From f14ebea65738a61b075dbddc6fddde0c609bdcf5 Mon Sep 17 00:00:00 2001 From: Mars Ultor Date: Thu, 4 Sep 2025 23:22:19 -0500 Subject: [PATCH] went back to the flashy styles --- general-template/homework.tex | 539 ---------------------- general-template/letterfonts.tex | 136 ++++++ general-template/macros.tex | 33 ++ general-template/preamble.tex | 746 +++++++++++++++++++++++++++++++ general-template/template.tex | 128 ++++++ labs-set-1/homework.pdf | Bin 60077 -> 0 bytes labs-set-1/homework.tex | 541 ---------------------- labs-set-1/letterfonts.tex | 136 ++++++ labs-set-1/macros.tex | 33 ++ labs-set-1/preamble.tex | 745 ++++++++++++++++++++++++++++++ labs-set-1/template.pdf | Bin 0 -> 105686 bytes labs-set-1/template.tex | 44 ++ labs-set-1/template.toc | 3 + 13 files changed, 2004 insertions(+), 1080 deletions(-) delete mode 100644 general-template/homework.tex create mode 100644 general-template/letterfonts.tex create mode 100644 general-template/macros.tex create mode 100644 general-template/preamble.tex create mode 100644 general-template/template.tex delete mode 100644 labs-set-1/homework.pdf delete mode 100644 labs-set-1/homework.tex create mode 100644 labs-set-1/letterfonts.tex create mode 100644 labs-set-1/macros.tex create mode 100644 labs-set-1/preamble.tex create mode 100644 labs-set-1/template.pdf create mode 100644 labs-set-1/template.tex create mode 100644 labs-set-1/template.toc diff --git a/general-template/homework.tex b/general-template/homework.tex deleted file mode 100644 index 872e6f5..0000000 --- a/general-template/homework.tex +++ /dev/null @@ -1,539 +0,0 @@ -\documentclass{article} - -\usepackage{fancyhdr} -\usepackage{extramarks} -\usepackage{amsmath} -\usepackage{amsthm} -\usepackage{amsfonts} -\usepackage{tikz} -\usepackage[plain]{algorithm} -\usepackage{algpseudocode} - -\usetikzlibrary{automata,positioning} - -% -% Basic Document Settings -% - -\topmargin=-0.45in -\evensidemargin=0in -\oddsidemargin=0in -\textwidth=6.5in -\textheight=9.0in -\headsep=0.25in - -\linespread{1.1} - -\pagestyle{fancy} -\lhead{\hmwkAuthorName} -\chead{\hmwkClass\ (\hmwkClassInstructor\ \hmwkClassTime): \hmwkTitle} -\rhead{\firstxmark} -\lfoot{\lastxmark} -\cfoot{\thepage} - -\renewcommand\headrulewidth{0.4pt} -\renewcommand\footrulewidth{0.4pt} - -\setlength\parindent{0pt} - -% -% Create Problem Sections -% - -\newcommand{\enterProblemHeader}[1]{ - \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} - \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} -} - -\newcommand{\exitProblemHeader}[1]{ - \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} - \stepcounter{#1} - \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} -} - -\setcounter{secnumdepth}{0} -\newcounter{partCounter} -\newcounter{homeworkProblemCounter} -\setcounter{homeworkProblemCounter}{1} -\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} - -% -% Homework Problem Environment -% -% This environment takes an optional argument. When given, it will adjust the -% problem counter. This is useful for when the problems given for your -% assignment aren't sequential. See the last 3 problems of this template for an -% example. -% -\newenvironment{homeworkProblem}[1][-1]{ - \ifnum#1>0 - \setcounter{homeworkProblemCounter}{#1} - \fi - \section{Problem \arabic{homeworkProblemCounter}} - \setcounter{partCounter}{1} - \enterProblemHeader{homeworkProblemCounter} -}{ - \exitProblemHeader{homeworkProblemCounter} -} - -% -% Homework Details -% - Title -% - Due date -% - Class -% - Section/Time -% - Instructor -% - Author -% - -\newcommand{\hmwkTitle}{Homework\ \#2} -\newcommand{\hmwkDueDate}{February 12, 2014} -\newcommand{\hmwkClass}{Calculus} -\newcommand{\hmwkClassTime}{Section A} -\newcommand{\hmwkClassInstructor}{Professor Isaac Newton} -\newcommand{\hmwkAuthorName}{\textbf{Josh Davis} \and \textbf{Davis Josh}} - -% -% Title Page -% - -\title{ - \vspace{2in} - \textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\ - \normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate\ at 3:10pm}\\ - \vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}} - \vspace{3in} -} - -\author{\hmwkAuthorName} -\date{} - -\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\} - -% -% Various