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1\relax
2\newlabel{prop:fta}{{1}{1}}
3\newlabel{eq:fubini}{{2}{1}}
4\gdef \@abspage@last{2}
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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
9\title{Mathematical software - homework 1}
10\author{Sebastiano Tronto}
11
12\newtheorem{thm}{Theorem}
13\newtheorem{prop}[thm]{Proposition}
14
15\theoremstyle{definition}
16\newtheorem{ex}{Exercise}
17
18\theoremstyle{definition}
19\newtheorem*{remark}{Remark}
20
21\begin{document}
22
23\noindent\hrulefill
24
25\begin{center}
26\Huge{\textbf{Mathematical Software - Homework 1}}
27\end{center}
28
29\noindent\hrulefill
30\begin{center}
31\begin{tabular}{lcr}
32\texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad &
33\textbf{Deadline: Sunday, March 28th}
34\end{tabular}
35\end{center}
36
37\vspace{1cm}
38
39\begin{center}
40 \emph{\large
41 For each of the following exercises submit a .tex and a .pdf file.
42 }
43\end{center}
44
45\vspace{1cm}
46
47\begin{ex}
48 Write a short Latex document that contains the following theorem-like
49 environments using the \texttt{\textbackslash newtheorem} command of the
50 \texttt{amsthm} package (the box around the text is not needed):
51 \begin{center}
52 \fbox{\parbox{0.95\textwidth}{
53 \begin{prop}[Fundamental Theorem of Algebra]
54 \label{prop:fta}
55 Let \(p(x)\) be a non-constant polynomial with coefficients in
56 $\mathbb C$. Then there is \(z\in\mathbb C\) such that $p(z)=0$.
57 \end{prop}
58
59 \begin{remark}
60 Proposition \ref{prop:fta} is not true for polynomials with
61 coefficients in $\mathbb R$. For example
62 \begin{align}
63 p(x) = x^2+1
64 \end{align}
65 does not have real roots.
66 \end{remark}
67
68 \begin{thm}
69 If $X$ and $Y$ are $\sigma$-finite measure spaces and $f:X\times Y\to
70 \mathbb R$ is measurable and such that
71 \begin{align*}
72 \int_{X\times Y}|f(x,y)|\mathrm d(x,y) < \infty
73 \end{align*}
74 then
75 \begin{align}
76 \label{eq:fubini}
77 \int_X\left(\int_Yf(x,y)\mathrm d y\right)\mathrm d x =
78 \int_Y\left(\int_Xf(x,y)\mathrm d x\right)\mathrm d y =
79 \int_{X\times Y} f(x,y)\mathrm d(x,y)\,.
80 \end{align}
81 \end{thm}
82
83 \begin{remark}
84 In practice, equation \eqref{eq:fubini} means that we can switch the
85 order of integration in a double integral.
86 \end{remark}
87 }}
88 \end{center}
89 Notice that Propositions, Theorems and some of the equations are numbered,
90 and some of them are referred to in the Remarks. This numbering should change
91 accordingly if more numbered Theorems and equations are added before this
92 part of the text.
93\end{ex}
94
95\vspace{0.8cm}
96
97\begin{ex}
98[This is basically a ``do whatever you want, just do something'' exercise]
99Write a Mathematical document in Latex, where ``Mathematical'' means any text
100that contains both inline and displaystyle symbols and formulas, and where most
101of the sentences contain a Mathematical symbol.
102For example, you can submit (part of) your homework or notes for another course
103that you are following at the moment.
104
105You must follow the following rules:
106\begin{itemize}
107 \item Your document must be at least 2 pages long.
108 \item Use the following in your preamble:
109 \begin{verbatim}
110\documentaclass[12pt,a4paper]{article}
111\usepackage[left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm]{geometry}
112 \end{verbatim}
113\end{itemize}
114The document itself can be about any area of Mathematics, Physics or similar.
115The correctness of the Mathematics itself will not be judged for this course.
116\end{ex}
117
118\section*{Grading}
119
120This homework assignment is worth 25\% of your final grade.
121
122\vspace{0.3cm}
123\textbf{Exercise 1 (10 points).}
124\begin{itemize}
125 \item A correct use of the \texttt{\textbackslash newtheorem} command is
126 worth 4 out of 10 points.
127 \item A correct use of the labelling and reference system is worth 4 points.
128 \item Reproducing correctly the mathematical formulas is worth 2 points.
129\end{itemize}
130
131\textbf{Exercise 2 (10 points).}
132\begin{itemize}
133 \item A resulting pdf file that looks readable and correct is worth 6 out of
134 10 points.
135 \item The correct use of latex commands and environments is worth 4 out of
136 10 points. \emph{Examples of mistakes in this sense: adjusting manually
137 the size of parenthesis instead of using \texttt{\textbackslash left}
138 and \texttt{\textbackslash right}; using \texttt{\textbackslash Sigma}
139 when you actually want to write a sum symbol.}
140\end{itemize}
141
142
143\end{document} \ No newline at end of file

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