 + y ** 2&quot;&quot;&quot;</span>
953
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mf">1.</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">)</span>
954
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mf">1.</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">gamma</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">r2</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">gamma</span><span class="p">)</span>
955
</pre></div>
956

    
957
</div>
958
</div>
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</div>
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<div class="cell border-box-sizing code_cell vbox">
961
<div class="input hbox">
962
<div class="prompt input_prompt">In&nbsp;[10]:</div>
963
<div class="input_area box-flex1">
964
<div class="highlight"><pre><span class="k">def</span> <span class="nf">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">norm</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
965
    <span class="sd">&quot;&quot;&quot;2D King PDF dP(r) / dr at r = sqrt(x ** 2 + y ** 2)&quot;&quot;&quot;</span>
966
    <span class="n">N</span> <span class="o">=</span> <span class="n">norm</span> <span class="o">*</span> <span class="n">r</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mf">1.</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">)</span>
967
    <span class="k">return</span> <span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mf">1.</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">gamma</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="n">sigma</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">gamma</span><span class="p">)</span>
968
</pre></div>
969

    
970
</div>
971
</div>
972
</div>
973
<div class="text_cell_render border-box-sizing rendered_html">
974
<h2>
975
  Checks / Plots
976
</h2>
977
</div>
978
<div class="text_cell_render border-box-sizing rendered_html">
979
<h3>
980
  For dP(r) / (dx dy)
981
</h3>
982
</div>
983
<div class="cell border-box-sizing code_cell vbox">
984
<div class="input hbox">
985
<div class="prompt input_prompt">In&nbsp;[11]:</div>
986
<div class="input_area box-flex1">
987
<div class="highlight"><pre><span class="c"># Check normalizations</span>
988
<span class="n">r_step</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c"># deg</span>
989

    
990
<span class="k">for</span> <span class="n">r_max</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">]:</span> <span class="c"># deg</span>
991
    <span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span><span class="p">)</span>
992
    <span class="n">pdf_gauss</span> <span class="o">=</span> <span class="n">gauss</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
993
    <span class="n">pdf_king_2</span> <span class="o">=</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
994
    <span class="n">pdf_king_3</span> <span class="o">=</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
995
    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="p">[</span><span class="n">pdf_gauss</span><span class="p">,</span> <span class="n">pdf_king_2</span><span class="p">,</span> <span class="n">pdf_king_3</span><span class="p">]:</span>
996
        <span class="n">norm</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">r</span> <span class="o">*</span> <span class="n">r_step</span> <span class="o">*</span> <span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
997
        <span class="k">print</span> <span class="n">norm</span>
998
</pre></div>
999

    
1000
</div>
1001
</div>
1002
<div class="vbox output_wrapper">
1003
<div class="output vbox">
1004
<div class="hbox output_area">
1005
<div class="prompt output_prompt"></div>
1006
<div class="output_subarea output_stream output_stdout">
1007
<pre>0.999787429515
1008
0.67113834294
1009
0.961792040763
1010
0.999791640617
1011
0.996466292832
1012
0.999861093167
1013
</pre>
1014
</div>
1015
</div>
1016
</div>
1017
</div>
1018
</div>
1019
<div class="cell border-box-sizing code_cell vbox">
1020
<div class="input hbox">
1021
<div class="prompt input_prompt">In&nbsp;[12]:</div>
1022
<div class="input_area box-flex1">
1023
<div class="highlight"><pre><span class="c"># Plot PDF dP / (dx dy)</span>
1024
<span class="c"># Note that this only contains 67 % of the events for the King profile (see previous cell)</span>
1025
<span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span> <span class="c"># deg</span>
1026
<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span><span class="p">)</span>
1027

    
1028
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">gauss</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
1029
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
1030
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
1031
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;r (deg)&#39;</span><span class="p">)</span>
1032
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP / (dx dy) (deg^-2)&#39;</span><span class="p">)</span>
1033
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">);</span>
1034
</pre></div>
1035