Helper Commands -% - -% Useful for algorithms -\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} - -% For derivatives -\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} - -% For partial derivatives -\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} - -% Integral dx -\newcommand{\dx}{\mathrm{d}x} - -% Alias for the Solution section header -\newcommand{\solution}{\textbf{\large Solution}} - -% Probability commands: Expectation, Variance, Covariance, Bias -\newcommand{\E}{\mathrm{E}} -\newcommand{\Var}{\mathrm{Var}} -\newcommand{\Cov}{\mathrm{Cov}} -\newcommand{\Bias}{\mathrm{Bias}} - -\begin{document} - -\maketitle - -\pagebreak - -\begin{homeworkProblem} - Give an appropriate positive constant \(c\) such that \(f(n) \leq c \cdot - g(n)\) for all \(n > 1\). - - \begin{enumerate} - \item \(f(n) = n^2 + n + 1\), \(g(n) = 2n^3\) - \item \(f(n) = n\sqrt{n} + n^2\), \(g(n) = n^2\) - \item \(f(n) = n^2 - n + 1\), \(g(n) = n^2 / 2\) - \end{enumerate} - - \textbf{Solution} - - We solve each solution algebraically to determine a possible constant - \(c\). - \\ - - \textbf{Part One} - - \[ - \begin{split} - n^2 + n + 1 &= - \\ - &\leq n^2 + n^2 + n^2 - \\ - &= 3n^2 - \\ - &\leq c \cdot 2n^3 - \end{split} - \] - - Thus a valid \(c\) could be when \(c = 2\). - \\ - - \textbf{Part Two} - - \[ - \begin{split} - n^2 + n\sqrt{n} &= - \\ - &= n^2 + n^{3/2} - \\ - &\leq n^2 + n^{4/2} - \\ - &= n^2 + n^2 - \\ - &= 2n^2 - \\ - &\leq c \cdot n^2 - \end{split} - \] - - Thus a valid \(c\) is \(c = 2\). - \\ - - \textbf{Part Three} - - \[ - \begin{split} - n^2 - n + 1 &= - \\ - &\leq n^2 - \\ - &\leq c \cdot n^2/2 - \end{split} - \] - - Thus a valid \(c\) is \(c = 2\). - -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Let \(\Sigma = \{0, 1\}\). Construct a DFA \(A\) that recognizes the - language that consists of all binary numbers that can be divided by 5. - \\ - - Let the state \(q_k\) indicate the remainder of \(k\) divided by 5. For - example, the remainder of 2 would correlate to state \(q_2\) because \(7 - \mod 5 = 2\). - - \begin{figure}[h] - \centering - \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] - \node[state, accepting, initial] (q_0) {$q_0$}; - \node[state] (q_1) [right=of q_0] {$q_1$}; - \node[state] (q_2) [right=of q_1] {$q_2$}; - \node[state] (q_3) [right=of q_2] {$q_3$}; - \node[state] (q_4) [right=of q_3] {$q_4$}; - \path[->] - (q_0) - edge [loop above] node {0} (q_0) - edge node {1} (q_1) - (q_1) - edge node {0} (q_2) - edge [bend right=-30] node {1} (q_3) - (q_2) - edge [bend left] node {1} (q_0) - edge [bend right=-30] node {0} (q_4) - (q_3) - edge node {1} (q_2) - edge [bend left] node {0} (q_1) - (q_4) - edge node {0} (q_3) - edge [loop below] node {1} (q_4); - \end{tikzpicture} - \caption{DFA, \(A\), this is really beautiful, ya know?} - \label{fig:multiple5} - \end{figure} - - \textbf{Justification} - \\ - - Take a given binary number, \(x\). Since there are only two inputs to our - state machine, \(x\) can either become \(x0\) or \(x1\). When a 0 comes - into the state machine, it is the same as taking the binary number and - multiplying it by two. When a 1 comes into the machine, it is the same as - multipying by two and adding one. - \\ - - Using this knowledge, we can construct a transition table that tell us - where to go: - - \begin{table}[ht] - \centering - \begin{tabular}{c || c | c | c | c | c} - & \(x \mod 5 = 0\) - & \(x \mod 5 = 1\) - & \(x \mod 5 = 2\) - & \(x \mod 5 = 3\) - & \(x \mod 5 = 4\) - \\ - \hline - \(x0\) & 0 & 2 & 4 & 1 & 3 \\ - \(x1\) & 1 & 3 & 0 & 2 & 4 \\ - \end{tabular} - \end{table} - - Therefore on state \(q_0\) or (\(x \mod 5 = 0\)), a transition line should - go to state \(q_0\) for the input 0 and a line should go to state \(q_1\) - for input 1. Continuing this gives us the Figure~\ref{fig:multiple5}. -\end{homeworkProblem} - -\begin{homeworkProblem} - Write part of \alg{Quick-Sort($list, start, end$)} - - \begin{algorithm}[] - \begin{algorithmic}[1] - \Function{Quick-Sort}{$list, start, end$} - \If{$start \geq end$} - \State{} \Return{} - \EndIf{} - \State{} $mid \gets \Call{Partition}{list, start, end}$ - \State{} \Call{Quick-Sort}{$list, start, mid - 1$} - \State{} \Call{Quick-Sort}{$list, mid + 1, end$} - \EndFunction{} - \end{algorithmic} - \caption{Start of QuickSort} - \end{algorithm} -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Suppose we would like to fit a straight line through the origin, i.e., - \(Y_i = \beta_1 x_i + e_i\) with \(i = 1, \ldots, n\), \(\E [e_i] = 0\), - and \(\Var [e_i] = \sigma^2_e\) and \(\Cov[e_i, e_j] = 0, \forall i \neq - j\). - \\ - - \part - - Find the least squares esimator for \(\hat{\beta_1}\) for the slope - \(\beta_1\). - \\ - - \solution - - To find the least squares estimator, we should minimize our Residual Sum - of Squares, RSS: - - \[ - \begin{split} - RSS &= \sum_{i = 1}^{n} {(Y_i - \hat{Y_i})}^2 - \\ - &= \sum_{i = 1}^{n} {(Y_i - \hat{\beta_1} x_i)}^2 - \end{split} - \] - - By taking the partial derivative in respect to \(\hat{\beta_1}\), we get: - - \[ - \pderiv{ - \hat{\beta_1} - }{RSS} - = -2 \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} - = 0 - \] - - This gives us: - - \[ - \begin{split} - \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} - &= \sum_{i = 1}^{n} {x_i Y_i} - \sum_{i = 1}^{n} \hat{\beta_1} x_i^2 - \\ - &= \sum_{i = 1}^{n} {x_i Y_i} - \hat{\beta_1}\sum_{i = 1}^{n} x_i^2 - \end{split} - \] - - Solving for \(\hat{\beta_1}\) gives the final estimator for \(\beta_1\): - - \[ - \begin{split} - \hat{\beta_1} - &= \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - } - \end{split} - \] - - \pagebreak - - \part - - Calculate the bias and the variance for the estimated slope - \(\hat{\beta_1}\). - \\ - - \solution - - For the bias, we need to calculate the expected value - \(\E[\hat{\beta_1}]\): - - \[ - \begin{split} - \E[\hat{\beta_1}] - &= \E \left[ \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - }\right] - \\ - &= \frac{ - \sum {x_i \E[Y_i]} - }{ - \sum x_i^2 - } - \\ - &= \frac{ - \sum {x_i (\beta_1 x_i)} - }{ - \sum x_i^2 - } - \\ - &= \frac{ - \sum {x_i^2 \beta_1} - }{ - \sum x_i^2 - } - \\ - &= \beta_1 \frac{ - \sum {x_i^2 \beta_1} - }{ - \sum x_i^2 - } - \\ - &= \beta_1 - \end{split} - \] - - Thus since our estimator's expected value is \(\beta_1\), we can conclude - that the bias of our estimator is 0. - \\ - - For the variance: - - \[ - \begin{split} - \Var[\hat{\beta_1}] - &= \Var \left[ \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - }\right] - \\ - &= - \frac{ - \sum {x_i^2} - }{ - \sum x_i^2 \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - \sum {x_i^2} - }{ - \sum x_i^2 \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - 1 - }{ - \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - 1 - }{ - \sum x_i^2 - } \sigma^2 - \\ - &= - \frac{ - \sigma^2 - }{ - \sum x_i^2 - } - \end{split} - \] - -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Prove a polynomial of degree \(k\), \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots - + a_1n^1 + a_0n^0\) is a member of \(\Theta(n^k)\) where \(a_k \hdots a_0\) - are nonnegative constants. - - \begin{proof} - To prove that \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + - a_0n^0\), we must show the following: - - \[ - \exists c_1 \exists c_2 \forall n \geq n_0,\ {c_1 \cdot g(n) \leq - f(n) \leq c_2 \cdot g(n)} - \] - - For the first inequality, it is easy to see that it holds because no - matter what the constants are, \(n^k \leq a_kn^k + a_{k - 1}n^{k - 1} + - \hdots + a_1n^1 + a_0n^0\) even if \(c_1 = 1\) and \(n_0 = 1\). This - is because \(n^k \leq c_1 \cdot a_kn^k\) for any nonnegative constant, - \(c_1\) and \(a_k\). - \\ - - Taking the second inequality, we prove it in the following way. - By summation, \(\sum\limits_{i=0}^k a_i\) will give us a new constant, - \(A\). By taking this value of \(A\), we can then do the following: - - \[ - \begin{split} - a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + a_0n^0 &= - \\ - &\leq (a_k + a_{k - 1} \hdots a_1 + a_0) \cdot n^k - \\ - &= A \cdot n^k - \\ - &\leq c_2 \cdot n^k - \end{split} - \] - - where \(n_0 = 1\) and \(c_2 = A\). \(c_2\) is just a constant. Thus the - proof is complete. - \end{proof} -\end{homeworkProblem} - -\pagebreak - -% -% Non sequential homework problems -% - -% Jump to