├── notes ├── rudin │ ├── ch-02.tex │ ├── commands.tex │ └── ch-01.tex ├── workbook-1 │ ├── ch-07.tex │ ├── commands.tex │ ├── ch-06-rings.tex │ └── ch-06.tex ├── sheets │ ├── multivariate.tex │ ├── trig-identities.tex │ ├── commands.tex │ ├── types-of-integrals.tex │ └── study-guide.tex ├── definitions │ └── cross-product.tex └── scripts │ └── euclid.py ├── code ├── src │ ├── talk.js │ ├── factor.js │ ├── min.js │ ├── polynomial.js │ └── improve.js └── package.json ├── .gitignore └── README.md /notes/rudin/ch-02.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | 3 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 4 | \include{commands} 5 | 6 | \begin{document} 7 | 8 | what 9 | 10 | \end{document} 11 | -------------------------------------------------------------------------------- /notes/workbook-1/ch-07.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 3 | \include{commands} 4 | \begin{document} 5 | 6 | \section{Additional Topics} 7 | 8 | \end{document} 9 | -------------------------------------------------------------------------------- /code/src/talk.js: -------------------------------------------------------------------------------- 1 | var net = require('net'); 2 | var socket = new net.Socket(); 3 | 4 | socket.connect(2323, 'challenge2.airtime.com', function () { 5 | 6 | }); 7 | 8 | socket.on('data', function (data) { 9 | console.log(data); 10 | }); 11 | 12 | socket.on('close', function () { 13 | 14 | }); -------------------------------------------------------------------------------- /code/src/factor.js: -------------------------------------------------------------------------------- 1 | function factor (n) { 2 | var primes = {}, k; 3 | n = Math.abs(n); 4 | k=2; 5 | 6 | while (k <= n) { 7 | if (n % k === 0) { 8 | n /= k; 9 | primes[k] = k in primes ? primes[k]+1 : 1; 10 | } 11 | else 12 | k++; 13 | } 14 | 15 | return primes; 16 | } 17 | 18 | module.exports = factor; -------------------------------------------------------------------------------- /code/package.json: -------------------------------------------------------------------------------- 1 | { 2 | "name": "Erdos-Euler-Code", 3 | "version": "0.0.0", 4 | "description": "", 5 | "main": "src/improve.js", 6 | "directories": { 7 | "test": "tests" 8 | }, 9 | "scripts": { 10 | "test": "echo \"Error: no test specified\" && exit 1" 11 | }, 12 | "author": "", 13 | "license": "UNLICENSE", 14 | "dependencies": { 15 | "underscore": "^1.7.0" 16 | } 17 | } 18 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | node_modules 2 | .DS_Store 3 | *.acn 4 | *.acr 5 | *.alg 6 | *.aux 7 | *.bbl 8 | *.blg 9 | *.dvi 10 | *.fdb_latexmk 11 | *.fls 12 | *.glg 13 | *.glo 14 | *.gls 15 | *.idx 16 | *.ilg 17 | *.ind 18 | *.ist 19 | *.lof 20 | *.log 21 | *.lot 22 | *.maf 23 | *.mtc 24 | *.mtc0 25 | *.nav 26 | *.nlo 27 | *.out 28 | *.pdf 29 | *.pdfsync 30 | *.ps 31 | *.snm 32 | *.synctex.gz 33 | *.toc 34 | *.vrb 35 | *.xdy 36 | *.tdo 37 | -------------------------------------------------------------------------------- /notes/sheets/multivariate.tex: -------------------------------------------------------------------------------- 1 | \documentclass[11pt,twoside,a4paper]{article} 2 | 3 | \title{Multivariate Calc for the GRE} 4 | \author{Matt Owen} 5 | 6 | \begin{document} 7 | 8 | \maketitle 9 | \tableofcontents 10 | \cleardoublepage 11 | 12 | \section{Intro} 13 | \label{intro} 14 | 15 | 2 16 | % Green's Theorem 17 | % 18 | % 19 | 20 | \section{Green's Theorem} 21 | \label{green-s-theorem} 22 | 23 | Crucial Theorem in Multivariate Calculus. Allows for the connection between rectangular-integrals and path-integrals. 24 | 25 | \subsection{Statement} 26 | ... 27 | 28 | \subsection{Proof} 29 | ... 30 | 31 | \subsection{Uses} 32 | ... 33 | 34 | \end{document} -------------------------------------------------------------------------------- /notes/sheets/trig-identities.tex: -------------------------------------------------------------------------------- 1 | \documentclass[11pt,twoside,a4paper]{article} 2 | 3 | 4 | \usepackage{mathtools} 5 | 6 | \title{Trig Identities} 7 | \author{Matt Owen} 8 | 9 | \begin{document} 10 | 11 | \maketitle 12 | \tableofcontents 13 | \cleardoublepage 14 | 15 | \section{Differentiation} 16 | 17 | \begin{itemize} 18 | \item $ \frac{d}{dx}(\sin x) = \cos x$ 19 | \item $ \frac{d}{dx}(\cos x) = -\sin x$ 20 | \item $ \frac{d}{dx}(\tan x) = \frac{1}{\cos^2 x}$ 21 | \item $ \frac{d}{dx}(\csc x) = -\csc x \cot x$ 22 | \item $ \frac{d}{dx}(\sec x) = \sec x \tan x$ 23 | \item $ \frac{d}{dx}(\cot x) = -\csc^2 x$ 24 | \end{itemize} 25 | 26 | 27 | \end{document} -------------------------------------------------------------------------------- /notes/rudin/commands.tex: -------------------------------------------------------------------------------- 1 | % Set builder notation 2 | \newcommand{\set}[2]{ 3 | \{\ #1 \mid #2\ \} 4 | } 5 | 6 | % ?= 7 | \newcommand{\qeq}{\stackrel{?}{=}} 8 | 9 | % def= 10 | \newcommand{\defeq}{\stackrel{\text{def}}{=}} 11 | 12 | % Definition block with label 13 | \newcommand{\DEFINITION}[1]{ 14 | \label{def-#1} 15 | {\noindent \bf Definition #1} 16 | } 17 | 18 | % Theorem block with label 19 | \newcommand{\THEOREM}[1]{ 20 | \label{theorem-#1} 21 | {\noindent \bf #1 Theorem} 22 | } 23 | 24 | % Generator: 25 | \newcommand{\gen}[1]{ 26 | \langle #1 \rangle 27 | } 28 | 29 | % Modulo: (mod 3) 30 | \newcommand{\modulo}[1]{ 31 | \ (\textrm{mod}\ #1) 32 | } 33 | -------------------------------------------------------------------------------- /notes/sheets/commands.tex: -------------------------------------------------------------------------------- 1 | % Set builder notation 2 | \newcommand{\set}[2]{ 3 | \{\ #1\ \mid\ #2\ \} 4 | } 5 | 6 | % ?= 7 | \newcommand{\qeq}{\stackrel{?