    
1036
</div>
1037
</div>
1038
<div class="vbox output_wrapper">
1039
<div class="output vbox">
1040
<div class="hbox output_area">
1041
<div class="prompt output_prompt"></div>
1042
<div class="output_subarea output_display_data">
1043
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXwAAAEMCAYAAADHxQ0LAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz
1044
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1045
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1046
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1047
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1048
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1049
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1050
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1051
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1052
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1053
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1054
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1055
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1056
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1057
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1058
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1059
QQP27t37zP1bt27lhRde0Mvfe+DAARo1aqR97unpyb59+3Kcd/78eW3qxtnZGTc3N4KDg7XHK1as
1060
yNWrV/USo6HJXDpClFFHjx4lNTWVl19++Znnubm5cejQISpXrsz69et5/fXXuXbtGi4uLjnO1Wg0
1061
2tXqLl++zIIFCwgKCqJy5crcvHmTjIyMXPcDnDx5kkmTJhXo7wgPD2fp0qW5Hm/VqhUvv/wyt27d
1062
wt7eXrvf3t6eK1eu5Di/R48e7NixA1AXYYqKisLNzU173NPTk+Dg4Gz7ygpp8IXQE12t4lnY6Xpi
1063
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1064
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1065
2rQ8/97Y2FgsLCy0z83MzEhISMhxnqmpKR4eHgBs27aNZs2a4eXlpT3u4OCgXcejrJGUjhB6oii6
1066
2QrLycmJmJgYsrKytPuOHDnCw4cPcXJy0jbc/v7+eHt74+DggIODA6GhocTExORZvpubG3PmzGHq
1067
1Km4uLgwZMgQbW/5afsBMjMzs5Xx4MEDxowZg7+/P/v27aNTp06sXr1a29gXhK2tbbZJxZKTk3F0
1068
dMz1/NjYWFauXMnq1auz7be0tCQtLa3A9ZcG0sMXooxq3bo15ubmbN68mX79+j31nBs3bvD222+z
1069
d+9eWrdujUajwdvbW9twWllZZRubHhUVRfXq1bXPhwwZwpAhQ3j06BGjR4/m448/xt/fP9f9JibZ
1070
m5wFCxYwbtw4bc88NTUVKyurbOfkN6VTp04dgoKCtPtjYmJo2rTpU1+jKArTp09n2bJl2NjYcOPG
1071
DWrWrAmo1z6e9UVRmkmDL0QZZW9vz5QpU/Dz80NRFLp27Yq1tTVnz54lMTERgMTERDQaDc7OzmRl
1072
ZeHv709oaKi2DC8vL9asWcPXX3/Nn3/+yYEDB2jRogWgpmJu3bpF27ZtMTc3x8LCAkVRct0P6tKn
1073
CQkJ2NjYAPDo0SPthdbz58/j7u6Oqalptr8jvymdDh06MHHiRO3zkJAQvv32WwCuXbtG7dq1temk
1074
efPmMWDAAFJSUjhx4gTJycnaBj8qKoqGDRsW/A0vDQo9vqeIDFi1EDpRWj7Da9asUVq0aKFYWVkp
1075
FStWVFq2bKksXbpUSUtLUxRFUT777DPF0dFRcXZ2VsaPH6/4+voqy5cvVxRFUYKCghR3d3fF1tZW
1076
GTp0qPLqq68qn3/+uaIoinL27FmlRYsWiq2treLo6Kj06tVLiYqKynW/oijK8uXLlT179mhjCw8P
1077
V+bMmaNs2LBBmTNnjpKenl6kv9Xf31/56quvlC+++EJZvXq1dr+3t7cSEhKiKIqiHDx4UDEyMlI0
1078
Go2i0WgUIyMj5datW9pz33zzTSU5OblIcehCbp+vonzuZAEUIQpJPsMFFxsby8yZM/n6668NHcpT
1079
paSkMGnSJL7//ntDh1L2FkCJS4kzZPVCiGJmb2+Ps7Nzvi4KG8K6desYPXq0ocPQG4M2+G7z3Gi1
1080
rBVf7f+KkKgQ6S0JUQ689957bNq0ydBh5BAZGYmDgwP169c3dCh6Y9CUTmpGKgdvHGTblW1svrSZ
1081
5lWbs6rPKixMLPIuQAgDk5SO0Cd9pHRKTA4/JSOFkX+M5GbcTf4Y/AfOVs6GCEuIfJMGX+hTmcvh
1082
/5OFiQVr+q3Bp6YPrZe35sr9nLdECyGEKLwS08P/p8VBi5l+eDqnR5/GzsKumCMTIn+khy/0qUyn
1083
dP7Nb5sfcalxrOm3phijEiL/pMEX+lSmUzr/NrPrTE5FnWL12dV5nyyEECJPJbbBtzK14pdXfuGD
1084
XR8Q/jA87xcIIYR4Jr2mdFxdXalQoQLGxsaYmppy4sSJJxXn82fJ7KOzWX9hPYdGHsLYyFhfoQpR
1085
YJLSEfpU6lI6Go2GwMBATp06la2xL4j3Wr2HscZYUjtCCFFEek/pFLUHZKQx4tvO3/KfwP+QkpGi
1086
o6iEEKL80ev0yBqNhs6dO2NsbMzo0aN56623sh2fOnWq9t++vr74+vo+tZy2NdriVdmLhScXMr71
1087
eD1GLIQoiuvXr1OrVi1Dh1HiRUVFYWdnl2Pu/6cJDAwkMDBQNxUXep7NfLhz546iKIpy7949xdPT
1088
Uzlw4ID2WEGrDv0rVKn4XUUlNjlWpzEKUVh6/t9H79zd3ZX9+/frrLxr164pa9eu1Vl5ZVl6eroy
1089
ZcqUZ56T2+erKJ87vaZ0qlSpAqirwPft27fQeXwA90ruvFTvJb478p2uwhOizHN1dWXPnj3a5+vW
1090
rcPR0ZGDBw8SGhpKhw4ddFbX4sWLGTJkiM7K+6fNmzfzzTffMH36dH7++edczzt9+jQTJkzQSwyF
