From 6a169af5129a663ecf04d3a45947d63f86733f25 Mon Sep 17 00:00:00 2001 From: Ivan Bushchik Date: Fri, 28 Jul 2023 16:28:16 +0300 Subject: [PATCH] Add latex_template --- latex_template/README.md | 24 + latex_template/letterfonts.tex | 178 ++++++++ latex_template/macros.tex | 88 ++++ latex_template/preamble.tex | 773 +++++++++++++++++++++++++++++++++ latex_template/template.tex | 128 ++++++ 5 files changed, 1191 insertions(+) create mode 100644 latex_template/README.md create mode 100644 latex_template/letterfonts.tex create mode 100644 latex_template/macros.tex create mode 100644 latex_template/preamble.tex create mode 100644 latex_template/template.tex diff --git a/latex_template/README.md b/latex_template/README.md new file mode 100644 index 0000000..8f6b780 --- /dev/null +++ b/latex_template/README.md @@ -0,0 +1,24 @@ +# Latex Template + +A personal template I have copied from [Charlie's Math Template](https://github.com/SeniorMars/dotfiles/tree/master/latex_template). + +To use: +1. Copy this directory to the same directory as your LaTeX file. + +```bash +$ cp ./letterfonts.tex ./macros.tex ./preamble.tex ./template.tex [some_dir] + +or + +$ ltcp +``` + +ltcp if alias, embedded into my `.zshrc` + + +2. Edit the file "template.tex" to your liking. + + +## Note + +I'm using texlive with it. \ No newline at end of file diff --git a/latex_template/letterfonts.tex b/latex_template/letterfonts.tex new file mode 100644 index 0000000..e28675f --- /dev/null +++ b/latex_template/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/latex_template/macros.tex b/latex_template/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/latex_template/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/latex_template/preamble.tex b/latex_template/preamble.tex new file mode 100644 index 0000000..48ed565 --- /dev/null +++ b/latex_template/preamble.tex @@ -0,0 +1,773 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[utf8]{inputenc} +\usepackage[T2A]{fontenc} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{1cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{Theorem} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{Theorem} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{Corollary} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{Corollary} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{Proposition} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{Proposition} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{Claim} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{Exercise} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{Exercise} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{Example} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{Example} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{Question}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{Question}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{Wrong Concept}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Note:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=1.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + + + + +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} + + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% +\begingroup +\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% +\pdfsave +\pdfsetmatrix{1 0 0.2 1}% +\rlap{\usebox{\diffdbox}}% +\pdfrestore +\hskip\wd\diffdbox +\endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% +\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% +{\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience +\DeclareMathOperator{\E}{E} % expected + +% Since the amsthm package isn't loaded + +% I prefer the slanted \leq +\let\oldleq\leq % save them in case they're every wanted +\let\oldgeq\geq +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +% % redefine matrix env to allow for alignment, use r as default +% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep +% \let\@ifnextchar\new@ifnextchar +% \array{*\c@MaxMatrixCols #1}} + + +%\usepackage{framed} +%\usepackage{titletoc} +%\usepackage{etoolbox} +%\usepackage{lmodern} + + +%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{} + +%\renewenvironment{leftbar} +%{\def\FrameCommand{\hspace{6em}% +% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}% +% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt% +%} +%{\endMakeFramed} + +%\titlecontents{chapter} +%[0em]{\vspace*{2\baselineskip}} +%{\parbox{4.5em}{% +% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}% +% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily} +%{}{\endleftbar} +%\titlecontents{section} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} +%\titlecontents{subsection} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage{tikz} +\definecolor{doc}{RGB}{0,0,0} +\usepackage{titletoc} +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=black,draw=black] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Chapter\ \thecontentslabel};% + \end{tikzpicture}\color{black}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Page \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=black,draw=black] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \center{\Huge{\sc\bfseries Contents}} + \@starttoc{toc}} +\makeatother + diff --git a/latex_template/template.tex b/latex_template/template.tex new file mode 100644 index 0000000..429956f --- /dev/null +++ b/latex_template/template.tex @@ -0,0 +1,128 @@ +\documentclass{report} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\Huge{Some Class}\\Random Examples} +\author{\huge{Тест}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[]{}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\chapter{} +\section{Random Examples} +\dfn{Limit of Sequence in $\bs{\bbR}$}{Let $\{s_n\}$ be a sequence in $\bbR$. We say $$\lim_{n\to\infty}s_n=s$$ where $s\in\bbR$ if $\forall$ real numbers $\eps>0$ $\exists$ natural number $N$ such that for $n>N$ $$s-\eps<s_n<s+\eps\text{ i.e. }|s-s_n|<\eps$$} +\qs{}{Is the set ${x-}$axis${\setminus\{\text{Origin}\}}$ a closed set} +\sol We have to take its complement and check whether that set is a open set i.e. if it is a union of open balls +\nt{We will do topology in Normed Linear Space (Mainly $\bbR^n$ and occasionally $\bbC^n$)using the language of Metric Space} +\clm{Topology}{}{Topology is cool} +\ex{Open Set and Close Set}{ + \begin{tabular}{rl} + Open Set: & $\bullet$ $\phi$ \\ + & $\bullet$ $\bigcup\limits_{x\in X}B_r(x)$ (Any $r>0$ will do) \\[3mm] + & $\bullet$ $B_r(x)$ is open \\ + Closed Set: & $\bullet$ $X,\ \phi$ \\ + & $\bullet$ $\overline{B_r(x)}$ \\ + & $x-$axis $\cup$ $y-$axis + \end{tabular}} +\thm{}{If $x\in$ open set $V$ then $\exists$ $\delta>0$ such that $B_{\delta}(x)\subset V$} +\begin{myproof}By openness of $V$, $x\in B_r(u)\subset V$ + \begin{center} + \begin{tikzpicture} + \draw[red] (0,0) circle [x radius=3.5cm, y radius=2cm] ; + \draw (3,1.6) node[red]{$V$}; + \draw [blue] (1,0) circle (1.45cm) ; + \filldraw[blue] (1,0) circle (1pt) node[anchor=north]{$u$}; + \draw (2.9,0.4) node[blue]{$B_r(u)$}; + \draw [green!40!black] (1.7,0) circle (0.5cm) node [yshift=0.7cm]{$B_{\delta}(x)$} ; + \filldraw[green!40!black] (1.7,0) circle (1pt) node[anchor=west]{$x$}; + \end{tikzpicture} + \end{center} + + Given $x\in B_r(u)\subset V$, we want $\delta>0$ such that $x\in B_{\delta} (x)\subset B_r(u)\subset V$. Let $d=d(u,x)$. Choose $\delta $ such that $d+\delta<r$ (e.g. $\delta<\frac{r-d}{2}$) + + If $y\in B_{\delta}(x)$ we will be done by showing that $d(u,y)<r$ but $$d(u,y)\leq d(u,x)+d(x,y)<d+\delta<r$$ +\end{myproof} + +\cor{}{By the result of the proof, we can then show...} +\mlenma{}{Suppose $\vec{v_1}, \dots, \vec{v_n} \in \RR[n]$ is subspace of $\RR^n$.} +\mprop{}{$1 + 1 = 2$.