day2/session2.tex
author Puneeth Chaganti <punchagan@fossee.in>
Thu, 08 Oct 2009 01:30:59 +0530
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Tutorial slides on Python.
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%
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% Author: Prabhu Ramachandran <prabhu at aero.iitb.ac.in>
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% Copyright (c) 2005-2009, Prabhu Ramachandran
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\documentclass[14pt,compress]{beamer}
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%\documentclass[draft]{beamer}
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%\documentclass[compress,handout]{beamer}
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%\usepackage{pgfpages} 
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%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
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% Modified from: generic-ornate-15min-45min.de.tex
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\mode<presentation>
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{
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  \usetheme{Warsaw}
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  \useoutertheme{split}
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  \setbeamercovered{transparent}
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}
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\usepackage[english]{babel}
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\usepackage[latin1]{inputenc}
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%\usepackage{times}
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\usepackage[T1]{fontenc}
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% Taken from Fernando's slides.
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\usepackage{ae,aecompl}
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\usepackage{mathpazo,courier,euler}
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\usepackage[scaled=.95]{helvet}
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\definecolor{darkgreen}{rgb}{0,0.5,0}
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\usepackage{listings}
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\lstset{language=Python,
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    basicstyle=\ttfamily\bfseries,
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    commentstyle=\color{red}\itshape,
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  stringstyle=\color{darkgreen},
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  showstringspaces=false,
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  keywordstyle=\color{blue}\bfseries}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Macros
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\setbeamercolor{emphbar}{bg=blue!20, fg=black}
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\newcommand{\emphbar}[1]
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{\begin{beamercolorbox}[rounded=true]{emphbar} 
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      {#1}
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 \end{beamercolorbox}
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}
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\newcounter{time}
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\setcounter{time}{0}
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\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
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\newcommand{\typ}[1]{\lstinline{#1}}
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\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
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%%% This is from Fernando's setup.
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% \usepackage{color}
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% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
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% % Use and configure listings package for nicely formatted code
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% \usepackage{listings}
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% \lstset{
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%    language=Python,
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%    basicstyle=\small\ttfamily,
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%    commentstyle=\ttfamily\color{blue},
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%    stringstyle=\ttfamily\color{orange},
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%    showstringspaces=false,
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%    breaklines=true,
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%    postbreak = \space\dots
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% }
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Title page
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\title[]{Numerical Computing with Numpy \& Scipy}
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\author[FOSSEE Team] {Asokan Pichai\\Prabhu Ramachandran}
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\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
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\date[] {11, October 2009}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
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%\logo{\pgfuseimage{iitmlogo}}
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%% Delete this, if you do not want the table of contents to pop up at
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%% the beginning of each subsection:
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\AtBeginSubsection[]
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{
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  \begin{frame}<beamer>
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    \frametitle{Outline}
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    \tableofcontents[currentsection,currentsubsection]
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  \end{frame}
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}
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\AtBeginSection[]
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{
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  \begin{frame}<beamer>
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    \frametitle{Outline}
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    \tableofcontents[currentsection,currentsubsection]
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  \end{frame}
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}
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% If you wish to uncover everything in a step-wise fashion, uncomment
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% the following command: 
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%\beamerdefaultoverlayspecification{<+->}
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%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DOCUMENT STARTS
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\begin{document}
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\begin{frame}
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  \maketitle
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Broadcasting}
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  Try it!
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  \begin{lstlisting}
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    >>> a = np.arange(4)
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    >>> b = np.arange(5)
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    >>> a+b
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    >>> a+3
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    >>> c=np.array([3])
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    >>> a+c
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    >>> b+c
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  \end{lstlisting}
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  \begin{itemize}
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    \item Enter Broadcasting!