1091
rbNOnTqYm5vj4uKCv78/ACYmJvTs2VP7vNgU+qsiD4mJiUp8fLyiKIqSkJCgtGnTRtm1a5f2eGGq
1092
vhl7U3H81lG5HX9bZ3EKUVh6/N9HZ1xdXbULjqxcuVJxcnJSjh49qvN6Tp8+rcydO1fn5SqKosTG
1093
xipNmzbVPm/VqpUSHR2d47xZs2Ypffv2VUaMGKGXOJ4mP3UuWbJEuXHjxlMXdxk6dGiur8vt81WU
1094
z53eevh//fUX7du3x8vLi5YtW/LSSy/RtWvXIpVZ3a46wz2HM/PITB1FKUTZpygKixcvZsKECQQE
1095
BNCqVSsgZ+/f1dWVWbNm4enpib29PYMHDyY1NRVQlwv09vamQoUKDBw4kEGDBvH5559rX7t161Ze
1096
eOEFvcR/4MAB7TKIAJ6enuzbty/HeePHj+fll1/WSwy5yU+dZmZm1KhRI8d6vqBmP65evaqv8HLQ
1097
20XbWrVqcfr0aZ2X+17L92i6pClTfadSwbyCzssXoqxZuHAhhw8fZu/evTRu3Fi7X6PRaNd4ffz8
1098
t99+Y9euXZibm9O2bVtWrlzJyJEj6du3LxMmTMDPz48tW7YwePBgPv74Y+1rT548yaRJkwoUV34X
1099
J7916xb29vba/fb29ly58vTJFZUijgrMb0wFqfPkyZOkpqYSHx9PvXr16N27t/aYp6cnwcHBuLm5
1100
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1101
775L5cqVAejVqxenT5/m2LFjZGZm8n//938A9O3bV7uQ+WNJSUnZvjxAXeR88uTJREdHExQUhK+v
1102
Lz179mTMmDFA/hcnj42NxcLiyRoZZmZmJCQkPPXcf8fwb/Hx8YwaNYqQkBD69u3LjBkziIyMJDIy
1103
kjZt2uQ7poLU2alTJ/r27Quoi8J36NBB+wXm4OBAWFhYgeorilLX4AOMbz2eQRsG8U6LdzAxKpV/
1104
gigHitJQ64pGo2HRokV89dVXjBo1iuXLlz/z/MeNPYCVlRV37twhKiqKqlWrZjuvevXq2Xq2mZmZ
1105
2Y4/ePCAMWPGsH37diwsLOjTpw+rVq3Czq7gs9/a2tpy//597fPk5GRcXFyeem5evW1/f3/mzZuH
1106
i4sLmzdv5siRI9y9e5d+/foVOK781vnPXwQODg4EBgbSp08fACwtLUlLSyt03QVVKlvLFlVbUNW2
1107
KpsvbaZ/o/6GDkeIEs3FxYU9e/bg4+ODn58fCxcuLNDrq1Spwu3bt7Ptu3nzZrY0xL/z0wsWLGDc
1108
uHHannlqamqOMef5TZ/UqVOHoKAg7f6YmBiaNm361Nfk1dseO3YsxsbqFC19+vRh+vTp2e7/KUxK
1109
51l1rl69mi1btrB+/XoAEhMTs71XcXFxODo6PjNmXSqVDT7AB60+YNbRWdLgC5EPVapU0Tb648eP
1110
5/vvv8/zNY97rq1bt8bY2Jj58+czZswYtm3bxsmTJ7NdpK1cuTIJCQnY2NgA8OjRI+2F1vPnz+Pu
1111
7o6pqWm28vObPunQoQMTJ07UPg8JCeHbb78F4Nq1a9SuXVvb6D6tt33lyhXq1KmDkZGRtrF/LCIi
1112
QnsRuyAx/dPT6nwcl6urqzaFlZSURHR0dLb3LSoqioYNGxaovqIosbNl5qVPgz7cTbjLsVvHDB2K
1113
EKVC9erV2bt3Lxs2bGDSpEl59oYfX9Q1NTXl999/Z/ny5Tg4OLBmzRpeeuklzMzMtOf6+Phku89m
1114
7NixBAQEsHHjRnbv3s306dMLHbe1tTUTJ07k66+/5ssvv2TixIlUqlQJgAEDBmgHh8yfP5+ffvqJ
1115
wMBAvvjiC+Lj4wHo3bs3AQEBTy27efPmhY7rWXU+jqtdu3ZERUUxZ84cPvvsM9atW5ftl87p06dp
1116
27ZtkWIoiBK7AEp+/HDsB47cOsKv/X/VUVRC5F95ni2zZcuW+Pn5MXz4cEC9sDpz5ky+/vprA0eW
1117
U1paGsePH6d9+/bZ9gcFBXHv3j169OhhkLhSUlKYNGlSrr+2St1smfo2wmsEAdcCiE6MNnQoQpRp
1118
Bw4c4O7du2RkZLBq1SpCQ0N58cUXtcft7e1xdnYmJibGgFE+3aZNm2jTpk2O/RcuXMDHx8cAEanW
1119
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1120
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1386
"></img>
1387
</div>
1388
</div>
1389
</div>
1390
</div>
1391
</div>
1392
<div class="cell border-box-sizing code_cell vbox">
1393
<div class="input hbox">
1394
<div class="prompt input_prompt">In&nbsp;[13]:</div>
1395
<div class="input_area box-flex1">
1396
<div class="highlight"><pre><span class="c"># Same plot as previous cell, but with log y axis scale</span>
1397
<span class="c"># With a log scale for y it can be clearly seen that at large r</span>
1398
<span class="c"># the Gauss profile is a parabola and the King profile is a line,</span>
1399
<span class="c"># i.e. has a power-law decrease.</span>
1400
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">gauss</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
1401
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
1402
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
1403
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;r (deg)&#39;</span><span class="p">)</span>
1404
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP(r) / (dx dy) (deg^-2)&#39;</span><span class="p">)</span>
1405
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">)</span>
1406
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mf">1e-3</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
1407
<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">();</span>
1408
</pre></div>
1409