}{=}} 8 | 9 | % def= 10 | \newcommand{\defeq}{\stackrel{\text{def}}{=}} 11 | 12 | % Definition block with label 13 | \newcommand{\DEFINITION}[1]{ 14 | \label{def-#1} 15 | {\noindent \bf Definition #1} 16 | } 17 | 18 | % Theorem block with label 19 | \newcommand{\THEOREM}[1]{ 20 | \label{theorem-#1} 21 | {\noindent \bf #1 Theorem} 22 | } 23 | 24 | % Generator: 25 | \newcommand{\gen}[1]{ 26 | \langle #1 \rangle 27 | } 28 | 29 | % Modulo: (mod 3) 30 | \newcommand{\modu}[1]{ 31 | \ (\textrm{mod}\ #1) 32 | } 33 | -------------------------------------------------------------------------------- /notes/workbook-1/commands.tex: -------------------------------------------------------------------------------- 1 | % Set builder notation 2 | \newcommand{\set}[2]{ 3 | \{\ #1\ \mid\ #2\ \} 4 | } 5 | 6 | % ?= 7 | \newcommand{\qeq}{\stackrel{?}{=}} 8 | 9 | % def= 10 | \newcommand{\defeq}{\stackrel{\text{def}}{=}} 11 | 12 | % Definition block with label 13 | \newcommand{\DEFINITION}[1]{ 14 | \label{def-#1} 15 | {\noindent \bf Definition #1} 16 | } 17 | 18 | % Theorem block with label 19 | \newcommand{\THEOREM}[1]{ 20 | \label{theorem-#1} 21 | {\noindent \bf #1 Theorem} 22 | } 23 | 24 | % Generator: 25 | \newcommand{\gen}[1]{ 26 | \langle #1 \rangle 27 | } 28 | 29 | % Modulo: (mod 3) 30 | \newcommand{\modu}[1]{ 31 | \ (\textrm{mod}\ #1) 32 | } 33 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Mathematics and Review 2 | 3 | *"Reunion of Broken Parts"* 4 | 5 | ## About 6 | 7 | A review of Mathematical material by Matt Hammerstadt. My life is git now. 8 | It's probably crucial to accept that, and move forward using it to an advantage. 9 | 10 | This work is written in LaTeX as a tool used for reflection. It probably has 11 | more archival strength than print, and - tbh - I might actually review it (which 12 | I wouldn't if it was in Five Stars). 13 | 14 | ## Manifest 15 | 16 | * `/rudin` : A study using *Principles of Mathematics by Rudin* 17 | * `/workbook-1` : First GRE Workbook 18 | * `/exams` : Lessons learned from practice exam, and explanation of tricky solutions. 19 | 20 | ## Manifesto 21 | 22 | In addition to actually practicing my Math skills, this is also become on of my 23 | many tools for reflection. 24 | -------------------------------------------------------------------------------- /notes/definitions/cross-product.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 3 | \newcommand{\mat}[4]{ 4 | \left| \begin{array}{cc} 5 | #1 & #2 \\ 6 | #3 & #4 7 | \end{array} \right| 8 | } 9 | 10 | \begin{document} 11 | 12 | $ i, j, k \in \mathbb{R}^3 $ 13 | 14 | 15 | \begin{align*} 16 | \vec{a} \times \vec{b} & = \vec{c}\\ 17 | & = 18 | \left| \begin{array}{ccc} 19 | \vec{i} & \vec{j} & \vec{k} \\ 20 | a_0 & a_1 & a_2 \\ 21 | b_0 & b_1 & b_2 22 | \end{array} \right| \\ 23 | & = \vec{i} \mat{a_1}{a_2}{b_1}{b_2} - \vec{j} \mat{a_0}{a_2}{b_0}{b_2} + \vec{k}\mat{a_0}{a_1}{b_0}{b_1} \\ 24 | & = 25 | \left[ \begin{array}{ccc} 26 | a_1 b_2 - a_2 b_1 \\ 27 | a_2 b_0 - a_0 b_2 \\ 28 | a_0 b_1 - a_1 b_0 29 | \end{array} \right] 30 | \end{align*} 31 | 32 | \end{document} 33 | -------------------------------------------------------------------------------- /notes/scripts/euclid.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/python 2 | 3 | # Greatest Common Divisor 4 | # Computes the Greatest Common Divisor between two integers using Euclid's 5 | # Algorithm. If the innards are wrapped in more efficient methods, then it'd 6 | # likely work quickly for very large integers! 7 | def gcd (a, b): 8 | # Let us compute the gcd of negative numbers 9 | if a < 0: a = -a 10 | if b < 0: b = -b 11 | # Ensure that b is the larger number 12 | if b < a: (a, b) = (b, a) 13 | 14 | # Do the first iteration by hand 15 | r = b % a 16 | 17 | while r != 0: 18 | # Note: b = a * m + r 19 | b = a 20 | a = r 21 | m = b // a 22 | r = b % a 23 | 24 | return a 25 | 26 | # Least Common Multiple 27 | # Computes the least common multiple between two integers. This method is not 28 | # good for very large integers 29 | def lcm (a, b): return (a*b)/gcd(a, b) 30 | 31 | print gcd(72, 22) 32 | print lcm(72, 22) 33 | 34 | -------------------------------------------------------------------------------- /code/src/min.js: -------------------------------------------------------------------------------- 1 | function MIN (n) { 2 | 3 | if (n < 1) 4 | return undefined; 5 | 6 | 7 | var min_val = n*(n+1); 8 | 9 | for (var i=0; i < 2000; i++) { 10 | var total = 0; 11 | var vals = random_array(n); 12 | for (var k=0; k < n; k++) { 13 | total += vals[k] * (k+1); 14 | } 15 | 16 | if (total >= 0 && total < min_val) { 17 | min_val = total; 18 | } 19 | } 20 | 21 | /** 22 | * [random_array description] 23 | * @param {[type]} num [description] 24 | * @return {[type]} [description] 25 | */ 26 | function random_array (num) { 27 | var vals = new Array(num); 28 | for (var i=0; i < vals.length; i++) 29 | vals[i] = Math.random() > 0.5 ? +1 : -1; 30 | return vals; 31 | } 32 | 33 | return min_val; 34 | } 35 | 36 | 37 | 38 | // var sum = 0; 39 | 40 | // for (var i=1; i < 10; i++) { 41 | // sum += MIN(i); 42 | // } 43 | 44 | // console.log(MIN(10)); 45 | // console.log(sum); 46 | 47 | module.exports = MIN; -------------------------------------------------------------------------------- /code/src/polynomial.js: -------------------------------------------------------------------------------- 1 | var _ = require('underscore'); 2 | var factor = require('./factor.js'); 3 | 4 | function p (x) { 5 | // return Math.pow(x, 3) + x; 6 | return Math.pow(h(x), 1/6.0); 7 | } 8 | 9 | function deriv (f, delta) { 10 | delta = delta || 0.00000001; 11 | return function (x) { 12 | return (f(x+delta) - f(x-delta))/delta/2.0; 13 | }; 14 | } 15 | 16 | function solve (F, x0) { 17 | 18 | var error; 19 | 20 | x0 = x0 || 0; 21 | 22 | f = deriv(F); 23 | 24 | do { 25 | x1 = x0 - F(x0)/f(x0); 26 | x0 = x1; 27 | error = F(x1); 28 | } while (error < 0.000000001); 29 | 30 | return x1; 31 | } 32 | 33 | // var guess = solve(p, 10000000.0); 34 | 35 | // console.log(Math.pow(guess, 6)); 36 | 37 | function g (y) { 38 | return Math.pow(y, 6); 39 | } 40 | 41 | function h (x) { 42 | return Math.pow(x, 6) + 8 * Math.pow(x, 4) - 6 * Math.pow(x, 2) + 8;; 43 | } 44 | 45 | function works (x, y) { 46 | return g(y) === h(x); 47 | } 48 | 49 | 50 | var BEGIN = -10; 51 | var END = +100; 52 | 53 | for (var n=BEGIN; n <= END; n++) { 54 | 55 | var x = Math.pow(h(n), 1/6.0); 56 | 57 | if (x) 58 | console.log(n, x); 59 | } 60 | 61 | 62 | 63 | -------------------------------------------------------------------------------- /notes/sheets/types-of-integrals.tex: -------------------------------------------------------------------------------- 1 | \documentclass[11pt,twoside,a4paper]{article} 2 | 3 | \newcommand{\curl}{ 4 | \text{curl}\ 5 | } 6 | 7 | % \newcommand{\vec}[1]{ 8 | % \mathbf{\1} 9 | % } 10 | 11 | \usepackage{mathtools} 12 | 13 | \title{Multivariate Integrals} 14 | \author{Matt Owen} 15 | 16 | \begin{document} 17 | 18 | \maketitle 19 | \tableofcontents 20 | \cleardoublepage 21 | 22 | \section{Definitions} 23 | \label{definitions} 24 | 25 | \subsection{Fundamental Theorem of Calc} 26 | \label{fun-thm-single-calc} 27 | 28 | $\int_a^b F'(x) = F(b) - F(a)$ 29 | 30 | \subsection{Fundamental Theorem for Line Integrals} 31 | \label{fun-thm-line-integral} 32 | 33 | Let $C = \{ \vec{r}(t) | a \leq t \leq b \}$ 34 | $$ \int_C \nabla f \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) $$ 35 | 36 | \subsection{Green's Theorem} 37 | \label{greens-theorem} 38 | 39 | $$ \iint_D{\frac{\partial\mathbf{Q}}{\partial x} - \frac{\partial\mathbf{P}}{\partial y} } = \int_C \mathbf{P}dx + \mathbf{Q}dy$$ 40 | 41 | \subsection{Stokes' Theorem} 42 | 43 | $$ \iint_S{\curl\mathbf{F}\cdot d\mathbf{S}} = \int_C \mathbf{F} \cdot d\mathbf{r} $$ 44 | 45 | 46 | \subsection{Divergence Theorem} 47 | 48 | Note: This is a higher-dimensional Green`s Theorem, relating parametric surfaces to curl. 49 | 50 | 51 | 52 | \end{document} -------------------------------------------------------------------------------- /code/src/improve.js: -------------------------------------------------------------------------------- 1 | var _ = require('underscore'); 2 | 3 | var MIN = require('./min.js') 4 | 5 | function improve (n, vec) { 6 | var sub_array = [0]; 7 | 8 | if (typeof vec === 'undefined') 9 | vec = number_array(n); 10 | 11 | var current_total = 0; 12 | var iterations = 0; 13 | 14 | 15 | do { 16 | current_total = vec.reduce(function (a, b) { return a + b; }); 17 | current_total2 = n*(n+1)/2.0 - sub_array.reduce(function (a, b) { return a+b; }); 18 | // console.log(current_total2, current_total); 19 | iterations++; 20 | } while (iterate(vec)); 21 | 22 | // console.log(delta_list); 23 | // console.log('best =', best_delta); 24 | 25 | // console.log('iterations =', iterations); 26 | return current_total; 27 | 28 | /** 29 | * [iterate description] 30 | * @param {[type]} vec [description] 31 | * @return {[type]} [description] 32 | */ 33 | function iterate (vec) { 34 | var min_index = -1; 35 | var min_delta = Infinity; 36 | var max_delta = -Infinity; 37 | 38 | // 1 ... n 39 | // 40 | 41 | _.each(vec, function (val, index) { 42 | var next_val = current_total - 2 * val; 43 | var delta = 2*val; 44 | 45 | if (next_val >= 0 && delta > max_delta) { 46 | min_index = index; 47 | max_delta = delta; 48 | } 49 | }); 50 | 51 | 52 | 53 | // console.log('max delta >', max_delta); 54 | vec[min_index] *= -1; 55 | sub_array.unshift(max_delta); 56 | 57 | // console.log(sub_array[0]/2, vec[min_index]); 58 | 59 | // console.log('best delta =', min_delta); 60 | 61 | return max_delta > 0; 62 | 63 | } 64 | } 65 | 66 | function D (n) { 67 | return n % 2; 68 | } 69 | 70 | function number_array (n) { 71 | var arr = new Array(n); 72 | for (var i=0; i < arr.length; i++) { 73 | arr[i] = i+1; 74 | } 75 | return arr; 76 | } 77 | 78 | console.log(summa); 79 | // console.log(improve(4)); 80 | // console.log(MIN(702)); -------------------------------------------------------------------------------- /notes/workbook-1/ch-06-rings.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | 3 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 4 | \include{commands} 5 | 6 | \begin{document} 7 | 8 | \section{Rings} 9 | 10 | % RING HOMOMORPHISMS 11 | % ... 12 | % ... 13 | \subsection{Ring Homomorphisms} 14 | 15 | $\phi$ is a \emph{ring homomorphism}, if: 16 | \begin{itemize} 17 | \item $\phi(a+b)=\phi(a)\oplus\phi(b)$ 18 | \item $\phi(a \cdot b)=\phi(a)\otimes\phi(b)$ 19 | \end{itemize} 20 | 21 | % FEATURES OF RING HOMOMORPHISMS 22 | % ... 23 | % ... 24 | \subsection{Features of Ring Homomorphisms} 25 | 26 | 27 | \begin{enumerate} 28 | \item 29 | The \textbf{kernel} of a ring homomorphism is the set:\\ 30 | $\ker \phi \defeq \set{a\in\mathbf{R}}{\phi(a)=0'}$ \\ 31 | Note that $\ker\phi: \mathbf{R} \mapsto \mathbf{R'}$ is a subring of 32 | $\mathbf{R}$ 33 | \item 34 | The image of $\mathbf{R}$, $\phi(\mathbf(R)$ is a subring of $\mathbf{R'}$ 35 | \item 36 | The image of $0_+\in\mathbf{R}$ is $0_+'\in\mathbf{R'}$.\\ 37 | Note that this means: $\phi(-a)=-\phi(a)$ 38 | 39 | \end{enumerate} 40 | 41 | 42 | \subsection{Extensions of Rings} 43 | 44 | {\noindent}Rings are sets $\mathbf{R}$ where: 45 | \begin{itemize} 46 | \item $(\mathbf{R}, +)$ is an Abelian group 47 | \item $(\mathbf{R}, \cdot)$ is a semigroup (multiplication is associative) 48 | \item 49 | The \emph{distributive law} holds: 50 | $a \cdot (b + c) = (a + b) \cdot c = a \cdot c + a \cdot b$ 51 | \end{itemize} 52 | 53 | {\noindent}Types of rings, and elemts: 54 | \begin{itemize} 55 | \item 56 | A {\bf Ring with Unity} is a a ring where $(R, \cdot)$ is a 57 | monoid (closed, associative, identity) 58 | \item 59 | A {\bf Unit} is a \emph{ring element} that has a multiplicative 60 | inverse. 61 | \item 62 | A {\bf Division Ring} is a ring where every nonzero element has a 63 | \emph{multiplicative inverse}. 64 | \item 65 | An {\bf Integral Domain} is a commutative ring, where: 66 | $a \cdot b = 0 \iff a=0 \lor b=0$ 67 | \item 68 | A {\bf Field} is a commutative \emph{division ring}. 