Homework2.tex (6416B)

1 \documentclass[12pt,a4paper]{article} 2 \usepackage[utf8]{inputenc} 3 \usepackage{amsmath} 4 \usepackage{amsfonts} 5 \usepackage{amssymb} 6 \usepackage{amsthm} 7 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} 8 \usepackage{enumitem} 9 \usepackage{tikz} 10 \usepackage{tikz-cd} 11 \usepackage{url} 12 13 \title{Mathematical software - homework 2} 14 \author{Sebastiano Tronto} 15 16 \newtheorem{thm}{Theorem} 17 \newtheorem{prop}[thm]{Proposition} 18 19 \theoremstyle{definition} 20 \newtheorem{ex}{Exercise} 21 22 \theoremstyle{definition} 23 \newtheorem*{remark}{Remark} 24 25 \newcommand{\bs}{\textbackslash} 26 27 \begin{document} 28 29 \noindent\hrulefill 30 31 \begin{center} 32 \Huge{\textbf{Mathematical Software - Homework 2}} 33 \end{center} 34 35 \noindent\hrulefill 36 \begin{center} 37 \begin{tabular}{lcr} 38 \texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad & 39 \textbf{Deadline: Sunday, April 18th} 40 \end{tabular} 41 \end{center} 42 43 \vspace{1cm} 44 45 \begin{center} 46 \emph{\large 47 For each of the following exercises submit a .tex and a .pdf file. 48 } 49 \end{center} 50 51 \vspace{1cm} 52 53 \begin{ex} 54 Create a Latex document containing the following pictures: 55 \begin{enumerate}[label=(\alph*)] 56 \item The regular polygon with $N$-sides centered at the origin of the 57 plane (see below). 58 The number $N$ of sides must be easy to change at will: you should 59 use the \texttt{\textbackslash pgfmathsetmacro} command to set a 60 value for $N$ at the beginning, so that changing only that number 61 makes the whole picture change accordingly. 62 \begin{center} 63 \begin{tikzpicture}[scale=1] 64 \pgfmathsetmacro{\N}{6} 65 \pgfmathsetmacro{\an}{360/\N} 66 \pgfmathsetmacro{\r}{4} 67 68 \draw[lightgray!30,thin] (-6,-6) grid (6,6); 69 \draw[->] (-5,0) -- (5,0); 70 \draw[->] (0,-5) -- (0,5); 71 72 \filldraw[draw=green,fill=green!30,thick] (0.3*\r,0) 73 arc[radius=0.3*\r,start angle=0, end angle=\an] 74 -- node[right] {$\alpha=\an$}(0,0) -- cycle; 75 \draw[thin] (\an:\r) -- (0,0) circle[radius=\r] -- (\r,0); 76 \draw[line width=1.1pt] 77 (\r,0) \foreach \x in {1,2,...,\N} { -- (\x*\an:\r) }; 78 \foreach \x in {1,...,\N} { 79 \filldraw[red] (\x*\an:\r) circle[radius=0.1]; 80 }; 81 \end{tikzpicture} 82 \end{center} 83 \item The following commutative diagram: 84 \begin{tikzcd} 85 0 \ar[r] & A' \ar[r,hook] \ar[d] & A \ar[r,two heads] \ar[d,"\sim"] & 86 A'' \ar[r] \ar[d] \ar[l,dashed,"s"',bend right] & 0 \\ 87 0 \ar[r] & A' \ar[r,hook,"i_{A'}"'] & A'\oplus A'' 88 \ar[r,two heads,"\pi_{A''}"'] & A'' \ar[r] & 0 89 \end{tikzcd} 90 \end{enumerate} 91 \end{ex} 92 93 \vspace{0.8cm} 94 95 \begin{ex} 96 Suppose you have to give a short presentation (10 minutes) on a topic of 97 your choice related to your study programme (you will not be asked to 98 actually perform this presentation). You can choose to talk about a theorem 99 you find important, a result you have seen in class or something else (see 100 below for a list of possible topics). For example, if you talk about an 101 important theorem you can give the theorem statement, explain why this 102 theorem is important and/or possible applications of this result, and 103 optionally an idea of the proof; but you can also deviate from this and talk 104 for example about the historical background that lead to the development of 105 this theorem. 106 107 Your task is to prepare slides for such a presentation using Beamer (Latex). 108 Since the time for the (imaginary) presentation is very short, you should 109 write 5-8 slides (you can have more if some contain very few or no 110 words). 111 112 If you feel like certain slides do not make sense without your explanation 113 (for example if you have one slide with just one picture and you plan to 114 talk with the picture in background), you can write some comments in the 115 .tex file. 116 117 If you can't think of a topic that you like, you can pick one of the 118 following: 119 \begin{itemize} 120 \item The fundamental theorem of arithmetic (about prime numbers) 121 \item The central limit theorem (probability theory) 122 \item Differential equations (what they are, applications, 123 methods to solve them...) 124 \end{itemize} 125 \end{ex} 126 127 \newpage 128 \section*{Grading} 129 130 This homework assignment is worth 25\% of your final grade. 131 132 \vspace{0.3cm} 133 \noindent\textbf{Exercise 1 (10 points).} 134 Part (a) is worth 5 points, divided as follows: 135 \begin{itemize} 136 \item 3 points for obtaining a regular polygon whose number of sides 137 can be changed by setting a variable with \texttt{\bs pgfmathsetmacro} 138 (or in a similarly easy way). 139 \item 1.5 points for other features of the picture (verteces, angle) that 140 also change accordingly to the same variable. 141 \item 0.5 points for the style of the other elements of the picture. This is 142 a matter of personal preference and it does not need to be exactly the 143 same as the picture, but some key features should remain (e.g. 144 the grid lines should be less visible than the rest of the picture, 145 the circle line style should be different from the polygon). 146 \end{itemize} 147 Part (b) is worth 5 points, divided as follows: 148 \begin{itemize} 149 \item 3 points if the nodes and arrows of the diagram are correct from a 150 mathematical point of view (that is, the arrows point to the correct 151 object). 152 \item 1 point if the labels of the arrows are placed as shown in the picture 153 above. 154 \item 1 point for the correct style of the arrows (dashed, curved). 155 \end{itemize} 156 157 \vspace{0.2cm} 158 \noindent\textbf{Exercise 2 (10 points).} 159 \begin{itemize} 160 \item Producing a presentation that contains at least 4 slides is worth 161 5 points (but within reason: for example, the slides must not be empty). 162 \item Up to 3 more points are given if the presentation is of a suitable 163 length (watch out: both a presentation too short and one too long can 164 loose points!). 165 Comments in the .tex file can help me understand how long you plan to 166 spend on each slide. If you are not sure how long your presentation is 167 going to take, try it and write down how long each slide took. 168 \item Up to 2 more points will be given if the slides ``look nice'' from 169 the audience's perspective (e.g. not too many words on the same slides, 170 are there nice pictures, etc). 171 \end{itemize} 172 173 174 \end{document}