%\documentclass[10pt,aspectratio=169]{beamer}
\documentclass[10pt]{beamer}
\mode<presentation>

  %\usetheme{Warsaw}
  \usetheme{Madrid}
  %\usetheme{Frankfurt}
  \usecolortheme{spruce}
  \usecolortheme{crane}
  %\setbeamertemplate{background canvas}[vertical shading] [bottom=white,top=white]

%\usepackage[dvipsnames]{xcolor}

\usepackage[T1]{fontenc} 

\usepackage{algorithm}
\usepackage{algpseudocode}

\newfloat{algorithm}{t}{loa}
\providecommand{\algorithmname}{Algorithm}
\floatname{algorithm}{\protect\algorithmname}


\usepackage{fancybox}
\usepackage{cancel}
\usepackage{setspace}
\usepackage{ulem}
\usepackage{pdflscape}
\usepackage{geometry}
\usepackage{framed}
\usepackage{graphicx}
\usepackage{pgfplots}
\usepackage{xcolor}
\usepackage{textpos}

\setbeamertemplate{navigation symbols}{}

\usepackage{amssymb,amsmath,amsfonts,amsthm,latexsym,color}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\definecolor{dred}{rgb}{0.9, 0, 0}
\definecolor{dgreen}{rgb}{0, 0.6, 0}
\definecolor{dblue}{rgb}{0, 0, 0.7}
\newcommand{\red}{\color{red}}
\newcommand{\blue}{\color{blue}}
\newcommand{\dgreen}{\color{dgreen}}
\newcommand{\dblue}{\color{dblue}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ds}{\displaystyle}



\begin{document}

\author{ Michael Monagan and Jamie Mulholland }

\date{}

\input{lec16cont}

\end{document}