    
1410
</div>
1411
</div>
1412
<div class="vbox output_wrapper">
1413
<div class="output vbox">
1414
<div class="hbox output_area">
1415
<div class="prompt output_prompt"></div>
1416
<div class="output_subarea output_display_data">
1417
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1796
"></img>
1797
</div>
1798
</div>
1799
</div>
1800
</div>
1801
</div>
1802
<div class="text_cell_render border-box-sizing rendered_html">
1803
<h3>
1804
  For dP(r^2) / dr^2
1805
</h3>
1806
</div>
1807
<div class="cell border-box-sizing code_cell vbox">
1808
<div class="input hbox">
1809
<div class="prompt input_prompt">In&nbsp;[14]:</div>
1810
<div class="input_area box-flex1">
1811
<div class="highlight"><pre><span class="c"># Check normalizations</span>
1812
<span class="n">r2_step</span> <span class="o">=</span> <span class="mf">0.001</span> <span class="c"># deg^2</span>
1813

    
1814
<span class="k">for</span> <span class="n">r2_max</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1000</span><span class="p">]:</span> <span class="c"># deg^2</span>
1815
    <span class="n">r2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r2_max</span><span class="p">,</span> <span class="n">r2_step</span><span class="p">)</span>
1816
    <span class="n">pdf_gauss</span> <span class="o">=</span> <span class="n">gauss_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
1817
    <span class="n">pdf_king_2</span> <span class="o">=</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
1818
    <span class="n">pdf_king_3</span> <span class="o">=</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
1819
    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="p">[</span><span class="n">pdf_gauss</span><span class="p">,</span> <span class="n">pdf_king_2</span><span class="p">,</span> <span class="n">pdf_king_3</span><span class="p">]:</span>
1820
        <span class="n">norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">r2_step</span> <span class="o">*</span> <span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
1821
        <span class="k">print</span> <span class="n">norm</span>
1822
</pre></div>
1823

    
1824
</div>
1825
</div>
1826
<div class="vbox output_wrapper">
1827
<div class="output vbox">
1828
<div class="hbox output_area">
1829
<div class="prompt output_prompt"></div>
1830
<div class="output_subarea output_stream output_stdout">
1831
<pre>1.00625927081
1832
0.674687757878
1833
0.966684146376
1834
1.0062630208
1835
0.991133876887
1836
1.0041752896
1837
</pre>
1838
</div>
1839
</div>
1840
</div>
1841
</div>
1842
</div>
1843
<div class="cell border-box-sizing code_cell vbox">
1844
<div class="input hbox">
1845
<div class="prompt input_prompt">In&nbsp;[15]:</div>
1846
<div class="input_area box-flex1">
1847
<div class="highlight"><pre><span class="c"># Plot PDF dP(r^2) / dr^2</span>
1848
<span class="n">r2_max</span><span class="p">,</span> <span class="n">r2_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.001</span> <span class="c"># deg</span>
1849
<span class="n">r2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r2_max</span><span class="p">,</span> <span class="n">r2_step</span><span class="p">)</span>
1850