problem 18 -\begin{homeworkProblem}[18] - Evaluate \(\sum_{k=1}^{5} k^2\) and \(\sum_{k=1}^{5} (k - 1)^2\). -\end{homeworkProblem} - -% Continue counting to 19 -\begin{homeworkProblem} - Find the derivative of \(f(x) = x^4 + 3x^2 - 2\) -\end{homeworkProblem} - -% Go back to where we left off -\begin{homeworkProblem}[6] - Evaluate the integrals - \(\int_0^1 (1 - x^2) \dx\) - and - \(\int_1^{\infty} \frac{1}{x^2} \dx\). -\end{homeworkProblem} - -\end{document} diff --git a/general-template/letterfonts.tex b/general-template/letterfonts.tex new file mode 100644 index 0000000..cf91649 --- /dev/null +++ b/general-template/letterfonts.tex @@ -0,0 +1,136 @@ +% 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}}} + +%--------------------------------------- +% 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/general-template/macros.tex b/general-template/macros.tex new file mode 100644 index 0000000..5ebc567 --- /dev/null +++ b/general-template/macros.tex @@ -0,0 +1,33 @@ +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\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{\der}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} +% \newcommand{\ddd}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d}^{#3} #2}} +\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{\id}{\text{id}} + + +\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/general-template/preamble.tex b/general-template/preamble.tex new file mode 100644 index 0000000..bc65870 --- /dev/null +++ b/general-template/preamble.tex @@ -0,0 +1,746 @@ +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage{gensymb} + +\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[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} +\usepackage{chngcntr} +\usepackage{tikz} +\usepackage{titletoc} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} +\tikzstyle{c} = [circle,fill=black,scale=0.5] +\tikzstyle{b} = [draw, thick, black, -] +\tikzset{ + vertex/.style={ + circle, + draw, + minimum size=6mm, + inner sep=0pt + } +} + +\renewcommand\qedsymbol{$\Laughey$} + +% \usepackage{xparse} +% \usepackage{pgffor} +% \usepackage{emoji} +% +% \NewDocumentCommand{\memoji}{m o}{% +% \IfNoValueTF{#2} +% {\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi} % Single emoji case +% {\foreach \index in {1,...,#2}{\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi}} % Repeated emoji case +% } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\definecolor{doc}{RGB}{0,60,110} +\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{mylemmabg}{HTML}{FFFAF8} +\definecolor{mylemmafr}{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} + + +%================================ +% LEMMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lemma}{Lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lemma}{lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{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} + + +%================================ +% 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} + + + +%================================ +% 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} + + +%================================ +% EXERCISE BOX +%================================ + +\newcounter{questioncounter} +\counterwithin{questioncounter}{chapter} +% \counterwithin{questioncounter}{section} + +\makeatletter +\newtcbtheorem[use counter=questioncounter]{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{\mlemma}[2]{\begin{Lemma}{#1}{}#2\end{Lemma}} +\newcommand{\mer}[2]{\begin{Exercise}{#1}{}#2\end{Exercise}} +\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: ]}{} +\newenvironment{iclaim}[1][\claimname]{\bfseries #1\mdseries:}{} +\newcommand{\iclm}[2]{\begin{iclaim}[#1]#2\end{iclaim}} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + +% 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} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.7cm,y=0.2cm]{\color{white}\Large\sc\bfseries Chapter\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Page \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (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}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/general-template/template.tex b/general-template/template.tex new file mode 100644 index 0000000..970913a --- /dev/null +++ b/general-template/template.tex @@ -0,0 +1,128 @@ +\documentclass{report} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\Huge{Some Class}\\Random Examples} +\author{\huge{Your Name}} +\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} diff --git a/labs-set-1/homework.pdf b/labs-set-1/homework.pdf deleted file mode 100644 index c8b27ff9381bb23ad6f35075317f50ded493baf1..