\ProvidesPackage{recap} \usepackage{amsmath,amssymb} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsopn} \usepackage{mathtools} \usepackage{stmaryrd} \usepackage{marvosym} \usepackage{float} \usepackage{enumerate} \usepackage{scalerel} \usepackage{stackengine} \usepackage{wasysym} \usepackage{iftex} \usepackage[minimal]{yhmath} \PassOptionsToPackage{usenames,svgnames,dvipsnames,table}{xcolor} \usepackage{xcolor} \usepackage[colorlinks=true]{hyperref} \hypersetup{urlcolor=RubineRed,linkcolor=RoyalBlue,citecolor=ForestGreen} \usepackage[nameinlink]{cleveref} \usepackage{amsthm} \usepackage{thmtools} \usepackage[framemethod=TikZ]{mdframed} \usepackage{listings} \usepackage{mathrsfs} \usepackage{textcomp} \usepackage[shortlabels]{enumitem} \usepackage[obeyFinal,textsize=scriptsize,shadow]{todonotes} \usepackage{textcomp} \usepackage{multirow} \usepackage{ellipsis} \usepackage{mathtools} \usepackage{microtype} \usepackage{xstring} \usepackage{wrapfig} \usepackage[headsepline]{scrlayer-scrpage} \hfuzz=\maxdimen \tolerance=10000 \hbadness=10000 \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 \let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}} \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{\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{\cc}{\mathcal{C}} \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 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}} \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{\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{\mapstoby}[1]{\xmapsto{#1}} \newcommand{\oplusperp}{\oplus^\perp} % 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 \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{\nsg}{\mathrel{\unlhd}} % 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}{\textit{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{physics}{ \let\vec\oldvec } \ProcessOptions\relax \title{} \author{Gabriel Antonio Videtta} \date{\today} \mdfdefinestyle{mdbluebox}{% roundcorner=10pt, linewidth=1pt, skipabove=12pt, innerbottommargin=9pt, skipbelow=2pt, linecolor=blue, nobreak=true, backgroundcolor=TealBlue!5, } \declaretheoremstyle[ headfont=\sffamily\bfseries\color{MidnightBlue}, mdframed={style=mdbluebox}, headpunct={\\[3pt]}, postheadspace={0pt} ]{thmbluebox} \mdfdefinestyle{mdbluebox2}{% roundcorner=10pt, linewidth=1pt, skipabove=12pt, innerbottommargin=9pt, skipbelow=2pt, linecolor=blue, nobreak=true, backgroundcolor=BlueViolet!9, } \declaretheoremstyle[ headfont=\sffamily\bfseries\color{RoyalPurple}, mdframed={style=mdbluebox2}, headpunct={\\[3pt]}, postheadspace={0pt} ]{thmbluebox2} \mdfdefinestyle{mdredbox}{% linewidth=0.5pt, skipabove=12pt, frametitleaboveskip=5pt, frametitlebelowskip=0pt, skipbelow=2pt, frametitlefont=\bfseries, innertopmargin=4pt, innerbottommargin=8pt, nobreak=true, backgroundcolor=Salmon!5, linecolor=RawSienna, } \declaretheoremstyle[ headfont=\bfseries\color{RawSienna}, mdframed={style=mdredbox}, headpunct={\\[3pt]}, postheadspace={0pt}, ]{thmredbox} \mdfdefinestyle{mdredbox2}{% roundcorner=10pt, linewidth=1pt, skipabove=12pt, innerbottommargin=9pt, skipbelow=2pt, linecolor=red, nobreak=true, backgroundcolor=WildStrawberry!5, } \declaretheoremstyle[ headfont=\sffamily\bfseries\color{Maroon}, mdframed={style=mdredbox2}, headpunct={\\[3pt]}, postheadspace={0pt} ]{thmredbox2} \mdfdefinestyle{mdgreenbox}{% skipabove=8pt, linewidth=2pt, rightline=false, leftline=true, topline=false, bottomline=false, linecolor=ForestGreen, backgroundcolor=ForestGreen!5, } \declaretheoremstyle[ headfont=\bfseries\sffamily\color{ForestGreen!70!black}, bodyfont=\normalfont, spaceabove=2pt, spacebelow=1pt, mdframed={style=mdgreenbox}, headpunct={ --- }, ]{thmgreenbox} \mdfdefinestyle{mdgreenbox2}{% roundcorner=10pt, linewidth=1pt, skipabove=12pt, innerbottommargin=9pt, skipbelow=2pt, linecolor=ForestGreen, nobreak=true, backgroundcolor=ForestGreen!5, } \declaretheoremstyle[ headfont=\sffamily\bfseries\color{ForestGreen!70!black}, mdframed={style=mdgreenbox2}, headpunct={\\[3pt]}, postheadspace={0pt} ]{thmgreenbox2} \mdfdefinestyle{mdblackbox}{% skipabove=8pt, linewidth=3pt, rightline=false, leftline=true, topline=false, bottomline=false, linecolor=black, backgroundcolor=RedViolet!5!gray!10, } \declaretheoremstyle[ headfont=\bfseries, bodyfont=\normalfont\small, spaceabove=0pt, spacebelow=0pt, mdframed={style=mdblackbox} ]{thmblackbox} \mdfdefinestyle{mdblackbox2}{% skipabove=8pt, linewidth=3pt, rightline=false, leftline=true, topline=false, bottomline=false, linecolor=gray, backgroundcolor=RedViolet!5!gray!10, } \declaretheoremstyle[ headfont=\bfseries, bodyfont=\normalfont\small, spaceabove=0pt, spacebelow=0pt, mdframed={style=mdblackbox2} ]{thmblackbox2} \newcommand{\listhack}{$\empty$\vspace{-2em}} \declaretheorem[% style=thmbluebox,name=Teorema,numberwithin=section]{theorem} \declaretheorem[style=thmgreenbox2,name=Lemma,sibling=theorem]{lemma} \declaretheorem[style=thmredbox2,name=Proposizione,sibling=theorem]{proposition} \declaretheorem[style=thmbluebox2,name=Corollario,sibling=theorem]{corollary} \declaretheorem[style=thmbluebox,name=Teorema,numbered=no]{theorem*} \declaretheorem[style=thmgreenbox2,name=Lemma,numbered=no]{lemma*} \declaretheorem[style=thmredbox2,name=Proposizione,numbered=no]{proposition*} \declaretheorem[style=thmbluebox2,name=Corollario,numbered=no]{corollary*} \declaretheorem[style=thmgreenbox,name=Nota,numbered=no]{note} \declaretheorem[style=thmgreenbox,name=Claim,sibling=theorem]{claim} \declaretheorem[style=thmgreenbox,name=Claim,numbered=no]{claim*} \declaretheorem[style=thmredbox,name=Esempio,sibling=theorem]{example} \declaretheorem[style=thmredbox,name=Esempio,numbered=no]{example*} \declaretheorem[style=thmblackbox, name=Definizione,sibling=theorem]{definition} \declaretheorem[style=thmblackbox, name=Definizione,numbered=no]{definition*} \declaretheorem[style=thmblackbox2,name=Osservazione,sibling=theorem]{remark} \declaretheorem[style=thmblackbox2,name=Osservazione,numbered=no]{remark*} \declaretheorem[name=Congettura,sibling=theorem]{conjecture} \declaretheorem[name=Congettura,numbered=no]{conjecture*} \declaretheorem[name=Esercizio,sibling=theorem]{exercise} \declaretheorem[name=Esercizio,numbered=no]{exercise*} \declaretheorem[name=Asserzione,sibling=theorem]{fact} \declaretheorem[name=Asserzione,numbered=no]{fact*} \declaretheorem[name=Problema,sibling=theorem]{problem} \declaretheorem[name=Problema,numbered=no]{problem*} \declaretheorem[name=Domanda,sibling=theorem]{ques} \declaretheorem[name=Domanda,numbered=no]{ques*} \Crefname{claim}{Claim}{Claim} \Crefname{conjecture}{Congettura}{Congetture} \Crefname{exercise}{Esercizio}{Esercizi} \Crefname{fact}{Asserzione}{Asserzioni} \Crefname{problem}{Problema}{Problemi} \Crefname{ques}{Domanda}{Domande} 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