    
1851
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">gauss_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
1852
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
1853
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
1854
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;$r^2$ (deg$^2$)&#39;</span><span class="p">)</span>
1855
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP(r^2) / dr^2 (deg^-2)&#39;</span><span class="p">)</span>
1856
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">);</span>
1857
</pre></div>
1858

    
1859
</div>
1860
</div>
1861
<div class="vbox output_wrapper">
1862
<div class="output vbox">
1863
<div class="hbox output_area">
1864
<div class="prompt output_prompt"></div>
1865
<div class="output_subarea output_display_data">
1866
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1939
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1940
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1941
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1942
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1943
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1944
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1945
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1946
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1947
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1948
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1949
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1950
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1951
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1952
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1953
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1954
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1955
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1956
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1957
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1958
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1959
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1960
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1961
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1962
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1963
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1964
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1965
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1966
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1967
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1968
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1969
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1970
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1971
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1972
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1973
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1974
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1975
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1976
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1977
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1978
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1979
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1980
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1981
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1982
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1983
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1984
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1985
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1986
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1987
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1988
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1989
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1990
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1991
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1992
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1993
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1994
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1995
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1996
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1997
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1998
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1999
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2000
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2001
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2002
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2003
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2004
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2005
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2006
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2007
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2008
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2009
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2010
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2011
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2012
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2013
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2014
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2015
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2016
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2017
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2018
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2019
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2020
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2021
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2022
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2023
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2024
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2025
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2026
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2027
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2028
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2029
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2030
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2031
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2032
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2033
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2034
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2035
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2036
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2037
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2038
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2039
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2040
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2041
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2042
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2043
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2044
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2045
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2046
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2047
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2048
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2049
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2050
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2051
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2052
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2053
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2054
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2055
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2056
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2057
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2058
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2059
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2060
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2061
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2062
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2063
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2064
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2065
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2066
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2067
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2068
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2069
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2070
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2071
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2088
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2089
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2090
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2091
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2092
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2093
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2094
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2095
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2096
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2097
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2098
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2099
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2100
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2101
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2102
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2103
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2104
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2105
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2107
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2109
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2110
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2111
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2112
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2113
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2115
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2119
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2121
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2122
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2123
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2124
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2125
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2126
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2127
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2128
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2129
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2130
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2131
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2132
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2133
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2142
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2143
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2144
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2145
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2147
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2148
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2149
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2150
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2151
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2152
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2153
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2154
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2155
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2156
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2157
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2158
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2159
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2160
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2161
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2162
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2163
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2164
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2165
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2166
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2167
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2168
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2169
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2170
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2171
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2172
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2173
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2174
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2175
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2176
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2177
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2178
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2179
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2180
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2181
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2182
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2183
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2184
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2186
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2187
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2192
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2193
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2194
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2195
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2196
QmCC
2197
"></img>
2198
</div>
2199
</div>
2200
</div>
2201
</div>
2202
</div>
2203
<div class="cell border-box-sizing code_cell vbox">
2204
<div class="input hbox">
2205
<div class="prompt input_prompt">In&nbsp;[16]:</div>
2206
<div class="input_area box-flex1">
2207
<div class="highlight"><pre><span class="c"># Plot PDF dP(r^2) / dr^2</span>
2208
<span class="n">r2_max</span><span class="p">,</span> <span class="n">r2_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.001</span> <span class="c"># deg</span>
2209
<span class="n">r2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r2_max</span><span class="p">,</span> <span class="n">r2_step</span><span class="p">)</span>
2210

    
2211
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">gauss_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
2212
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
2213
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">king_B</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
2214
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;$r^2$ (deg$^2$)&#39;</span><span class="p">)</span>
2215
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP(r^2) / dr^2 (deg^-2)&#39;</span><span class="p">)</span>
2216
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">)</span>
2217
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mf">1e-3</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
2218
<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">();</span>
2219
</pre></div>
2220

    
2221
</div>
2222
</div>
2223
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2224
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2225
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2226
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2227
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"></img>
2597
</div>
2598
</div>
2599
</div>
2600
</div>
2601
</div>
2602
<div class="text_cell_render border-box-sizing rendered_html">
2603
<h3>
2604
  For dP(r) / dr
2605
</h3>
2606
</div>
2607
<div class="cell border-box-sizing code_cell vbox">
2608
<div class="input hbox">
2609
<div class="prompt input_prompt">In&nbsp;[17]:</div>
2610
<div class="input_area box-flex1">
2611
<div class="highlight"><pre><span class="c"># Check normalizations</span>
2612
<span class="n">r_step</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c"># deg</span>
2613

