From d6c61d988bfa4255baf9cdae42db59ebee38363f Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Tue, 25 May 2021 17:10:49 +0200 Subject: Added files --- src/Homework2/Homework2.tex | 174 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 174 insertions(+) create mode 100644 src/Homework2/Homework2.tex (limited to 'src/Homework2/Homework2.tex') diff --git a/src/Homework2/Homework2.tex b/src/Homework2/Homework2.tex new file mode 100644 index 0000000..e36214d --- /dev/null +++ b/src/Homework2/Homework2.tex @@ -0,0 +1,174 @@ +\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 -- cgit v1.3