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Broadcasting}
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  \begin{columns}
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    \column{0.65\textwidth}
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    \hspace*{-1.5in}
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    \begin{lstlisting}
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      >>> a = np.arange(4)
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      >>> a+3
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      array([3, 4, 5, 6])
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    \end{lstlisting}
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    \column{0.35\textwidth}
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    \includegraphics[height=0.7in, interpolate=true]{data/broadcast_scalar}
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  \end{columns}
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  \begin{itemize}
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    \item Allows functions to take inputs not of the same shape
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    \item 2 rules -
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      \begin{enumerate}
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      \item 1 is (repeatedly) prepended to shapes of smaller arrays
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      \item Size 1 in a dimension changed to Largest size in that dimension
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      \end{enumerate}
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Broadcasting in 3D}
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    \begin{lstlisting}
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      >>> x = np.ones((3, 5))
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      >>> y = np.ones(8)
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      >>> (x[..., None] + y).shape
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      (3, 5, 8)
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    \end{lstlisting}
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    \begin{figure}
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      \begin{center}
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      \includegraphics[height=1.5in, interpolate=true]{data/array_3x5x8}        
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      \end{center}
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    \end{figure}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Copies \& Views}
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  Try it!
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  \begin{lstlisting}
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    >>> a = np.array([[1,2,3],[4,5,6]])
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    >>> b = a
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    >>> b is a
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    >>> b[0,0]=0; print a
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    >>> c = a.view()
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    >>> c is a
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    >>> c.base is a
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    >>> c.flags.owndata
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    >>> d = a.copy()
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    >>> d.base is a
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    >>> d.flags.owndata
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Copies \& Views}
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  Try it!
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  \begin{lstlisting}
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    >>> a = np.arange(1,9)
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    >>> a.shape=3,3
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    >>> b = a[0,1:3]
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    >>> c = a[0::2,0::2]
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    >>> a.flags.owndata
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    >>> b.flags.owndata
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    >>> b.base
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    >>> c.base is a
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  \end{lstlisting}
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  \begin{itemize}
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  \item Slicing and Striding just reference the same memory
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  \item They produce views of the data, not copies
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Copies contd \ldots}
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  \begin{lstlisting}
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    >>> b = a[np.array([0,1,2])]
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    array([[1, 2, 3],
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           [4, 5, 6],
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           [7, 8, 9]])
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    >>> b.flags.owndata
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    >>> abool=np.greater(a,2)
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    >>> c = a[abool]
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    >>> c.flags.owndata
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  \end{lstlisting}
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  \begin{itemize}
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  \item Indexing arrays or Boolean arrays produce copies
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  \end{itemize}
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\inctime{15}
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\end{frame}
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\begin{frame}
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    {Intro to SciPy}
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  \begin{itemize}
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  \item \url{http://www.scipy.org}
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  \item Open source scientific libraries for Python
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  \item Based on NumPy
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    \end{itemize}
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\end{frame}
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\begin{frame}
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  \frametitle{SciPy}
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  \begin{itemize}
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  \item Provides:
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    \begin{itemize}
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    \item Linear algebra
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    \item Numerical integration
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    \item Fourier transforms
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    \item Signal processing
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    \item Special functions
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    \item Statistics
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    \item Optimization
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    \item Image processing
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    \item ODE solvers
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    \end{itemize}
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  \item Uses LAPACK, QUADPACK, ODEPACK, FFTPACK etc. from netlib
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  \end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{SciPy - Functions \& Submodules}
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  \begin{itemize}
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    \item All \typ{numpy} functions are in \typ{scipy} namespace
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    \item Domain specific functions organized into subpackages
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    \item Subpackages need to be imported separately
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  \end{itemize}
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  \begin{lstlisting}
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    >>> from scipy import linalg
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Linear Algebra}
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  Try it!
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  \begin{lstlisting}
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    >>> import scipy as sp
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    >>> from scipy import linalg
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    >>> A=sp.mat(np.arange(1,10))
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    >>> A.shape=3,3
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    >>> linalg.inv(A)
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    >>> linalg.det(A)
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    >>> linalg.norm(A)
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    >>> linalg.expm(A) #logm
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    >>> linalg.sinm(A) #cosm, tanm, ...
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Linear Algebra ...}
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  Try it!