    
2614
<span class="k">for</span> <span class="n">r_max</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">]:</span> <span class="c"># deg</span>
2615
    <span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span><span class="p">)</span>
2616
    <span class="n">pdf_gauss</span> <span class="o">=</span> <span class="n">gauss_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
2617
    <span class="n">pdf_king_2</span> <span class="o">=</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
2618
    <span class="n">pdf_king_3</span> <span class="o">=</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
2619
    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="p">[</span><span class="n">pdf_gauss</span><span class="p">,</span> <span class="n">pdf_king_2</span><span class="p">,</span> <span class="n">pdf_king_3</span><span class="p">]:</span>
2620
        <span class="n">norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">r_step</span> <span class="o">*</span> <span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
2621
        <span class="k">print</span> <span class="n">norm</span>
2622
</pre></div>
2623

    
2624
</div>
2625
</div>
2626
<div class="vbox output_wrapper">
2627
<div class="output vbox">
2628
<div class="hbox output_area">
2629
<div class="prompt output_prompt"></div>
2630
<div class="output_subarea output_stream output_stdout">
2631
<pre>0.999787429515
2632
0.67113834294
2633
0.961792040763
2634
0.999791640617
2635
0.996466292832
2636
0.999861093167
2637
</pre>
2638
</div>
2639
</div>
2640
</div>
2641
</div>
2642
</div>
2643
<div class="cell border-box-sizing code_cell vbox">
2644
<div class="input hbox">
2645
<div class="prompt input_prompt">In&nbsp;[18]:</div>
2646
<div class="input_area box-flex1">
2647
<div class="highlight"><pre><span class="c"># Plot PDF dP(r) / dr</span>
2648
<span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span> <span class="c"># deg</span>
2649
<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span><span class="p">)</span>
2650

    
2651
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">gauss_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
2652
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
2653
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
2654
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;r (deg)&#39;</span><span class="p">)</span>
2655
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP / dr (deg^-1)&#39;</span><span class="p">)</span>
2656
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">);</span>
2657
</pre></div>
2658

    
2659
</div>
2660
</div>
2661
<div class="vbox output_wrapper">
2662
<div class="output vbox">
2663
<div class="hbox output_area">
2664
<div class="prompt output_prompt"></div>
2665
<div class="output_subarea output_display_data">
2666
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2740
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2746
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2747
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2748
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2749
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2750
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2751
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2752
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2756
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"></img>
3093
</div>
3094
</div>
3095
</div>
3096
</div>
3097
</div>
3098
<div class="cell border-box-sizing code_cell vbox">
3099
<div class="input hbox">
3100
<div class="prompt input_prompt">In&nbsp;[19]:</div>
3101
<div class="input_area box-flex1">
3102
<div class="highlight"><pre><span class="c"># Plot PDF dP(r) / dr</span>
3103
<span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span> <span class="c"># deg</span>
3104
<span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r_max</span><span class="p">,</span> <span class="n">r_step</span><span class="p">)</span>
3105

    
3106
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">gauss_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;Gauss$(\sigma=0.2)$&#39;</span><span class="p">);</span>
3107
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.1, \gamma=1.5)$&#39;</span><span class="p">);</span>
3108
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">king_C</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;King$(\sigma=0.2, \gamma=3)$&#39;</span><span class="p">);</span>
3109
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;r (deg)&#39;</span><span class="p">)</span>
3110
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP / dr (deg^-1)&#39;</span><span class="p">)</span>
3111
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">)</span>
3112
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mf">1e-3</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
3113

    
3114

    
3115
<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">();</span>
3116
</pre></div>
3117

    
3118
</div>
3119
</div>
3120
<div class="vbox output_wrapper">
3121
<div class="output vbox">
3122
<div class="hbox output_area">
3123
<div class="prompt output_prompt"></div>
3124
<div class="output_subarea output_display_data">
3125
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3420
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3428
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3429
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3430
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3431
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3444
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3445
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3446
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3447
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3448
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3450
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3451
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3452
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3453
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3454
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3455
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3457
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3458
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3459
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3460
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3461
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3462
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3463
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3466
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3467
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3469
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3470
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3471
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3472
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3473
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3474
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3475
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3478
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3481
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3482
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3483
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3488
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3489
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3490
xpggXDAYY4wJwgWDMcaYIP8FsriuL0Zd8AcAAAAASUVORK5CYII=
3491
"></img>
3492
</div>
3493
</div>
3494
</div>
3495
</div>
3496
</div>
3497
<div class="text_cell_render border-box-sizing rendered_html">
3498
<h2>
3499
  Gammalib
3500
</h2>
3501
</div>
3502
<div class="text_cell_render border-box-sizing rendered_html">
3503
<p>We want to use gammalib for most of our simulations / fitting.</p>
3504
<p>At the moment the Fermi LAT PSF is fully implemented in gammalib, for HESS there is the <code>GCTAPsf2D</code> which implements a triple-Gauss model:
3505
http://gammalib.sourceforge.net/doxygen/classGCTAPsf2D.html</p>
3506
</div>
3507
<div class="cell border-box-sizing code_cell vbox">
3508
<div class="input hbox">
3509
<div class="prompt input_prompt">In&nbsp;[120]:</div>
3510
<div class="input_area box-flex1">
3511
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">gammalib</span>
3512
</pre></div>
3513

