\ProvidesPackage{notes_2023} \usepackage{amsmath,amssymb} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsopn} \usepackage{bookmark} \usepackage{faktor} \usepackage[utf8]{inputenc} \usepackage{mathtools} \usepackage{nicefrac} \usepackage{stmaryrd} \usepackage{marvosym} \usepackage{float} \usepackage{enumerate} \usepackage{scalerel} \usepackage{stackengine} \usepackage{wasysym} \usepackage{tikz-cd} \usepackage[italian]{babel} \usepackage{tabularx} % Setup preliminari \setlength{\extrarowheight}{4pt} \newcommand{\system}[1]{\begin{cases} #1 \end{cases}} \newcommand{\wip}{\begin{center}\textit{Questo avviso sta ad indicare che questo documento è ancora una bozza e non è da intendersi né completo, né revisionato.}\end{center}} \newcommand\hr{\vskip 0.05in \par\vspace{-.5\ht\strutbox}\noindent\hrulefill\par} % Modalità matematica/fisica \newcommand{\SMatrix}[1]{\begin{psmallmatrix}#1\end{psmallmatrix}} \let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}} \newcommand{\con}{\text{con }} \newcommand{\dove}{\text{dove }} \newcommand{\E}{\text{ e }} \newcommand{\altrimenti}{\text{altrimenti}} \newcommand{\se}{\text{se }} \newcommand{\tc}{\text{ t.c. }\!} \newcommand{\epari}{\text{ è pari}} \newcommand{\edispari}{\text{ è dispari}} \newcommand{\nl}{\ \\} \newcommand{\bigmid}{\;\middle\vert\;} \newcommand{\lemmaref}[1]{\textit{Lemma \ref{#1}}} \newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} \newcommand{\li}[0]{$\blacktriangleright\;\;$} \newcommand{\tends}[1]{\xrightarrow[\text{$#1$}]{}} \newcommand{\tendsto}[1]{\xrightarrow[\text{$x \to #1$}]{}} \newcommand{\tendstoy}[1]{\xrightarrow[\text{$y \to #1$}]{}} \newcommand{\tendston}[0]{\xrightarrow[\text{$n \to \infty$}]{}} \setlength\parindent{0pt} % Principio di induzione e setup dimostrativi. \newcommand{\basestep}{\mbox{(\textit{passo base})}\;} \newcommand{\inductivestep}{\mbox{(\textit{passo induttivo})}\;} \newcommand{\rightproof}{\mbox{($\implies$)}\;} \newcommand{\leftproof}{\mbox{($\impliedby$)}\;} % Spesso utilizzati al corso di Fisica 1. \newcommand{\dx}{\dot{x}} \newcommand{\ddx}{\ddot{x}} \newcommand{\dv}{\dot{v}} \newcommand{\del}{\partial} \newcommand{\tendstot}[0]{\xrightarrow[\text{$t \to \infty$}]{}} \newcommand{\grad}{\vec{\nabla}} \DeclareMathOperator{\rot}{rot} \newcommand{\ihat}{\hat{i}} \newcommand{\jhat}{\hat{j}} \newcommand{\khat}{\hat{k}} \newcommand{\der}[1]{\frac{d#1}{dx}} \newcommand{\parx}{\frac{\del}{\del x}} \newcommand{\pary}{\frac{\del}{\del y}} \newcommand{\parz}{\frac{\del}{\del z}} % Spesso utilizzati al corso di Analisi 1. %\newcommand{\liminf}{\lim_{x \to \infty}} \newcommand{\liminfty}{\lim_{x \to \infty}} \newcommand{\liminftym}{\lim_{x \to -\infty}} \newcommand{\liminftyn}{\lim_{n \to \infty}} \newcommand{\limzero}{\lim_{x \to 0}} \newcommand{\limzerop}{\lim_{x \to 0^+}} \newcommand{\limzerom}{\lim_{x \to 0^-}} \newcommand{\xbar}{\overline{x}} \newcommand{\ybar}{\overline{y}} \newcommand{\tbar}{\overline{t}} \newcommand{\zbar}{\overline{z}} \newcommand{\RRbar}{\overline{\RR}} % Spesso utilizzati al corso di Geometria 2 \newcommand{\PP}{\mathbb{P}} \newcommand{\PPGL}{\mathbb{P}\!\GL} % Spesso utilizzati al corso di Geometria 1. \DeclareMathOperator{\OO}{O} % gruppo ortogonale \DeclareMathOperator{\SOO}{SO} % gruppo ortogonale speciale \newcommand{\proj}[1]{\Matrix{#1 \\[0.03in] \hline 1}} \newcommand{\projT}[1]{\Matrix{#1 & \rvline & 1}^\top} \newcommand{\cc}{\mathcal{C}} \let\AA\undefined \DeclareMathOperator{\Iso}{Iso} \newcommand{\AA}{\mathcal{A}} \newcommand{\MM}{\mathcal{M}} \newcommand{\KKxn}{\KK[x_1, \ldots, x_n]} \let\ext\faktor \newcommand{\quot}[1]{/{#1}} \newcommand{\Aa}{\mathcal{A}} \DeclareMathOperator{\An}{\mathcal{A}_n} \DeclareMathOperator{\AnK}{\mathcal{A}_n(\KK)} \DeclareMathOperator{\Giac}{Giac} \DeclareMathOperator{\IC}{IC} \DeclareMathOperator{\Aff}{Aff} \DeclareMathOperator{\Orb}{Orb} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Gr}{Gr} \newcommand{\vvec}[1]{\overrightarrow{#1}} \newcommand{\conj}[1]{\overline{#1}} \DeclareMathOperator{\PH}{PH} \DeclareMathOperator{\PS}{PS} \let\imm\Im \let\Im\undefined \DeclareMathOperator{\Im}{Im} \DeclareMathOperator{\Rad}{Rad} \newcommand{\restr}[2]{ #1\arrowvert_{#2} } \newcommand{\innprod}[1]{\langle #1 \rangle} \newcommand{\zerovecset}{\{\vec 0\}} \newcommand{\bigzero}{\mbox{0}} \newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} \newcommand{\Idv}{\operatorname{Id}_V} \newcommand{\Idw}{\operatorname{Id}_W} \newcommand{\IdV}[1]{\operatorname{Id}_{#1}} \DeclareMathOperator{\CI}{CI} \DeclareMathOperator{\Bil}{Bil} \DeclareMathOperator{\Mult}{Mult} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\Cof}{Cof} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\Sp}{sp} \newcommand{\dperp}{{\perp\perp}} \newcommand{\Eigsp}[0]{V_{\lambda}} \newcommand{\Gensp}[0]{\widetilde{V_{\lambda}}} \newcommand{\eigsp}[1]{V_{\lambda_{#1}}} \newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} \newcommand{\genspC}[1]{\widetilde{V_{#1}}} \DeclareMathOperator{\val}{val} \DeclareMathOperator{\Span}{Span} \newcommand{\charpoly}[1]{p_{#1}} \newcommand{\charpolyrestr}[2]{p_{#1\arrowvert_#2}\hspace{-1pt}(\lambda)} \newcommand{\minpoly}[1]{\varphi_{#1}} \newcommand{\valf}{\val_f} \newcommand{\valfv}{\val_{f,\V}} \newcommand{\e}[1]{\vec{e_{#1}}} \newcommand{\V}{\vec{v}} \newcommand{\VV}[1]{\vec{v_{#1}}} \newcommand{\basisdual}{\dual{\basis}} \newcommand{\vecdual}[1]{\vec{\dual{#1}}} \newcommand{\vecbidual}[1]{\vec{\bidual{#1}}} \newcommand{\NMatrix}[1]{\begin{matrix} #1 \end{matrix}} \newcommand{\Matrix}[1]{\begin{pmatrix} #1 \end{pmatrix}} \newcommand{\Vector}[1]{\begin{pmatrix} #1 \end{pmatrix}} \DeclareMathOperator{\rg}{rg} \let\v\undefined \newcommand{\v}{\vec{v}} \newcommand{\vv}[1]{\vec{v_{#1}}} \newcommand{\w}{\vec{w}} \newcommand{\U}{\vec{u}} \newcommand{\ww}[1]{\vec{w_{#1}}} \newcommand{\uu}[1]{\vec{u_{#1}}} \newcommand{\x}{\vec{x}} \newcommand{\xx}[1]{\vec{x_{#1}}} \newcommand{\y}{\vec{y}} \newcommand{\yy}[1]{\vec{y_{#1}}} \newcommand{\mapstoby}[1]{\xmapsto{#1}} \newcommand{\oplusperp}{\oplus^\perp} % Spesso utilizzati durante il corso di Algebra 1 \newcommand{\bij}{\leftrightarrow} \newcommand{\ZZmod}[1]{\ZZ \quot #1 \ZZ} \newcommand{\cleq}[1]{\overline{#1}} \newcommand{\rotations}{\mathcal{R}} \newcommand{\gen}[1]{\langle #1 \rangle} \DeclareMathOperator{\Cl}{Cl} \newcommand{\actson}{\circlearrowleft} \newcommand{\Cyc}[1]{\left<#1\right>} % Comandi personali. \newcommand{\card}[1]{\left|#1\right|} \newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} \newcommand{\zeroset}{\{0\}} \newcommand{\setminuszero}{\setminus \{0\}} \newenvironment{solution} {\textit{Soluzione.}\,} \theoremstyle{definition} \let\abstract\undefined \let\endabstract\undefined \newtheorem*{abstract}{Abstract} \newtheorem*{algorithm}{Algoritmo} \newtheorem*{corollary}{Corollario} \newtheorem*{definition}{Definizione} \newtheorem*{example}{Esempio} \newtheorem{exercise}{Esercizio} \newtheorem{lemma}{Lemma} \newtheorem*{note}{Nota} \newtheorem*{remark}{Osservazione} \newtheorem*{proposition}{Proposizione} \newtheorem*{summary}{Sommario} \newtheorem*{theorem}{Teorema} \newcommand{\basis}{\mathcal{B}} \newcommand{\BB}{\mathcal{B}} \newcommand{\basisC}{\mathcal{B}} \newcommand{\HH}{\mathbb{H}} \newcommand{\FFp}[1]{\mathbb{F}_p} \newcommand{\FFpx}[1]{\mathbb{F}_p[x]} \newcommand{\CCx}{\mathbb{C}[x]} \newcommand{\KK}{\mathbb{K}} \newcommand{\KKx}{\mathbb{K}[x]} \newcommand{\QQx}{\mathbb{Q}[x]} \newcommand{\RRx}{\mathbb{R}[x]} \newcommand{\ZZi}{\mathbb{Z}[i]} \newcommand{\ZZp}{\mathbb{Z}_p} \newcommand{\ZZpx}{\mathbb{Z}_p[x]} \newcommand{\ZZx}{\mathbb{Z}[x]} \newcommand{\ii}{\mathbf{i}} \newcommand{\jj}{\mathbf{j}} \newcommand{\kk}{\mathbf{k}} \newcommand{\bidual}[1]{#1^{**}} \newcommand{\dual}[1]{#1^{*}} \newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} % L(V, W) \newcommand{\Ll}{\mathcal{L}} \newcommand{\nsg}{\triangleleft} % sottogruppo