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 60077 zcmY!laB<T$)HCN&y>R8|4K6bU1BLvgEG`=xE`6WWy!4U`1!Ds<1qk5McgxHvNi9;) zcgsmENp(q0&QD3@va{nVE-6Y)%;l=M)f(vQf7pO$@B8qGN0&TIy63J>Gv9Ztfm>*= zuMAfT&%tRc?;g2-JeNyKNNBo6<+l69^F;SFHNEcqr9SWa<d-qw4RJzI%wM@YB0EDG zRh4@`2JAnuX-9_LJlB<bb@MD9e78J(Ln-&s@&tvg?<FlAKSta)DsAv$dv3IO8Xu<@ 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-\renewcommand\headrulewidth{0.4pt} -\renewcommand\footrulewidth{0.4pt} - -\setlength\parindent{0pt} - -% -% Create Problem Sections -% - -\newcommand{\enterProblemHeader}[1]{ - \nobreak\extramarks{}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} - \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} -} - -\newcommand{\exitProblemHeader}[1]{ - \nobreak\extramarks{Problem \arabic{#1} (continued)}{Problem \arabic{#1} continued on next page\ldots}\nobreak{} - \stepcounter{#1} - \nobreak\extramarks{Problem \arabic{#1}}{}\nobreak{} -} - -\setcounter{secnumdepth}{0} -\newcounter{partCounter} -\newcounter{homeworkProblemCounter} -\setcounter{homeworkProblemCounter}{1} -\nobreak\extramarks{Problem \arabic{homeworkProblemCounter}}{}\nobreak{} - -% -% Homework Problem Environment -% -% This environment takes an optional argument. When given, it will adjust the -% problem counter. This is useful for when the problems given for your -% assignment aren't sequential. See the last 3 problems of this template for an -% example. -% -\newenvironment{homeworkProblem}[1][-1]{ - \ifnum#1>0 - \setcounter{homeworkProblemCounter}{#1} - \fi - \section{Problem \arabic{homeworkProblemCounter}} - \setcounter{partCounter}{1} - \enterProblemHeader{homeworkProblemCounter} -}{ - \exitProblemHeader{homeworkProblemCounter} -} - -% -% Homework Details -% - Title -% - Due date -% - Class -% - Section/Time -% - Instructor -% - Author -% - -\newcommand{\hmwkTitle}{Homework\ \#1} -\newcommand{\hmwkDueDate}{September 21, 2025} -\newcommand{\hmwkClass}{Calc III} -\newcommand{\hmwkClassTime}{$6^{\text{th}}$ Hour}} -\newcommand{\hmwkClassInstructor}{Professor Foresee} -\newcommand{\hmwkAuthorName}{\textbf{Krishna A.}} - -% -% Title Page -% - -\title{ - \vspace{2in} - \textmd{\textbf{\hmwkClass:\ \hmwkTitle}}\\ - \normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate\ at 3:10pm}\\ - \vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}} - \vspace{3in} -} - -\author{\hmwkAuthorName} -\date{} - -\renewcommand{\part}[1]{\textbf{\large Part \Alph{partCounter}}\stepcounter{partCounter}\\} - -% -% Various Helper Commands -% - -% Useful for algorithms -\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} - -% For derivatives -\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} - -% For partial derivatives -\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} - -% Integral dx -\newcommand{\dx}{\mathrm{d}x} - -% Alias for the Solution section header -\newcommand{\solution}{\textbf{\large Solution}} - -% Probability commands: Expectation, Variance, Covariance, Bias -\newcommand{\E}{\mathrm{E}} -\newcommand{\Var}{\mathrm{Var}} -\newcommand{\Cov}{\mathrm{Cov}} -\newcommand{\Bias}{\mathrm{Bias}} - -\begin{document} - -\maketitle - -\pagebreak - -\begin{comment} - -\begin{homeworkProblem} - Give an appropriate positive constant \(c\) such that \(f(n) \leq c \cdot - g(n)\) for all \(n > 1\). - - \begin{enumerate} - \item \(f(n) = n^2 + n + 1\), \(g(n) = 2n^3\) - \item \(f(n) = n\sqrt{n} + n^2\), \(g(n) = n^2\) - \item \(f(n) = n^2 - n + 1\), \(g(n) = n^2 / 2\) - \end{enumerate} - - \textbf{Solution} - - We solve each solution algebraically to determine a possible constant - \(c\). - \\ - - \textbf{Part One} - - \[ - \begin{split} - n^2 + n + 1 &= - \\ - &\leq n^2 + n^2 + n^2 - \\ - &= 3n^2 - \\ - &\leq c \cdot 