    
3514
</div>
3515
</div>
3516
</div>
3517
<div class="cell border-box-sizing code_cell vbox">
3518
<div class="input hbox">
3519
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3520
<div class="input_area box-flex1">
3521
<div class="highlight"><pre><span class="c"># As far as I can see the PSF has to be initialized from a FITS file, which is not nice for simulations / checks ...</span>
3522
<span class="c"># This is an exmple PSF, I think for HD / Hillas and some Crab run, but I&#39;m not sure ...</span>
3523
<span class="n">psf</span> <span class="o">=</span> <span class="n">gammalib</span><span class="o">.</span><span class="n">GCTAPsf2D</span><span class="p">(</span><span class="s">&#39;/Users/deil/code/gammalib/inst/cta/test/caldb/dc1/psf.fits&#39;</span><span class="p">)</span>
3524
</pre></div>
3525

    
3526
</div>
3527
</div>
3528
</div>
3529
<div class="cell border-box-sizing code_cell vbox">
3530
<div class="input hbox">
3531
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3532
<div class="input_area box-flex1">
3533
<div class="highlight"><pre><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">degrees</span><span class="p">(</span><span class="n">psf</span><span class="o">.</span><span class="n">delta_max</span><span class="p">(</span><span class="n">logE</span><span class="p">))</span> <span class="k">for</span> <span class="n">logE</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)]</span>
3534
</pre></div>
3535

    
3536
</div>
3537
</div>
3538
</div>
3539
<div class="cell border-box-sizing code_cell vbox">
3540
<div class="input hbox">
3541
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3542
<div class="input_area box-flex1">
3543
<div class="highlight"><pre><span class="c"># This is how you can evaluate the psf, i.e. compute the PDF (in radians^(-2))</span>
3544
<span class="c"># at a given offset &quot;delta&quot; (in radians) and a given logE (in TeV?)</span>
3545
<span class="n">delta</span><span class="p">,</span> <span class="n">logE</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span>
3546
<span class="n">psf</span><span class="p">(</span><span class="n">delta</span><span class="p">,</span> <span class="n">logE</span><span class="p">)</span>
3547
</pre></div>
3548

    
3549
</div>
3550
</div>
3551
</div>
3552
<div class="cell border-box-sizing code_cell vbox">
3553
<div class="input hbox">
3554
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3555
<div class="input_area box-flex1">
3556
<div class="highlight"><pre><span class="c"># Let&#39;s plot the psf at 1 TeV, i.e. at </span>
3557
<span class="n">x_max</span><span class="p">,</span> <span class="n">x_step</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span> <span class="c"># deg</span>
3558
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">x_step</span><span class="p">)</span>
3559
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
3560
<span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
3561
    <span class="c"># We use deg, gammalib uses radians, so convert before and after call</span>
3562
    <span class="n">y</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">180</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">psf</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">radians</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">ii</span><span class="p">]),</span> <span class="mi">0</span><span class="p">)</span>
3563
</pre></div>
3564

    
3565
</div>
3566
</div>
3567
</div>
3568
<div class="cell border-box-sizing code_cell vbox">
3569
<div class="input hbox">
3570
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3571
<div class="input_area box-flex1">
3572
<div class="highlight"><pre><span class="c"># Check normalization</span>
3573
<span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">pi</span> <span class="o">*</span> <span class="n">x</span> <span class="o">*</span> <span class="n">x_step</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
3574
</pre></div>
3575

    
3576
</div>
3577
</div>
3578
</div>
3579
<div class="cell border-box-sizing code_cell vbox">
3580
<div class="input hbox">
3581
<div class="prompt input_prompt">In&nbsp;[&nbsp;]:</div>
3582
<div class="input_area box-flex1">
3583
<div class="highlight"><pre><span class="c"># Plot</span>
3584
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&#39;gammalib&#39;</span><span class="p">);</span>
3585

    
3586
<span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">()</span>
3587
<span class="c">#plt.loglog()</span>
3588
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">&#39;PDF of 2D distributions on x-axis&#39;</span><span class="p">)</span>
3589
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;x (deg)&#39;</span><span class="p">)</span>
3590
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;dP / dx (deg^-2)&#39;</span><span class="p">)</span>
3591
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">);</span>
3592
</pre></div>
3593

