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{"nbformat_minor": 0, "cells": [{"source": "Question 2\n====\n\n> Consider a noisy oscillator given by\n> \n $$\n \\mathbf{x}  = \\alpha R \\mathbf{x} + \\mathbf{\\epsilon}\n $$\n>\n> where R is a unitary rotation matrix through a small angle and epsilon is a very small isotropic Gaussian noise term.\n> Initialise at the unit vector. What happens to this system over time if $\\alpha=1$? What if $alpha < 1$? Draw an\n> example picture\n\nSo I'm not going to draw it, it's going to draw itself.\n", "cell_type": "markdown", "metadata": {}}, {"execution_count": 1, "cell_type": "code", "source": "import numpy as np\n%matplotlib inline", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"execution_count": 2, "cell_type": "code", "source": "# initialise unit column vector\nx = np.ones((2,1))", "outputs": [], "metadata": {"collapsed": false, "trusted": true}}, {"execution_count": 3, "cell_type": "code", "source": "#define our update equation\ndef update(x, alpha, R, var_epsilon):\n    return alpha*np.

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name: fresh
channels: !!python/tuple
- defaults
dependencies:
- _nb_ext_conf=0.3.0=py27_0
- anaconda-client=1.5.1=py27_0
- backports=1.0=py27_0
- backports_abc=0.4=py27_0
- clyent=1.2.2=py27_0
- configparser=3.5.0=py27_0

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Collecting calysto-processing
  Using cached calysto_processing-2.0.2.zip
Collecting metakernel (from calysto-processing)
  Using cached metakernel-0.14.0-py2.py3-none-any.whl
Collecting html2text (from calysto-processing)
Requirement already up-to-date: IPython>=3.0 in ./conda/envs/fresh/lib/python2.7/site-packages (from metakernel->calysto-processing)
Building wheels for collected packages: calysto-processing
  Running setup.py bdist_wheel for calysto-processing ... error
  Complete output from command /opt/conda/envs/fresh/bin/python -u -c "import setuptools, tokenize;__file__='/tmp/pip-build-Zz1AnC/calysto-processing/setup.py';exec(compile(getattr(tokenize, 'open', open)(__file__).read().replace('\r\n', '\n'), __file__, 'exec'))" bdist_wheel -d /tmp/tmpa62Bwypip-wheel- --python-tag cp27:
  running bdist_wheel

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\documentclass[border={4pt 4pt 4pt 4pt},preview]{standalone}
\usepackage{mathrsfs,amsmath}
\usepackage{shortbold}
\begin{document}
$my_text
\end{document}

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$ \newcommand{\aB}{\mathbf{a}} \newcommand{\bB}{\mathbf{b}} \newcommand{\cB}{\mathbf{c}} \newcommand{\dB}{\mathbf{d}} \newcommand{\eB}{\mathbf{e}} \newcommand{\fB}{\mathbf{f}} \newcommand{\gB}{\mathbf{g}} \newcommand{\hB}{\mathbf{h}} \newcommand{\iB}{\mathbf{i}}