} + +\section{Random} +\dfn{Normed Linear Space and Norm $\boldsymbol{\|\cdot\|}$}{Let $V$ be a vector space over $\bbR$ (or $\bbC$). A norm on $V$ is function $\|\cdot\|\ V\to \bbR_{\geq 0}$ satisfying \begin{enumerate}[label=\bfseries\tiny\protect\circled{\small\arabic*}] + \item \label{n:1}$\|x\|=0 \iff x=0$ $\forall$ $x\in V$ + \item \label{n:2} $\|\lambda x\|=|\lambda|\|x\|$ $\forall$ $\lambda\in\bbR$(or $\bbC$), $x\in V$ + \item \label{n:3} $\|x+y\| \leq \|x\|+\|y\|$ $\forall$ $x,y\in V$ (Triangle Inequality/Subadditivity) + \end{enumerate}And $V$ is called a normed linear space. + + $\bullet $ Same definition works with $V$ a vector space over $\bbC$ (again $\|\cdot\|\to\bbR_{\geq 0}$) where \ref{n:2} becomes $\|\lambda x\|=|\lambda|\|x\|$ $\forall$ $\lambda\in\bbC$, $x\in V$, where for $\lambda=a+ib$, $|\lambda|=\sqrt{a^2+b^2}$ } + + +\ex{$\bs{p-}$Norm}{\label{pnorm}$V={\bbR}^m$, $p\in\bbR_{\geq 0}$. Define for $x=(x_1,x_2,\cdots,x_m)\in\bbR^m$ $$\|x\|_p=\Big(|x_1|^p+|x_2|^p+\cdots+|x_m|^p\Big)^{\frac1p}$$(In school $p=2$)} +\textbf{Special Case $\bs{p=1}$}: $\|x\|_1=|x_1|+|x_2|+\cdots+|x_m|$ is clearly a norm by usual triangle inequality. \par +\textbf{Special Case $\bs{p\to\infty\ (\bbR^m$ with $\|\cdot\|_{\infty})}$}: $\|x\|_{\infty}=\max\{|x_1|,|x_2|,\cdots,|x_m|\}$\\ +For $m=1$ these $p-$norms are nothing but $|x|$. +Now exercise +\qs{}{\label{exs1}Prove that triangle inequality is true if $p\geq 1$ for $p-$norms. (What goes wrong for $p<1$ ?)} +\sol{\textbf{For Property \ref{n:3} for norm-2} \subsubsection*{\textbf{When field is $\bbR:$}} We have to show\begin{align*} + & \sum_i(x_i+y_i)^2\leq \left(\sqrt{\sum_ix_i^2} +\sqrt{\sum_iy_i^2}\right)^2 \\ + \implies & \sum_i (x_i^2+2x_iy_i+y_i^2)\leq \sum_ix_i^2+2\sqrt{\left[\sum_ix_i^2\right]\left[\sum_iy_i^2\right]}+\sum_iy_i^2 \\ + \implies & \left[\sum_ix_iy_i\right]^2\leq \left[\sum_ix_i^2\right]\left[\sum_iy_i^2\right] + \end{align*}So in other words prove $\langle x,y\rangle^2 \leq \langle x,x\rangle\langle y,y\rangle$ where + $$\langle x,y\rangle =\sum\limits_i x_iy_i$$ + + \begin{note} + \begin{itemize} + \item $\|x\|^2=\langle x,x\rangle$ + \item $\langle x,y\rangle=\langle y,x\rangle$ + \item $\langle \cdot,\cdot\rangle$ is $\bbR-$linear in each slot i.e. \begin{align*} + \langle rx+x',y\rangle=r\langle x,y\rangle+\langle x',y\rangle \text{ and similarly for second slot} + \end{align*}Here in $\langle x,y\rangle$ $x$ is in first slot and $y$ is in second slot. + \end{itemize} + \end{note}Now the statement is just the Cauchy-Schwartz Inequality. For proof $$\langle x,y\rangle^2\leq \langle x,x\rangle\langle y,y\rangle $$ expand everything of $\langle x-\lambda y,x-\lambda y\rangle$ which is going to give a quadratic equation in variable $\lambda $ \begin{align*} + \langle x-\lambda y,x-\lambda y\rangle & =\langle x,x-\lambda y\rangle-\lambda\langle y,x-\lambda y\rangle \\ + & =\langle x ,x\rangle -\lambda\langle x,y\rangle -\lambda\langle y,x\rangle +\lambda^2\langle y,y\rangle \\ + & =\langle x,x\rangle -2\lambda\langle x,y\rangle+\lambda^2\langle y,y\rangle + \end{align*}Now unless $x=\lambda y$ we have $\langle x-\lambda y,x-\lambda y\rangle>0$ Hence the quadratic equation has no root therefore the discriminant is greater than zero. + + \subsubsection*{\textbf{When field is $\bbC:$}}Modify the definition by $$\langle x,y\rangle=\sum_i\overline{x_i}y_i$$Then we still have $\langle x,x\rangle\geq 0$} + +\section{Algorithms} +\begin{algorithm}[H] +\KwIn{This is some input} +\KwOut{This is some output} +\SetAlgoLined +\SetNoFillComment +\tcc{This is a comment} +\vspace{3mm} +some code here\; +$x \leftarrow 0$\; +$y \leftarrow 0$\; +\uIf{$ x > 5$} { + x is greater than 5 \tcp*{This is also a comment} +} +\Else { + x is less than or equal to 5\; +} +\ForEach{y in 0..5} { + $y \leftarrow y + 1$\; +} +\For{$y$ in $0..5$} { + $y \leftarrow y - 1$\; +} +\While{$x > 5$} { + $x \leftarrow x - 1$\; +} +\Return Return something here\; +\caption{what} +\end{algorithm} + +\end{document}