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+\documentclass[12pt,a4paper]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
+\usepackage{enumitem}
+\usepackage{tikz}
+\usepackage{tikz-cd}
+\usepackage{url}
+
+\title{Mathematical software - homework 2}
+\author{Sebastiano Tronto}
+
+\newtheorem{thm}{Theorem}
+\newtheorem{prop}[thm]{Proposition}
+
+\theoremstyle{definition}
+\newtheorem{ex}{Exercise}
+
+\theoremstyle{definition}
+\newtheorem*{remark}{Remark}
+
+\newcommand{\bs}{\textbackslash}
+
+\begin{document}
+
+\noindent\hrulefill
+
+\begin{center}
+\Huge{\textbf{Mathematical Software - Homework 2}}
+\end{center}
+
+\noindent\hrulefill
+\begin{center}
+\begin{tabular}{lcr}
+\texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad &
+\textbf{Deadline: Sunday, April 18th}
+\end{tabular}
+\end{center}
+
+\vspace{1cm}
+
+\begin{center}
+  \emph{\large
+    For each of the following exercises submit a .tex and a .pdf file.
+  }
+\end{center}
+
+\vspace{1cm}
+
+\begin{ex}
+  Create a Latex document containing the following pictures:
+  \begin{enumerate}[label=(\alph*)]
+    \item The regular polygon with $N$-sides centered at the origin of the
+          plane (see below).
+          The number $N$ of sides must be easy to change at will: you should
+          use the \texttt{\textbackslash pgfmathsetmacro} command to set a
+          value for $N$ at the beginning, so that changing only that number
+          makes the whole picture change accordingly.
+      \begin{center}
+        \begin{tikzpicture}[scale=1]
+                \pgfmathsetmacro{\N}{6} 
+                \pgfmathsetmacro{\an}{360/\N}
+                \pgfmathsetmacro{\r}{4}
+        
+                \draw[lightgray!30,thin] (-6,-6) grid (6,6);
+                \draw[->] (-5,0) -- (5,0);
+                \draw[->] (0,-5) -- (0,5);
+
+                \filldraw[draw=green,fill=green!30,thick] (0.3*\r,0)
+                  arc[radius=0.3*\r,start angle=0, end angle=\an]
+                  -- node[right] {$\alpha=\an$}(0,0) --  cycle;
+                \draw[thin] (\an:\r) -- (0,0) circle[radius=\r] -- (\r,0);
+                \draw[line width=1.1pt]
+                  (\r,0)  \foreach \x in {1,2,...,\N} { -- (\x*\an:\r) };
+                \foreach \x in {1,...,\N} {
+                        \filldraw[red] (\x*\an:\r) circle[radius=0.1];
+                };
+        \end{tikzpicture}
+      \end{center}
+    \item The following commutative diagram:
+      \begin{tikzcd}
+        0 \ar[r] & A' \ar[r,hook] \ar[d] & A \ar[r,two heads] \ar[d,"\sim"] &
+          A'' \ar[r] \ar[d] \ar[l,dashed,"s"',bend right] & 0 \\
+        0 \ar[r] & A' \ar[r,hook,"i_{A'}"'] & A'\oplus A''
+          \ar[r,two heads,"\pi_{A''}"'] & A'' \ar[r] & 0 
+      \end{tikzcd}
+  \end{enumerate}
+\end{ex}
+
+\vspace{0.8cm}
+
+\begin{ex}
+  Suppose you have to give a short presentation (10 minutes) on a topic of
+  your choice related to your study programme (you will not be asked to
+  actually perform this presentation). You can choose to talk about a theorem
+  you find important, a result you have seen in class or something else (see
+  below for a list of possible topics). For example, if you talk about an
+  important theorem you can give the theorem statement, explain why this
+  theorem is important and/or possible applications of this result, and
+  optionally an idea of the proof; but you can also deviate from this and talk
+  for example about the historical background that lead to the development of
+  this theorem.
+
+  Your task is to prepare slides for such a presentation using Beamer (Latex).
+  Since the time for the (imaginary) presentation is very short, you should
+  write 5-8 slides (you can have more if some contain very few or no
+  words).
+
+  If you feel like certain slides do not make sense without your explanation
+  (for example if you have one slide with just one picture and you plan to
+  talk with the picture in background), you can write some comments in the
+  .tex file.
+
+  If you can't think of a topic that you like, you can pick one of the
+  following:
+  \begin{itemize}
+    \item The fundamental theorem of arithmetic (about prime numbers)
+    \item The central limit theorem (probability theory)
+    \item Differential equations (what they are, applications,
+          methods to solve them...)
+  \end{itemize}
+\end{ex}
+
+\newpage
+\section*{Grading}
+
+This homework assignment is worth 25\% of your final grade.
+
+\vspace{0.3cm}
+\noindent\textbf{Exercise 1 (10 points).}
+Part (a) is worth 5 points, divided as follows:
+\begin{itemize}
+  \item 3 points for obtaining a regular polygon whose number of sides
+        can be changed by setting a variable with \texttt{\bs pgfmathsetmacro}
+        (or in a similarly easy way).
+  \item 1.5 points for other features of the picture (verteces, angle) that
+        also change accordingly to the same variable.
+  \item 0.5 points for the style of the other elements of the picture. This is
+        a matter of personal preference and it does not need to be exactly the
+        same as the picture, but some key features should remain (e.g.
+        the grid lines should be less visible than the rest of the picture,
+        the circle line style should be different from the polygon).
+\end{itemize}
+Part (b) is worth 5 points, divided as follows:
+\begin{itemize}
+  \item 3 points if the nodes and arrows of the diagram are correct from a
+        mathematical point of view (that is, the arrows point to the correct
+        object).
+  \item 1 point if the labels of the arrows are placed as shown in the picture
+        above.
+  \item 1 point for the correct style of the arrows (dashed, curved).
+\end{itemize}
+
+\vspace{0.2cm}
+\noindent\textbf{Exercise 2 (10 points).}
+\begin{itemize}
+  \item Producing a presentation that contains at least 4 slides is worth
+        5 points (but within reason: for example, the slides must not be empty).
+  \item Up to 3 more points are given if the presentation is of a suitable
+        length (watch out: both a presentation too short and one too long can
+        loose points!).
+        Comments in the .tex file can help me understand how long you plan to
+        spend on each slide. If you are not sure how long your presentation is
+        going to take, try it and write down how long each slide took.
+  \item Up to 2 more points will be given if the slides ``look nice'' from
+        the audience's perspective (e.g. not too many words on the same slides,
+        are there nice pictures, etc).
+\end{itemize}
+
+
+\end{document}
\ No newline at end of file