normale proprio \newcommand{\nsgeq}{\trianglelefteqslant} % sottogruppo normale % evan.sty original commands \newcommand{\cbrt}[1]{\sqrt[3]{#1}} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil} \newcommand{\mailto}[1]{\href{mailto:#1}{\texttt{#1}}} \newcommand{\eps}{\varepsilon} \newcommand{\vocab}[1]{\textbf{\color{blue}\sffamily #1}} \providecommand{\alert}{\vocab} \newcommand{\catname}{\mathsf} \providecommand{\arc}[1]{\wideparen{#1}} % From H113 "Introduction to Abstract Algebra" at UC Berkeley \newcommand{\CC}{\mathbb C} \newcommand{\FF}{\mathbb F} \newcommand{\NN}{\mathbb N} \newcommand{\QQ}{\mathbb Q} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\SL}{SL} % From Kiran Kedlaya's "Geometry Unbound" \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\dang}{\measuredangle} %% Directed angle \newcommand{\ray}[1]{\overrightarrow{#1}} \newcommand{\seg}[1]{\overline{#1}} % From M275 "Topology" at SJSU \newcommand{\Id}{\mathrm{Id}} \newcommand{\id}{\mathrm{id}} \newcommand{\taking}[1]{\xrightarrow{#1}} \newcommand{\inv}{^{-1}} \DeclareMathOperator{\ord}{ord} \newcommand{\defeq}{\overset{\mathrm{def}}{=}} \newcommand{\defiff}{\overset{\mathrm{def}}{\iff}} % From the USAMO .tex files \newcommand{\dg}{^\circ} \newcommand{\liff}{\leftrightarrow} \newcommand{\lthen}{\rightarrow} \newcommand{\opname}{\operatorname} \newcommand{\surjto}{\twoheadrightarrow} \newcommand{\injto}{\hookrightarrow} % Alcuni degli operatori più comunemente utilizzati. \DeclareMathOperator{\Char}{char} \DeclareMathOperator{\Dom}{Dom} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\existsone}{\exists !} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Imm}{Imm} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\MCD}{MCD} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\mcm}{mcm} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\tr}{tr} % Reimposta alcuni simboli presenti di default in LaTeX con degli analoghi % più comuni. \let\oldemptyset\emptyset \let\emptyset\varnothing % Trasforma alcuni simboli in operatori matematici. \let\oldcirc\circ \let\circ\undefined \DeclareMathOperator{\circ}{\oldcirc} \let\oldexists\exists \let\exists\undefined \DeclareMathOperator{\exists}{\oldexists} \let\oldforall\forall \let\forall\undefined \DeclareMathOperator{\forall}{\oldforall} \let\oldnexists\nexists \let\nexists\undefined \DeclareMathOperator{\nexists}{\oldnexists} \let\oldland\land \let\land\undefined \DeclareMathOperator{\land}{\oldland} \let\oldlnot\lnot \let\lnot\undefined \DeclareMathOperator{\lnot}{\oldlnot} \let\oldlor\lor \let\lor\undefined \DeclareMathOperator{\lor}{\oldlor} \DeclareOption{counter}{ \let\algorithm\@undefined \let\endalgorithm\@undefined \let\corollary\@undefined \let\endcorollary\@undefined \let\c@lemma\@undefined \let\lemma\@undefined \let\endlemma\@undefined \let\proposition\@undefined \let\endproposition\@undefined \let\theorem\@undefined \let\endtheorem\@undefined \newtheorem{algorithm}{Algoritmo}[chapter] \newtheorem{corollary}{Corollario}[chapter] \newtheorem{lemma}{Lemma}[chapter] \newtheorem{proposition}{Proposizione}[chapter] \newtheorem{theorem}{Teorema}[chapter] } \ProcessOptions\relax \author{di Gabriel Antonio Videtta} \date{\vspace{-0.5cm}}