2n^3 - \end{split} - \] - - Thus a valid \(c\) could be when \(c = 2\). - \\ - - \textbf{Part Two} - - \[ - \begin{split} - n^2 + n\sqrt{n} &= - \\ - &= n^2 + n^{3/2} - \\ - &\leq n^2 + n^{4/2} - \\ - &= n^2 + n^2 - \\ - &= 2n^2 - \\ - &\leq c \cdot n^2 - \end{split} - \] - - Thus a valid \(c\) is \(c = 2\). - \\ - - \textbf{Part Three} - - \[ - \begin{split} - n^2 - n + 1 &= - \\ - &\leq n^2 - \\ - &\leq c \cdot n^2/2 - \end{split} - \] - - Thus a valid \(c\) is \(c = 2\). - -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Let \(\Sigma = \{0, 1\}\). Construct a DFA \(A\) that recognizes the - language that consists of all binary numbers that can be divided by 5. - \\ - - Let the state \(q_k\) indicate the remainder of \(k\) divided by 5. For - example, the remainder of 2 would correlate to state \(q_2\) because \(7 - \mod 5 = 2\). - - \begin{figure}[h] - \centering - \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] - \node[state, accepting, initial] (q_0) {$q_0$}; - \node[state] (q_1) [right=of q_0] {$q_1$}; - \node[state] (q_2) [right=of q_1] {$q_2$}; - \node[state] (q_3) [right=of q_2] {$q_3$}; - \node[state] (q_4) [right=of q_3] {$q_4$}; - \path[->] - (q_0) - edge [loop above] node {0} (q_0) - edge node {1} (q_1) - (q_1) - edge node {0} (q_2) - edge [bend right=-30] node {1} (q_3) - (q_2) - edge [bend left] node {1} (q_0) - edge [bend right=-30] node {0} (q_4) - (q_3) - edge node {1} (q_2) - edge [bend left] node {0} (q_1) - (q_4) - edge node {0} (q_3) - edge [loop below] node {1} (q_4); - \end{tikzpicture} - \caption{DFA, \(A\), this is really beautiful, ya know?} - \label{fig:multiple5} - \end{figure} - - \textbf{Justification} - \\ - - Take a given binary number, \(x\). Since there are only two inputs to our - state machine, \(x\) can either become \(x0\) or \(x1\). When a 0 comes - into the state machine, it is the same as taking the binary number and - multiplying it by two. When a 1 comes into the machine, it is the same as - multipying by two and adding one. - \\ - - Using this knowledge, we can construct a transition table that tell us - where to go: - - \begin{table}[ht] - \centering - \begin{tabular}{c || c | c | c | c | c} - & \(x \mod 5 = 0\) - & \(x \mod 5 = 1\) - & \(x \mod 5 = 2\) - & \(x \mod 5 = 3\) - & \(x \mod 5 = 4\) - \\ - \hline - \(x0\) & 0 & 2 & 4 & 1 & 3 \\ - \(x1\) & 1 & 3 & 0 & 2 & 4 \\ - \end{tabular} - \end{table} - - Therefore on state \(q_0\) or (\(x \mod 5 = 0\)), a transition line should - go to state \(q_0\) for the input 0 and a line should go to state \(q_1\) - for input 1. Continuing this gives us the Figure~\ref{fig:multiple5}. -\end{homeworkProblem} - -\begin{homeworkProblem} - Write part of \alg{Quick-Sort($list, start, end$)} - - \begin{algorithm}[] - \begin{algorithmic}[1] - \Function{Quick-Sort}{$list, start, end$} - \If{$start \geq end$} - \State{} \Return{} - \EndIf{} - \State{} $mid \gets \Call{Partition}{list, start, end}$ - \State{} \Call{Quick-Sort}{$list, start, mid - 1$} - \State{} \Call{Quick-Sort}{$list, mid + 1, end$} - \EndFunction{} - \end{algorithmic} - \caption{Start of QuickSort} - \end{algorithm} -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Suppose we would like to fit a straight line through the origin, i.e., - \(Y_i = \beta_1 x_i + e_i\) with \(i = 1, \ldots, n\), \(\E [e_i] = 0\), - and \(\Var [e_i] = \sigma^2_e\) and \(\Cov[e_i, e_j] = 0, \forall i \neq - j\). - \\ - - \part - - Find the least squares esimator for \(\hat{\beta_1}\) for the slope - \(\beta_1\). - \\ - - \solution - - To find the least squares estimator, we should minimize our Residual Sum - of Squares, RSS: - - \[ - \begin{split} - RSS &= \sum_{i = 1}^{n} {(Y_i - \hat{Y_i})}^2 - \\ - &= \sum_{i = 1}^{n} {(Y_i - \hat{\beta_1} x_i)}^2 - \end{split} - \] - - By taking the partial derivative in respect to \(\hat{\beta_1}\), we get: - - \[ - \pderiv{ - \hat{\beta_1} - }{RSS} - = -2 \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} - = 0 - \] - - This gives us: - - \[ - \begin{split} - \sum_{i = 1}^{n} {x_i (Y_i - \hat{\beta_1} x_i)} - &= \sum_{i = 1}^{n} {x_i Y_i} - \sum_{i = 1}^{n} \hat{\beta_1} x_i^2 - \\ - &= \sum_{i = 1}^{n} {x_i Y_i} - \hat{\beta_1}\sum_{i = 