69 | \end{itemize} 70 | 71 | \end{document} 72 | -------------------------------------------------------------------------------- /notes/sheets/study-guide.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,twoside,a4paper]{article} 2 | 3 | \title{What to Study} 4 | 5 | \begin{document} 6 | 7 | \section*{vv/e} 8 | 9 | List of common topics in the GRE; topics that are easy to convert into problems. 10 | 11 | \section*{List} 12 | 13 | \small 14 | 15 | % \tableofcontents 16 | % \cleardoublepage 17 | 18 | \begin{description} 19 | \item[Calculus] ... 20 | \begin{itemize} 21 | \item u-substitution 22 | \item convergence theorems 23 | \item limit tests 24 | \item l'hopital rule 25 | \item power series 26 | \item partial-fraction decomposition 27 | \item fundamental theorem of calculus 28 | \item mean-value theorem, Cauchy's mean-value theorem 29 | \item riemann and lesbesgue integration 30 | \item improper integral 31 | \item integration identities 32 | \item inverse function 33 | \end{itemize} 34 | 35 | \item[Multivariate Calculus] ... 36 | \begin{itemize} 37 | \item surface of revolutions 38 | \item tangent planes 39 | \item chain-rule 40 | \item lagrange multipliers 41 | \item integration: exact equations, path integrals, arc-tangent integrals, green's theorem 42 | \item vector algebra: cross-products, norms, 43 | \end{itemize} 44 | 45 | \item[Differential Equations] ... 46 | \begin{itemize} 47 | \item nonlinear first-order 48 | \item linear n-order 49 | \item substitution techniques 50 | \item exact equation 51 | \item misc. techniques 52 | \end{itemize} 53 | 54 | \item[Linear Algebra] ... 55 | \begin{itemize} 56 | \item computing determinant 57 | \item rank-nullity theorem 58 | \item inversion 59 | \item subspaces 60 | \item eigenvalues 61 | \item change-of-base 62 | \end{itemize} 63 | 64 | \pagebreak 65 | 66 | \item[Trigonometry] ... 67 | \begin{itemize} 68 | \item identities: double-angle, half-angle 69 | \item integration: identities, u-substitution 70 | \item complex-number operations 71 | \end{itemize} 72 | 73 | \item[Complex Analysis] ... 74 | \begin{itemize} 75 | \item lambert series 76 | \item Residual Theorem 77 | \item logarithms, explonents, trig operators 78 | \end{itemize} 79 | 80 | \item[Abstract Algebra] ... 81 | \begin{itemize} 82 | \item definition of group, ring, field 83 | \item size of groups, rings, fields: number theory 84 | \item order of a group 85 | \item cyclic groups and generators 86 | \item diophantine equations 87 | \item homomorphism 88 | \end{itemize} 89 | 90 | \item[Topology] ... 91 | \begin{itemize} 92 | \item continuous functions 93 | \item inverse-functions 94 | \item homomorphism 95 | \item hausdorff, connected 96 | \item theory 97 | \item metric spaces 98 | \end{itemize} 99 | \end{description} 100 | 101 | 102 | 103 | \end{document} -------------------------------------------------------------------------------- /notes/rudin/ch-01.tex: -------------------------------------------------------------------------------- 1 | % Review + Learning Latex 2 | % 3 | % 4 | 5 | \documentclass{article} 6 | 7 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 8 | 9 | \newcommand{\set}[2]{ 10 | \{ #1 \mid #2 \} 11 | } 12 | \newcommand{\qeq}{\stackrel{?}{=}} 13 | \newcommand{\defeq}{\stackrel{\text{def}}{=}} 14 | 15 | 16 | \newcommand{\DEFINITION}[1]{ 17 | \label{def-#1} 18 | {\noindent \bf Definition #1} 19 | } 20 | 21 | \newcommand{\THEOREM}[1]{ 22 | \label{theorem-#1} 23 | {\noindent \bf #1 Theorem} 24 | } 25 | 26 | \begin{document} 27 | 28 | Chapter 1. 29 | 30 | \section{Introduction} 31 | 32 | \subsection{Rational Numbers} 33 | 34 | $ 35 | \forall p \in \mathbb{Q}, \exists m, n \in \mathbb{Z}, \mid p = \frac{m}{n} 36 | $ 37 | 38 | \subsection{Irrational Numbers} 39 | 40 | $ 41 | \exists p \in \mathbb{R}, s.t. \forall m, n \in \mathbb{Z}, p \neq \frac{m}{n} 42 | $ 43 | 44 | \subsubsection{Example} 45 | 46 | Suppose: 47 | \begin{align*} 48 | p \in \mathbb{Q} \\ 49 | m,n \in \mathbb{Z}, st. p =j \frac{m}{n} \\ 50 | gcd(m, n) = 1 \\ 51 | \end{align*} 52 | 53 | \begin{align*} 54 | p^2 & < 2 \\ 55 | \frac{m^2}{n^2} & < 2 \\ 56 | m^2 & < 2 n^2 57 | \end{align*} 58 | 59 | Then: 60 | \begin{align*} 61 | q & = p - \frac{p^2-2}{p+2} \\ 62 | q & = \frac{2p-2}{p+2} \\ 63 | q^2 & = \frac{4p^2 - 8p - 4}{(p+2)^2} \\ 64 | q^2 - 2 & = \frac{(4p^2 - 8p - 4) - 2 (p+2)^2}{(p+2)^2} \\ 65 | q^2 - 2 & = \frac{(4p^2 - 8p + 4) - (2p^2 + 8p + 8)}{(p+2)^2} \\ 66 | q^2 - 2 & = \frac{2p^2 - 16p - 4}{(p+2)^2} 67 | \end{align*} 68 | 69 | \section{Ordered Sets} 70 | 71 | {\bf Definition.} Let S be a set. An \emph{order} on S is a relation, denoted by <, with the following two properties: 72 | 73 | \begin{equation} 74 | If x,y \in S, \\ 75 | Then x < y, x = y, or y < x 76 | \end{equation} 77 | 78 | \begin{align*} 79 | If: & x, y, z \in S \\ 80 | And: & x < y, y < z \\ 81 | Then: & x < z 82 | \end{align*} 83 | 84 | {\bf Definition.} An \emph{ordered set} is a set S in which an order is defined. 85 | 86 | {\bf Definition.} 87 | Suppose S is an ordered set, and $ E \subset S $. 88 | If there $ \exists \beta \in S, st. \forall x \in E, x < \beta $, then E is \emph{bounded above} by $\beta$. 89 | 90 | {\bf Definition.} 91 | Suppose $\mathbf{S}$ is an ordered set, $\mathbf{E} \subset \mathbf{S}$, and $\mathbf{E}$ is bounded above. Suppose there exists an $\alpha \in S$ with the following properties: 92 | \begin{gather*} 93 | \forall x \in \mathbf{E}, \quad x < \alpha \\ 94 | If \quad \gamma < \alpha, \quad then \quad \gamma is not an upper bound of \mathbf{E} 95 | \end{gather*} 96 | 97 | Then: 98 | $$ \alpha = \sup \mathbf{E} $$ 99 | 100 | \subsection{Examples} 101 | (a) ***Ignored \\ 102 | (b) If $\alpha = \sup \mathbf{E}$ exists, then $\alpha$ may or may not be in $\mathbf{E}$. \\ 103 | (c) Let $\mathbf{E}$ consist of all numbers $\frac{1}{n}$, when n = 1\dots. Then, $\sup\mathbf{E} = 1$ and $\inf\mathbf{E} = 0$. 