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  \begin{lstlisting}
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    >>> A = sp.mat(np.arange(1,10))
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    >>> A.shape=3,3
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    >>> linalg.lu(A)
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    >>> linalg.eig(A)
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    >>> linalg.eigvals(A)
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Solving Linear Equations}
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  \begin{align*}
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    3x + 2y - z  & = 1 \\
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    2x - 2y + 4z  & = -2 \\
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    -x + \frac{1}{2}y -z & = 0
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  \end{align*}
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  To Solve this, 
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  \begin{lstlisting}
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    >>> A = sp.mat([[3,2,-1],[2,-2,4]
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                  ,[-1,1/2,-1]])
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    >>> B = sp.mat([[1],[-2],[0]])
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    >>> linalg.solve(A,B)
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  \end{lstlisting}
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\inctime{15}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Integrate}
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  \begin{itemize}
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    \item Integrating Functions given function object
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    \item Integrating Functions given fixed samples
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    \item Numerical integrators of ODE systems
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  \end{itemize}
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  Calculate $\int^1_0(sin(x) + x^2)dx$
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  \begin{lstlisting}
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    >>> def f(x):
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            return np.sin(x)+x**2
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    >>> integrate.quad(f, 0, 1)
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Integrate \ldots}
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  Numerically solve ODEs\\
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  \begin{align*}
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  \frac{dx}{dt}&=-e^{-t}x^2\\ 
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           x(0)&=2    
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  \end{align*}
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  \begin{lstlisting}
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    def dx_dt(x,t):
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        return -np.exp(-t)*x**2
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    x=integrate.odeint(dx_dt, 2, t)
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    plt.plot(x,t)
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  \end{lstlisting}
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\inctime{10}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Interpolation}
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  Try it!
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  \begin{lstlisting}
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    >>> from scipy import interpolate
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    >>> interpolate.interp1d?
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    >>> x = np.arange(0,2*np.pi,np.pi/4)
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    >>> y = np.sin(x)
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    >>> fl = interpolate.interp1d(x,y,kind='linear')
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    >>> fc = interpolate.interp1d(x,y,kind='cubic')
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    >>> fl(np.pi/3)
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    >>> fc(np.pi/3)
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  \end{lstlisting}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Interpolation - Splines}
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   363
  Cubic Spline of $sin(x)$
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  \begin{lstlisting}
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   365
    x = np.arange(0,2*np.pi,np.pi/4)
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    y = np.sin(x)
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    tck = interpolate.splrep(x,y)
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    X = np.arange(0,2*np.pi,np.pi/50)
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    Y = interpolate.splev(X,tck,der=0)
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    plt.plot(x,y,'o',x,y,X,Y)
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    plt.show()
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  \end{lstlisting}
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\inctime{10}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Signal \& Image Processing}
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    \begin{itemize}
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     \item Convolution
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   380
     \item B-splines
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     \item Filtering
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     \item Filter design
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   383
     \item IIR filter design
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   384
     \item Linear Systems
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   385
     \item LTI Reresentations
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     \item Waveforms
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     \item Window functions
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     \item Wavelets
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   389
    \end{itemize}
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\end{frame}
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   391
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\begin{frame}[fragile]
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  \frametitle{Signal \& Image Processing}
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   394
  Applying a simple median filter
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   395
  \begin{lstlisting}
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   396
    from scipy import signal, ndimage
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   397
    from scipy import lena
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   398
    A=lena().astype('float32')
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   399
    B=signal.medfilt2d(A)
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   400
    imshow(B)
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  \end{lstlisting}
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  Zooming an array - uses spline interpolation
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  \begin{lstlisting}
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    b=ndimage.zoom(A,0.5)
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    imshow(b)
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  \end{lstlisting}
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    \inctime{5}
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\end{frame}
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\begin{frame}[fragile]
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  \frametitle{Problems}
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  The Van der Pol oscillator is a type of nonconservative oscillator with nonlinear damping. It evolves in time according to the second order differential equation:
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  \begin{equation*}
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  \frac{d^2x}{dt^2}+\mu(x^2-1)\frac{dx}{dt}+x= 0
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  \end{equation*}
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\inctime{25}
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\end{frame}
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\end{document}
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- Numpy arrays (30 mins)
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    - Matrices
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    - random number generation.
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    - Image manipulation: jigsaw puzzle.
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    - Monte-carlo integration.
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