1}^{n} x_i^2 - \end{split} - \] - - Solving for \(\hat{\beta_1}\) gives the final estimator for \(\beta_1\): - - \[ - \begin{split} - \hat{\beta_1} - &= \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - } - \end{split} - \] - - \pagebreak - - \part - - Calculate the bias and the variance for the estimated slope - \(\hat{\beta_1}\). - \\ - - \solution - - For the bias, we need to calculate the expected value - \(\E[\hat{\beta_1}]\): - - \[ - \begin{split} - \E[\hat{\beta_1}] - &= \E \left[ \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - }\right] - \\ - &= \frac{ - \sum {x_i \E[Y_i]} - }{ - \sum x_i^2 - } - \\ - &= \frac{ - \sum {x_i (\beta_1 x_i)} - }{ - \sum x_i^2 - } - \\ - &= \frac{ - \sum {x_i^2 \beta_1} - }{ - \sum x_i^2 - } - \\ - &= \beta_1 \frac{ - \sum {x_i^2 \beta_1} - }{ - \sum x_i^2 - } - \\ - &= \beta_1 - \end{split} - \] - - Thus since our estimator's expected value is \(\beta_1\), we can conclude - that the bias of our estimator is 0. - \\ - - For the variance: - - \[ - \begin{split} - \Var[\hat{\beta_1}] - &= \Var \left[ \frac{ - \sum {x_i Y_i} - }{ - \sum x_i^2 - }\right] - \\ - &= - \frac{ - \sum {x_i^2} - }{ - \sum x_i^2 \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - \sum {x_i^2} - }{ - \sum x_i^2 \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - 1 - }{ - \sum x_i^2 - } \Var[Y_i] - \\ - &= - \frac{ - 1 - }{ - \sum x_i^2 - } \sigma^2 - \\ - &= - \frac{ - \sigma^2 - }{ - \sum x_i^2 - } - \end{split} - \] - -\end{homeworkProblem} - -\pagebreak - -\begin{homeworkProblem} - Prove a polynomial of degree \(k\), \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots - + a_1n^1 + a_0n^0\) is a member of \(\Theta(n^k)\) where \(a_k \hdots a_0\) - are nonnegative constants. - - \begin{proof} - To prove that \(a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + - a_0n^0\), we must show the following: - - \[ - \exists c_1 \exists c_2 \forall n \geq n_0,\ {c_1 \cdot g(n) \leq - f(n) \leq c_2 \cdot g(n)} - \] - - For the first inequality, it is easy to see that it holds because no - matter what the constants are, \(n^k \leq a_kn^k + a_{k - 1}n^{k - 1} + - \hdots + a_1n^1 + a_0n^0\) even if \(c_1 = 1\) and \(n_0 = 1\). This - is because \(n^k \leq c_1 \cdot a_kn^k\) for any nonnegative constant, - \(c_1\) and \(a_k\). - \\ - - Taking the second inequality, we prove it in the following way. - By summation, \(\sum\limits_{i=0}^k a_i\) will give us a new constant, - \(A\). By taking this value of \(A\), we can then do the following: - - \[ - \begin{split} - a_kn^k + a_{k - 1}n^{k - 1} + \hdots + a_1n^1 + a_0n^0 &= - \\ - &\leq (a_k + a_{k - 1} \hdots a_1 + a_0) \cdot n^k - \\ - &= A \cdot n^k - \\ - &\leq c_2 \cdot n^k - \end{split} - \] - - where \(n_0 = 1\) and \(c_2 = A\). \(c_2\) is just a constant. Thus the - proof is complete. - \end{proof} -\end{homeworkProblem} - -\pagebreak - -% -% Non sequential homework problems -% - -% Jump to problem 18 -\begin{homeworkProblem}[18] - Evaluate \(\sum_{k=1}^{5} k^2\) and \(\sum_{k=1}^{5} (k - 1)^2\). -\end{homeworkProblem} - -% Continue counting to 19 -\begin{homeworkProblem} - Find the derivative of \(f(x) = x^4 + 3x^2 - 2\) -\end{homeworkProblem} - -% Go back to where we left off -\begin{homeworkProblem}[6] - Evaluate the integrals - \(\int_0^1 (1 - x^2) \dx\) - and - \(\int_1^{\infty} \frac{1}{x^2} \dx\). -\end{homeworkProblem} -\end{comment} -\end{document} diff --git a/labs-set-1/letterfonts.tex b/labs-set-1/letterfonts.tex new file mode 100644 index 0000000..cf91649 --- /dev/null +++ b/labs-set-1/letterfonts.tex @@ -0,0 +1,136 @@ +% 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}}} + +%--------------------------------------- +% 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/labs-set-1/macros.tex b/labs-set-1/macros.tex new file mode 100644 index 0000000..5ebc567 --- /dev/null +++ b/labs-set-1/macros.tex @@ -0,0 +1,33 @@ +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\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{\der}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} +% \newcommand{\ddd}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d}^{#3} #2}} +\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{\id}{\text{id}} + + +\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/labs-set-1/preamble.tex