    
3594
</div>
3595
</div>
3596
</div>
3597
<div class="text_cell_render border-box-sizing rendered_html">
3598
<h2>
3599
  Backup Cells
3600
</h2>
3601
</div>
3602
<div class="cell border-box-sizing code_cell vbox">
3603
<div class="input hbox">
3604
<div class="prompt input_prompt">In&nbsp;[20]:</div>
3605
<div class="input_area box-flex1">
3606
<div class="highlight"><pre><span class="k">class</span> <span class="nc">Fermi_LAT_PSF</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
3607
    <span class="sd">&quot;&quot;&quot;Fermi LAT PSF model</span>
3608

    
3609
<span class="sd">    References:</span>
3610
<span class="sd">    [1] http://adsabs.harvard.edu/abs/2012ApJS..203....4A</span>
3611
<span class="sd">    [2] http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_LAT_IRFs/IRF_PSF.html</span>
3612
<span class="sd">    [3] http://gammalib.sourceforge.net/doxygen/classGLATPsfV3.html</span>
3613
<span class="sd">    &quot;&quot;&quot;</span>
3614

    
3615
    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c0</span><span class="o">=</span><span class="mf">3.32</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="mf">0.022</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="mf">0.80</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.2</span><span class="p">):</span>
3616
        <span class="bp">self</span><span class="o">.</span><span class="n">scale_pars</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">c0</span><span class="o">=</span><span class="n">c0</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="n">c1</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="n">beta</span><span class="p">)</span>
3617
        <span class="bp">self</span><span class="o">.</span><span class="n">radial_pars</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">sigma</span><span class="o">=</span><span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">gamma</span><span class="p">)</span>
3618

    
3619
    <span class="nd">@staticmethod</span>
3620
    <span class="k">def</span> <span class="nf">default</span><span class="p">():</span>
3621
        <span class="sd">&quot;&quot;&quot;Set default parameters&quot;&quot;&quot;</span>
3622
        <span class="c"># Front converting values from Table 13 in [1]</span>
3623
        <span class="n">c0</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">beta</span> <span class="o">=</span> <span class="mf">3.32</span><span class="p">,</span> <span class="mf">0.022</span><span class="p">,</span> <span class="mf">0.80</span>
3624
        <span class="c"># Random values</span>
3625
        <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">2</span>
3626
        <span class="k">return</span> <span class="n">Fermi_LAT_PSF</span><span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
3627
        
3628
    <span class="k">def</span> <span class="nf">scale_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">energy</span><span class="p">):</span>
3629
        <span class="sd">&quot;&quot;&quot;Scale factor containing most of the energy dependence</span>
3630
<span class="sd">        energy in MeV</span>
3631

    
3632
<span class="sd">        Reference: Equation (34) in [1].</span>
3633
<span class="sd">        &quot;&quot;&quot;</span>
3634
        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale_pars</span>
3635
        <span class="n">c0</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="n">beta</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="s">&#39;c0&#39;</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="s">&#39;c1&#39;</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="s">&#39;beta&#39;</span><span class="p">]</span>
3636
        <span class="n">term1</span> <span class="o">=</span> <span class="n">c0</span> <span class="o">*</span> <span class="p">(</span><span class="n">energy</span> <span class="o">/</span> <span class="mi">100</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span>
3637
        <span class="n">term2</span> <span class="o">=</span> <span class="n">term1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">c1</span> <span class="o">**</span> <span class="mi">2</span>
3638
        <span class="k">return</span> <span class="n">sort</span><span class="p">(</span><span class="n">term2</span><span class="p">)</span>
3639

    
3640
    <span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">energy</span><span class="p">):</span>
3641
        <span class="sd">&quot;&quot;&quot;Scaled angular deviation x.</span>
3642
<span class="sd">        Reference: Equation (35) in [1]</span>
3643
<span class="sd">        &quot;&quot;&quot;</span>
3644
        <span class="k">return</span> <span class="n">r</span> <span class="o">/</span> <span class="n">scale_factor</span><span class="p">(</span><span class="n">energy</span><span class="p">)</span>
3645

    
3646
    <span class="k">def</span> <span class="nf">K</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
3647
        <span class="sd">&quot;&quot;&quot;Radial PDF</span>
3648
<span class="sd">        Reference: Equation (36) in [1]</span>
3649
<span class="sd">        &quot;&quot;&quot;</span>
3650
        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radial_pars</span>
3651
        <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="s">&#39;sigma&#39;</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="s">&#39;gamma&#39;</span><span class="p">]</span>
3652
        <span class="k">return</span> <span class="n">king</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
3653
</pre></div>
3654

    
3655
</div>
3656
</div>
3657
</div>
3658

    
3659
</body>
3660
</html>