104 | 105 | {\bf Definition.} An ordered set $\mathbf{S}$ is said to have the \emph{least-upperbound property} if the following is true: \\ 106 | 107 | \begin{enumerate} 108 | \item 109 | If $\mathbf{E} \subset \mathbf{S}$, $\mathbf{E}$ is not empty, and $\mathbf{E}$ is bounded above, 110 | then $\sup\mathbf{E} \in \mathbf{S}$ 111 | \end{enumerate} 112 | 113 | 114 | {\bf Given.} 115 | $$ \mathbf{S} \text{is an ordered set with a least upper bound} $$ 116 | $$ \alpha := x $$ 117 | $$ \mathbf{B} \subset \mathbf{S} $$ 118 | $$ \mathbf{B} \neq \emptyset $$ 119 | $$ \mathbf{L} := \set{y \in \mathbf{S}}{y \leq x \quad \forall x \in \mathbf{B}} $$ 120 | 121 | {\bf Prove.} 122 | $$ \alpha = \sup\mathbf{L} $$ 123 | $$ \alpha = \inf\mathbf{B} $$ 124 | 125 | {\bf Proof.} 126 | $$ \text{Since $\mathbf{L}$ is bounded below, } \mathbf{L} \neq \emptyset $$ 127 | $$ \text{Every $\mathbf{B}$ is an upper-bound for $\mathbf{L}$} $$ 128 | $$ \text{$\mathbf{L}$ is bounded above, because $\mathbf{B}$ is nonempty} $$ 129 | $$ \sup\mathbf{L} \in \mathbf{S} \text{, because $\mathbf{S}$ has a least upperbound $\beta$, which is greater than everything in $\mathbf{S}$ and $\mathbf{L}$} $$ 130 | $$ \text{If } \gamma < \alpha, \gamma \text{ is not an upper-bound of } \mathbf{L} \text{, and } \gamma \notin \mathbf{B} $$ 131 | $$ \text{Thus } \alpha \leq x, \quad \forall x \in \mathbf{B} $$ 132 | $$ \text{Thus } \alpha \in \mathbf{L} $$ 133 | $$ \text{But } \alpha \notin \mathbf{B} $$ 134 | $$ \text{Then this kinda becomes too confusing for me to follow on computer, because $\gamma$ comes out of nowhere} $$ 135 | 136 | \section{Fields} 137 | 138 | {\bf Definition.} 139 | A \emph{field} is a set $\mathbf{F}$ with two operations, called \emph{addition} and \emph{multipleication}, which satisfy the field axioms: 140 | \begin{description} 141 | \item[\emph{closure addition}:] 142 | If $x \in \mathbf{F}$, then their sum $x + y \in \mathbf{BF}$ 143 | \item[\emph{commutative addition}:] 144 | $x + y = y + x$ 145 | \item[\emph{assosciative addition}:] 146 | $(x + y) + z = x + (y + z)$ 147 | \item[\emph{identity addition}:] 148 | $\mathbf{F}$ contains an element $0$ s.t. $0 + x = x, \quad \forall x\in F$ 149 | \item[\emph{Inverse addition}:] 150 | $\forall x \in \mathbf{F}, \quad \exists -x \in \mathbf{F}, \text{ s.t. } x + (-x) = 0$ 151 | \item[\emph{Closure (multiplication)}:] 152 | If $x \in \mathbf{F}$, then their product $x \cdot y \in \mathbf{BF}$ 153 | \item[\emph{Commutative (multiplication)}:] 154 | $x \cdot y = y \cdot x$ 155 | \item[\emph{Assosciative (multiplication)}:] 156 | $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ 157 | \item[\emph{Identity (multiplication)}:] 158 | $\mathbf{F}$ contains an element $1$ s.t. $1 \cdot x = x, \quad \forall x\in F$ 159 | \item[\emph{Inverse (multiplication)}:] 160 | $\forall x \in \mathbf{F\setminus\{0\}}, \quad \exists x^{-1} \in \mathbf{F}, \text{ s.t. } x \cdot (x^{-1}) = 1$ 161 | \item[\emph{Distribution}:] 162 | $x \cdot (y + z) = x \cdot y + x \cdot z, \quad \forall x, y, z \in \mathbf{F}$ 163 | 164 | \end{description} 165 | 166 | 167 | \subsection{Proposition} 168 | 169 | {\noindent\bf Proposition} 170 | 171 | \begin{enumerate} 172 | \item If $x + y = x + z$, then $y = z$ 173 | \item If $x + y = x$, then $y = 0$ 174 | \item If $x + y = 0$, then $y = -x$ 175 | \item $-(-x) = x$ 176 | \end{enumerate} 177 | 178 | {\bf Proof.} 179 | 180 | (a) 181 | \begin{align*} 182 | y & = 0 + y \\ 183 | & = (x + (-x)) + y\\ 184 | & = -x + (x + y) \\ 185 | & = -x + (x + z) \\ 186 | & = (-x + x) + z \\ 187 | & = 0 + z \\ 188 | & = z 189 | \end{align*} 190 | (b) 191 | \begin{align*} 192 | x + y & = x \\ 193 | x + y & = x + 0 \\ 194 | \text{Thus } y = 0 \text{ by the previous proof} 195 | \end{align*} 196 | 197 | (c) 198 | \begin{align*} 199 | \text{Skipped.} 200 | \end{align*} 201 | 202 | (d) 203 | \begin{align*} 204 | -(-x) & \qeq \dots \\ 205 | y & := -x \\ 206 | -y & = 0 - y \\ 207 | -y & = (x - x) - y \\ 208 | & = (x + y) - y \\ 209 | & = x + (y - y) \\ 210 | & = x + 0 \\ 211 | & = x \\ 212 | -(-x) & = x 213 | \end{align*} 214 | 215 | 216 | \pagebreak 217 | 218 | {\noindent\bf Definition.} 219 | An \emph{Ordered Field} is a \emph{Field} $\mathbf{F}$, which is also an \emph{ordered set}, such that: 220 | \begin{itemize} 221 | \item $x + y < x + z$, if $x, y, z \in \mathbf{F}$ and $y < z$ 222 | \item $x \cdot y > 0$ if $x, y \in \mathbf{F}$, $x > 0$, and $y > 0$ 223 | \end{itemize} 224 | 225 | 226 | \subsection{The Real Field} 227 | 228 | {\noindent\bf Theorem.} 229 | There exists an ordered field $\mathbf{R}$ which has the least upperbound property. 230 | $\mathbb{Q} \subset \mathbb{R}$ 231 | 232 | {\noindent\bf Theorem.} 233 | \begin{itemize} 234 | \item If $x, y \in \mathbb{R}$ and $x > 0$, then $\exists n \in \mathbb{Z}$ st. $nx > y$ 235 | \item If $x, y \in \mathbb{R}$ and $x < y$, then $\exists p \in \mathbb{Q}$ st. $x < p < y$ 236 | \end{itemize} 237 | 238 | {\noindent\bf Proof.} 239 | (a) 240 | $$ A := \set{nx}{n \in \mathbb{Z}} $$ 241 | $$ \text{Assume to the contrary that } \forall n \in \mathbb{Z}, nx \le y $$ 242 | $$ \text{Then } y \text{ is an upperbound of } \mathbf{A} $$ 243 | $$ \text{So define } \alpha := \sup A $$ 244 | $$ \alpha - x < \alpha \quad\because x > 0 $$ 245 | $$ \exists m \in \mathbb{Z} \text{, st. } \alpha - x < mx \quad\because a - x \text{ is not an upperbound of } \mathbf{A} $$ 246 | $$ \text{But then } \alpha < (m+1)x \in A \text{, which impossible because $\alpha$ is an upperbound for $\mathbf{A}$}$$ 247 | $$ \therefore \text{(a) is true, aka the contrary is absurd} $$ 248 | 249 | (b) 250 | \begin{align*} 251 | \text{\bf Given.} \\ 252 | & x, y \in \mathbb{R} \land x < y \\ 253 | \text{\bf Show.} \\ 254 | & \exists p \in \mathbb{Q}: x < p < y \\ 255 | \text{\bf Proof.