b/labs-set-1/preamble.tex new file mode 100644 index 0000000..5ba904f --- /dev/null +++ b/labs-set-1/preamble.tex @@ -0,0 +1,745 @@ +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage{gensymb} +\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[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} +\usepackage{chngcntr} +\usepackage{tikz} +\usepackage{titletoc} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} +\tikzstyle{c} = [circle,fill=black,scale=0.5] +\tikzstyle{b} = [draw, thick, black, -] +\tikzset{ + vertex/.style={ + circle, + draw, + minimum size=6mm, + inner sep=0pt + } +} + +\renewcommand\qedsymbol{$\Laughey$} + +% \usepackage{xparse} +% \usepackage{pgffor} +% \usepackage{emoji} +% +% \NewDocumentCommand{\memoji}{m o}{% +% \IfNoValueTF{#2} +% {\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi} % Single emoji case +% {\foreach \index in {1,...,#2}{\ifmmode \text{\emoji{#1}} \else \emoji{#1}\fi}} % Repeated emoji case +% } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS 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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} + + +%================================ +% LEMMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lemma}{Lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lemma}{lemma} +{% + enhanced, + breakable, + colback = mylemmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylemmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylemmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylemmafr}, +} +{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} + + +%================================ +% 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} + + + +%================================ +% 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} + + +%================================ +% EXERCISE BOX +%================================ + +\newcounter{questioncounter} +\counterwithin{questioncounter}{chapter} +% \counterwithin{questioncounter}{section} + +\makeatletter +\newtcbtheorem[use counter=questioncounter]{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{\mlemma}[2]{\begin{Lemma}{#1}{}#2\end{Lemma}} +\newcommand{\mer}[2]{\begin{Exercise}{#1}{}#2\end{Exercise}} +\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: ]}{} +\newenvironment{iclaim}[1][\claimname]{\bfseries #1\mdseries:}{} +\newcommand{\iclm}[2]{\begin{iclaim}[#1]#2\end{iclaim}} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + +% 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} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 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+\input{macros} +\input{letterfonts} + +\title{\Huge{Calculus III}\\Homework \# 1} +\author{\huge{Krishna Ayyalasomayajula}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\chapter{Lab 2 - 13.1 Apps} +\section{Work} + +\qs{}{ + Let $\vec{T_1}$ represent the tension of the leftmost cable, while $\vec{T_2}$ encodes the tension force experienced by the rightmost cable. Our coordinate system will originate at the intersection of the two cables. + + \begin{align*} + \vec{T_1} \coloneqq \langle \|\vec{T_1}\| ; 135\degree \rangle \\ + \vec{T_2} \coloneqq \langle \|\vec{T_2}|\| ; 15\degree \rangle \\ + 500=\|\vec{T_1}\|\sin{135\degree} + \|\vec{T_2}\|\sin{15} \\ + 0 = \|\vec{T_1}\|\cos{135\degree} + \|\vec{T_2}\|\cos{15} \\ + \implies 500=\|\vec{T_1}\|\tfrac{\sqrt{2}}{2} + \|\vec{T_2}\|\cdot0.258819045103 \\ + \implies0 = \|\vec{T_1}\|\tfrac{-\sqrt{2}}{2} + \|\vec{T_2}\|\cdot0.965925826289 \\ + \text{Solving the system numerically yields: }\\ + \|\vec{T_1}\| \Rightarrow 557.677\, \mathrm{lb}\\ + \|\vec{T_2}\| \Rightarrow 408.248\,\mathrm{lb} \\ + \text{This corresponds to answer choice A} + \end{align*} + +} + +\qs{}{ + +} +\end{document} diff --git a/labs-set-1/template.toc b/labs-set-1/template.toc new file mode 100644 index 0000000..853a91d --- /dev/null +++ b/labs-set-1/template.toc @@ -0,0 +1,3 @@ +\contentsline {chapter}{\numberline {1}Lab 2 - 13.1 Apps}{2}{chapter.1}% +\contentsline {section}{\numberline {1.1}Work}{2}{section.1.1}% +\contentsfinish