} \\ 256 | & y - x > 0 & \because x < y \\ 257 | & \exists n \in \mathbb{Z}: n \cdot (y-x) > 1 & \because proof (a) \\ 258 | & \text{Let $m_1$ be the integer st. } nx < m_1 \\ 259 | & \text{Let $m_2$ be the integer st. } -nx < m_2 \\ 260 | & -m_2 < nx < m_1 \\ 261 | & \exists m \in \mathbb{Z}: -m_2 \le m \le m_1 & \because \mathbb{Z}\text{is ordered} \\ 262 | & m - 1 \le nx < m \\ 263 | & nx < m \le nx + 1 < ny & (!!!) \\ 264 | & x < \frac{m}{n} < y & \because n > 0 \\ 265 | & \text{Let } p = \frac{m}{n} \\ 266 | & x < p < y \\ 267 | & \qed... ask about me 268 | \end{align*} 269 | 270 | 271 | 272 | \subsection{} 273 | 274 | 275 | % Theorem 276 | % 277 | % 278 | {\bf 1.21 Theorem} 279 | 280 | \begin{align*} 281 | \text{\bf Prove.} \\ 282 | & \forall x > 0 \in \mathbb{R}, \forall n > 0 \in \mathbb{Z}, \exists! y > 0 : y^n = x \\ 283 | \text{\bf Note.} \\ 284 | & \text{There is at most one $y > 0 : y^n = x$, because } \\ 285 | & 0 < y_1 < y_2 \implies y^n < y^m \\ 286 | \text{\bf Outline.} \\ 287 | & \text{1. Okay} \\ 288 | & \text{2. Okay} \\ 289 | \text{\bf Proof.} \\ 290 | & \mathbf{A} := \set{y \in \mathbb{R}}{ y > 0 \land y^n < x } \\ 291 | & \text{Let } \alpha = \sup\mathbf{A} & \because \text{$\mathbf{A}$ is bound by $y$, there is a supremum} \\ 292 | & \text{Assume to the contrary that $y^n < x$} & \text{Note: }\\ 293 | \end{align*} 294 | 295 | Fuck 296 | 297 | 298 | 299 | \pagebreak 300 | 301 | \section{Extended Real Number Line} 302 | 303 | {\noindent \bf Definition 1.23}\label{def-1-23} The extended real reals $\defeq \mathbb{R} \cup { -\infty, +\infty } $ 304 | 305 | {\noindent} $ \forall x \in \mathbb{R}, -\infty < x < \infty $ 306 | 307 | Note that the extended-reals do not form a field, however the following is "customary": 308 | 309 | \begin{itemize} 310 | \item 311 | \begin{itemize} 312 | \item $ x + \infty = +\infty $ 313 | \item $ x - \infty = -\infty $ 314 | \item $ \frac{x}{+\infty} = \frac{x}{-\infty} = 0 $ 315 | \end{itemize} 316 | \item 317 | \begin{itemize} 318 | \item If $x > 0$, then $x \cdot (+\infty) = +\infty$ 319 | \item If $x > 0$, then $x \cdot (-\infty) = -\infty$ 320 | \end{itemize} 321 | \item 322 | \begin{itemize} 323 | \item If $x < 0$, then $x \cdot (+\infty) = -\infty$ 324 | \item If $x < 0$, then $x \cdot (-\infty) = +\infty$ 325 | \end{itemize} 326 | \end{itemize} 327 | 328 | \section{The Complex Field $\mathbb{C}$} 329 | 330 | \DEFINITION{1.24} A \emph{complex} number is an ordered pair $(a, b)$ with $a, b \in \mathbb{R}$ 331 | 332 | We define, $ \forall x, y \in \mathbb{C} $: 333 | \begin{itemize} 334 | \item $ x + y = (a + c, b + d) $ 335 | \item $ x y = (ac - bd, ad + bc) $ 336 | \item $ 0 = (0, 0) $ 337 | \item $ I = (1, 0) $ 338 | \end{itemize} 339 | 340 | \THEOREM{1.25} $\mathbb{C}$ is a field 341 | 342 | \begin{align*} 343 | {\bf Prove.} \\ 344 | & \mathbb{C} \text{is a field} \\ 345 | {\bf Given.} \\ 346 | & x = (a, b), y = (c, d), z = (e, f) 347 | {\bf Proof.} \\ 348 | & \text{(A1)} x + 0 = (a, b) + (0, 0) = (a + 0, b + 0) = (a, b) \\ 349 | & \text{(A2)} x + y = (a, b) + (c, d) = (a + c, b + d) = (c + a, d + b) = y + x \\ 350 | & \text{(A3)} (x + y) + z = (a + c, b + d) + (e, f) = (a + c + e, b + d + f) = (a, b) + (c + e, d + f) = x + (y+z) 351 | & \text{\emph{Skipping a few...}} 352 | & \text{M2} xy = (ac - bd, ad + bc) = (ca - db, da + cd) = yx 353 | \end{align*} 354 | 355 | \pagebreak 356 | 357 | \DEFINITION{1.30} If $a, b \in \mathbb{R}$ and $z = a + bi$, then the complex number $\overline{z} = a - bi$ 358 | 359 | \THEOREM{1.31} 360 | \begin{itemize} 361 | \item $ \overline{z + w} = \overline{z} + \overline{w} $ 362 | \item $ \overline{zw} = \overline{z} \cdot \overline{w} $ 363 | \item $ z + \overline{z} = 2 \text{Re(z)}, z - \overline{z} = 2 i \text{Im(z)} $ 364 | \item $ z \overline{z} \in \mathbb{R}^{+} \text{unless} z = 0 $ 365 | \end{itemize} 366 | 367 | \DEFINITION{1.32} $ |z| = (z\overline{z})^{\frac{1}{2}} $ 368 | 369 | \THEOREM{1.33} $z, w \in \mathbb{C} $ 370 | \begin{itemize} 371 | \item $ |z| \geq 0; |z| = 0 \iff z = 0 $ 372 | \item $ |\overline{z}| = |z| $ 373 | \item $ |zw| = |z||w| $ 374 | \item $ |\text{Re(z)}| \leq |z| $ 375 | \item $ |z + w| \leq |z| + |w| $ 376 | \end{itemize} 377 | 378 | Note that 1, 2, 3, 4 are \emph{really} easy. So here's 5: 379 | 380 | \begin{align*} 381 | {\bf Prove.} \\ 382 | & |z + w| \leq |z| + |w| \\ 383 | {\bf Given.} \\ 384 | & z, w \in \mathbb{C} \\ 385 | {\bf Proof.} \\ 386 | & |z+w|^2 & = (z+w)(\overline{z+w}) = z\overline{z} + z\overline{w} + \overline{z}w + w\overline{w} \\ 387 | & & = |z|^2 + 2\text{Re($z\overline{w}$)} + |w|^2 & \because |z|^2 = z\overline{z} \land \overline{z\overline{w}} = \overline{z}w \\ 388 | & & \leq |z|^2 + 2\text{Re($z$)} + |w|^2 \\ 389 | & & = |z|^2 2 |z| |w| + |w|^2 = (|z| + |w|)^2 \\ 390 | & & \therefore |z+w| \leq |z| + |w| \because |z| + |w| \in \mathbb{R} > 0 391 | \end{align*} 392 | 393 | \THEOREM{1.35} If $ a_1, ..., a_n and b_1, ..., b_n \in \mathbb{C} $, then: \\ 394 | {\indent} $ |\sum_{j=1}^n a_j\overline{b_j}|^2 \leq \sum_{j=1}^n |a_j|^2 \sum_{j=1}^n |b_j|^2$ 395 | 396 | \pagebreak 397 | 398 | \begin{align*} 399 | {\bf Proof.} \\ 400 | & ... \\ 401 | {\bf Given.} \\ 402 | & A = \sum{a_j}^2 \\ 403 | & B = \sum{b_j}^2 \\ 404 | & C = \sum{a_j\overline{b_j}} \\ 405 | {\bf Prove.} \\ 406 | \sum{|B a_j - C b_j|}^2 & = \sum{(B a_j - C b_j)(B\overline{a_j} - \overline{C b_j})} \\ 407 | & = B^2 \sum{|a_j}^2 - B \overline{C} \sum{a_j\overline{b_j}} - B C \sum{\overline{a_j} b_j} + |C|^2 \sum{|b_j}^2 \\ 408 | & = B^2 - B |C|^2 \\ 409 | & = B (AB - |C|^2) 410 | & B (AB - |C|^2) > 0 & \because B>0 \\ 411 | & AB - |C|^2 > 0 & \because B>0 \\ 412 | & \therefore AB > |C|^2 \\ 413 | & |C|^2 = |\sum{a_j\overline{b_j}}|^2 \leq \sum{|a_j|}^2\sum{|b_j|}^2 - AB \qed \\ 414 | \end{align*} 415 | 416 | \pagebreak 417 | \section{Euclidean Spaces} 418 | 419 | \DEFINITION{1.36} $\mathbb{R}^k$ is the set of all ordered tuples, such that: 420 | $ \vec{x} = (x_1, x_2, \dots, x_k) : \forall k > 0, x_k \in \mathbb{R} $ 421 | 422 | Vectors form a field! With $+$ and $*$ 423 | 424 | In addition to field operations, vector-spaces contain \emph{scalar multiplication}, \emph{inner product}, and a \emph{norm}: 425 | 426 | \begin{itemize} 427 | \item $ \alpha \vec{x} = (\alpha x_1, \dots, \alpha x_k) $ 428 | \item $ \vec{x} \cdot \vec{y} = \sum_{i=1}^k{x_i y_i}$ 429 | \item $ |\vec{x}| = (\vec{x} \cdot \vec{x})^\frac{1}{2} $ 430 | \end{itemize} 431 | 432 | {\bf Proof.} Schwartz Inequality \\ 433 | 434 | $|x+y| \leq |x| + |y|, \forall x, y, z \in \mathbb{R}^k$ 435 | 436 | \begin{align*} 437 | |x + y|^2 & = (x+y) \cdot (x+y) \\ 438 | & = x \cdot x + 2 x \cdot y + y \cdot y \\ 439 | & \leq |x|^2 + 2|x||y| + |y|^2 \because x \cdot y \leq |x||y| \\ 440 | & = (|x| + |y|)^2 441 | \end{align*} 442 | 443 | \end{document} 444 | -------------------------------------------------------------------------------- /notes/workbook-1/ch-06.tex: -------------------------------------------------------------------------------- 1 | \documentclass{article} 2 | \usepackage{amssymb,amsmath,amsfonts,amsthm} 3 | \include{commands} 4 | 5 | \newcommand{\congr}[3]{ 6 | #1 \equiv #2\ (\textrm{mod}\ #3) 7 | } 8 | \begin{document} 9 | 10 | \section{Number Theory and Abstract Algebra} 11 | 12 | This takes up roughly 15\% of the test, but is partially involved in other 13 | questions. This is a common mix-in, to make a problem harder. 14 | 15 | \section{Divisibility} 16 | 17 | \subsection{Quick Rules for Factoring} 18 | 19 | \begin{itemize} 20 | \item by 2, iff the last digit is divisible by 2 21 | \item by 3, iff the sum of the digits is divisible by 3 22 | \item by 4, iff the last \emph{two} digitis is divisible by 4 23 | \item by 5, iff the last digit is 0 or 5 24 | \item by 8, iff the last \emph{three} digits are divisible by 8 25 | \item by 9, iff the sum of the digits is divisible by 9 26 | \end{itemize} 27 | 28 | \subsection{Division Algorithm} 29 | 30 | If $a, b \in \mathbb{Z}^+$, then $\exists q, r \in \mathbb{Z} : b = qa + r$ 31 | 32 | 33 | \subsection{Primes} 34 | 35 | $ \forall \in \mathbb{Z}^+, \exists \text{prime} p : k < p < 2k $ 36 | 37 | $ \sum{k=1}^n{\frac{1}{p_k}} $ divierges where $p_k$ is the k-th prime 38 | 39 | \subsection{GCD and LCD} 40 | 41 | Greatest-common-division and least-common-denominator come up in group size and 42 | and various algorithmic problems. 43 | 44 | \pagebreak 45 | \subsubsection{GCD} 46 | 47 | ... 48 | 49 | For any integers $a, b$, we can write: 50 | \begin{align*} 51 | a & = (p_1)^{a_1} (p_2)^{a_2} ... (p_k)^{a_k} \\ 52 | b & = (p_1)^{b_1} (p_2)^{b_2} ... (p_k)^{b_k} \\ 53 | m_i & \defeq min(a_i, b_i) \\ 54 | M_i & \defeq max(a_i, b_i) \\ 55 | gcd(a, b) & \defeq (p_1)^{m_1} (p_2)^{m_2} (\dots) (p_k)^{m_k} \\ 56 | lcM(a, b) & \defeq (p_1)^{M_1} (p_2)^{M_2} (\dots) (p_k)^{M_k} \\ 57 | \end{align*} 58 | 59 | As a result: 60 | $$ gcd(a, b) \cdot lcm(a, b) = a \cdot b $$ 61 | 62 | \subsubsection{Euclidean Algorithm} 63 | 64 | Algorithmically determine the greatest common divisor between two numbers. This 65 | divides the larger number by the smaller and continually checks whether it did 66 | it evenly. When $r = 0$ we know $ r | b_k, b_{k-1}, b_{k-2}, ..., b_0 $ and that 67 | it must be the largest such number that does it. 68 | 69 | \begin{verbatim} 70 | def gcd (a, b): 71 | if b < a: (a, b) = (b, a) 72 | 73 | r = b % a 74 | 75 | while r != 0: 76 | # Note: b = a * m + r 77 | b = a 78 | a = r 79 | m = b // a 80 | r = b % a 81 | 82 | return a 83 | \end{verbatim} 84 | 85 | \subsubsection{Diophantine Equation $ax +by = c$} 86 | 87 | The Simple Linear Diophantine Equations $ax + by = c$ has solutions of the form:\\ 88 | $ x = x_1 + \frac{b}{d} t $\\ 89 | $ y = y_1 - \frac{a}{d} t $ 90 | 91 | Finding $x_1$ and $y_1$ is the only real hard part. We know that we can \emph{always} find solutions of the form: \\ 92 | $ a x_0 + b y_0 = d $ where $d = gcd(a, b)$ 93 | 94 | After solving this equation, we multiply both sides of the equation by $\frac{c}{d}$ to get: 95 | $ a \frac{c}{d} x_0 + b \frac{c}{d} y_0 = d \frac{c}{d} $ 96 | 97 | Define: \\ 98 | $ x_1 = \frac{c}{d} x_0 $ \\ 99 | $ y_1 = \frac{c}{d} y_0 $ \\ 100 | 101 | Then plug in these values of $x_1$ and $y_1$ to attain solutions: \\ 102 | $ x = \frac{c}{d} x_0 + \frac{b}{d} t $ 103 | $ y = \frac{c}{d} y_0 - \frac{a}{d} t $ 104 | 105 | As an aside, note the negative in y's value. This seems to relate to the Euclidean algorithm. 106 | 107 | 108 | \pagebreak 109 | \subsection{Congruences} 110 | 111 | Rules for congruences: 112 | \begin{itemize} 113 | \item $\congr{a}{b}{n}$ and $\congr{b}{c}{n}$, then $\congr{a}{c}{n}$ 114 | 115 | \item 116 | $\congr{a}{b}{n}$, then $\forall c \in \mathbb{Z}$ 117 | \begin{itemize} 118 | \item $\congr{a \pm c}{b \pm c}{n}$ 119 | \item $\congr{a c}{b c}{n}$ 120 | \end{itemize} 121 | 122 | \item 123 | If $\congr{a_1}{b_1}{n}$ and $\congr{a_2}{b_2}{n}$, then: 124 | \begin{itemize} 125 | \item $\congr{a_1 \pm a_2}{b_1 \pm b_2}{n}$ 126 | \item $\congr{a_1 a_2}{b_1 b_2}{n}$ 127 | \end{itemize} 128 | 129 | \item 130 | $\forall\ 0 < c \in \mathbb{Z}$, the following are equivalent: 131 | \begin{itemize} 132 | \item $\congr{a}{b}{n}$ 133 | \item $\congr{a}{b + n}{cn}$ 134 | \item $\congr{a}{b + 2n}{cn}$ 135 | \item \dots 136 | \item $\congr{a}{b + (c-1)n}{cn}$ 137 | \end{itemize} 138 | 139 | \item 140 | If $\congr{ab}{ac}{n}$, then: 141 | \begin{itemize} 142 | \item $\congr{b}{c}{n}$ if $d\defeq$ gcd($a$, $n$) $= 1$ 143 | \item $\congr{b}{c}{\frac{n}{d}}$ if $d\defeq$ gcd($a$, $n$) $> 1$ 144 | \end{itemize} 145 | 146 | \item 147 | The linear congruence equation $\congr{ax}{b}{n}$ has a solution iff gcd($a$, $n$) $ = d | b$ and: 148 | \begin{itemize} 149 | \item if $d=1$, then the solution is unique mod $n$ 150 | \item if $d>1$, then the solution is unique mod $\frac{n}{d}$ 151 | \end{itemize} 152 | \end{itemize} 153 | 154 | \subsection{Congruence Equation $\congr{ax}{b}{n}$} 155 | 156 | Note that Linear Congruence Equations are \emph{equivalent} to 157 | Simple, Linear Diophantine Equations. 158 | 159 | Solving these requires applying rules 1-7 of the previous section. They are pretty arduous. 160 | 161 | \pagebreak 162 | \section{Abstract Algebra} 163 | 164 | For a group $\mathbf{G}$ the following must be true: 165 | \begin{description} 166 | \item[closure] 167 | $\forall g, h \in \mathbf{G}, g \cdot h \in \mathbf{G}$ 168 | \item[associate - semigroup] 169 | $\forall a, b, c \in \mathbf{G}, a \cdot (b \cdot c) = (a \cdot b) \cdot c$ 170 | \item[identity - monoid] 171 | $\exists e \in \mathbf{G} : \forall g \in \mathbf{G}, g \cdot e = e \cdot g = g$ 172 | \item[inverse - group] 173 | $\forall g \in \mathbf{G}, \exists h \in \mathbf{G} : g \cdot h = h \cdot g = e$ 174 | \end{description} 175 | 176 | Note that having closure is necessary. Adding associativity makes it a 177 | \emph{semigroup}. Adding an identity makes it a monoid. Adding an inverse, 178 | finally, makes a binary operator a \emph{\bf group}. 179 | 180 | \subsection{Cyclic Groups} 181 | 182 | Cyclic groups are defined by a generator, s.t.: \\ 183 | $\mathbf{G} = \gen{a} = \set{a^n}{n \in \mathbb{Z}}$ 184 | 185 | Furthermore, $\mathbb{Z}_n$ forms a group under modulo-addition. And the following is true:\\ 186 | $m$ is a generator iff $m$ is coprime with $n$, the order of $\mathbb{Z}_n$ 187 | 188 | \subsection{Subgroups} 189 | 190 | Let $(\mathbf{G},\cdot)$ be a group. If there exists $\mathbf{H} \subset 191 | \mathbf{G}$ s.t. $(H,\cdot)$ is also a group, then we call 192 | $(\mathbf{H},\cdot)$ a subgroup of $\mathbf{G}$. 193 | 194 | Notably, we define the center of a group to be: \\ 195 | $Z(\mathbf{G}) \defeq \set{z \in \mathbf{G}}{zg = gz \ \forall g \in \mathbf{G}}$ 196 | 197 | And know that this is \emph{always} a subgroup (not merely a subset). 198 | 199 | \pagebreak 200 | \subsection{Cyclic Subgroups} 201 | 202 | Let $a \in \mathbf{G}$. With $(G,\cdot)$ being a group.\\ 203 | $\gen{a} = \set{a^n}{n \in \mathbb{Z}}$ 204 | 205 | $|\gen{a}|$ is the order of $$, the lowest possible number, if finite, st\\ 206 | $a^n = e$ 207 | 208 | \begin{itemize} 209 | \item 210 | If $a \in \mathbf{G}$, then the cyclic subgroup $\gen{a}$ is the 211 | \emph{smallest} subgroup G containing $a$ 212 | \item 213 | If $\mathbf{I}$ is an indexing set of $\mathbf{G}$, then $\gen{\{a_i\}}$ is the 214 | subgroup containing all finite products of $a_i \in \mathbf{G}$ 215 | \end{itemize} 216 | 217 | \subsection{Generators and Relations} 218 | 219 | \DEFINITION{1} A \emph{\bf presentation} is a \emph{generating set} and a set 220 | of \emph{relations}. 221 | 222 | A \emph{presentation} can define a group. For example, Klein Group is defined by:\\ 223 | generating set: $\{a, b\}$; relations: $a^2 = e, b^2 = e, ab = ba$ 224 | 225 | The integers $\mathbb{Z} \modu{6}$ can be described:\\ 226 | $\mathbb{Z}_6 \defeq $ generating set: $\{a\}$, relations: $a^6=e$ 227 | 228 | \pagebreak 229 | \subsection{Some Theorems Concerning Subgroups} 230 | 231 | \begin{itemize} 232 | \item 233 | Let $\mathbf{G}$ be a finite, \emph{Abelian} group. If $\mathbf{H}$ is a 234 | subgroup of $\mathbf{G}$, the $|\mathbf{H}|$ divides $|\mathbf{G}|$ 235 | ({\bf Lagrange's Theorem}) 236 | 237 | \item 238 | Let $\mathbf{G}$ be a finite, abelian group of order $n$. 239 | Then $\mathbf{G}$ has at least one subgroup of order $d | n$, for every 240 | positive divisor $d$ of $n$. 241 | 242 | \item 243 | \emph{If $\mathbf{G}$ is cyclic, we can do better.}\\ 244 | Let $\mathbf{G}$ be a finite, cyclic group of order $n$. Then $\mathbf{G}$ 245 | has \emph{exactly one} subgroup - a cyclic group at that - foreach 246 | positive divisor $d$ of $n$. If $\mathbf{G}$ is generated by $\gen{a}$, 247 | then $b = a^m$ has order $d = \frac{n}{\gcd(m, n)}$. 248 | Note that, $\gcd(n, 0) = n$ 249 | 250 | \item 251 | Let $\mathbf{G}$ be a finite group of order $n$, and let $p$ be a prime 252 | that divides $n$ (that is, $p | n$). Then, $\mathbf{G}$ has at least one 253 | subgroup of order $p$. ({\bf Cauchy's Theorem}) 254 | 255 | \item 256 | Let $\mathbf{G}$ be a finite subgroup of order $n$. Let $n = p^k m$, where 257 | $p$ is prime and $p\not | m$. Then $\mathbf{G}$ has at least one subgroup 258 | of order $p^i \ \forall i \in [0 \dots k]$. ({\bf Sylow's first theorem}) 259 | \end{itemize} 260 | 261 | 262 | \subsection{Isomorphism} 263 | 264 | Structural properties of groups: 265 | \begin{itemize} 266 | \item Order 267 | \item Abelian-ness 268 | \item Cyclic 269 | \item \emph{et cetera} 270 | \end{itemize} 271 | 272 | Note that, $\mathbf{G} \subset \mathbf{M}_2(\mathbb{Z}, \times) \cong$ Klein Group, when $\mathbf{G}$ is:\\ 273 | $I = \begin{pmatrix} 274 | 1 & 0 \\ 275 | 0 & 1 276 | \end{pmatrix}$ 277 | $A = \begin{pmatrix} 278 | 1 & 0 \\ 279 | 0 & -1 280 | \end{pmatrix}$, -I, -A 281 | 282 | 283 | Thus $\mathbf{M}_2(\mathbb{Z}) \cong \mathbf{V}_4$. 284 | 285 | 286 | \subsection{Classification of Finite Abelian Groups} 287 | 288 | First, let $\mathbf{G}_1$, $\mathbf{G}_2$ be finite Abelian on the set:\\ 289 | $\mathbf{G}_1 \times \mathbf{G}_2 = \set{(a, b)}{a \in \mathbf{G}_1, b \in \mathbf{G}_2}$ 290 | 291 | Define a binary operations, *, by the equation:\\ 292 | $(a_1, a_2) * (b_1, b_2) = (a_1 *_1 b_1,\ a_2 *_2 b_2)$ 293 | 294 | Some facts: 295 | \begin{itemize} 296 | \item 297 | The direct sum $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic iff 298 | $\gcd(m, n) = 1$. If this is true, then 299 | $|\mathbb{Z}_m \times \mathbb{Z}_n| = m n$, and is isomorphic to $\mathbb{Z}_{mn}$. 300 | 301 | \item 302 | The above can be generalized: 303 | $\mathbb{Z}_{m_1} \times \mathbb{Z}_{m_2} \times \dotso \times \mathbb{Z}_{m_k}$ 304 | is cyclic iff $\gcd(m_i, m_j) = 1$ for every distinct pair. If this is 305 | true, then: 306 | $\mathbb{Z}_{m_1} \times \dots \times \mathbb{Z}_{m_k} \cong \mathbb{Z}_{m_1 \dotsm m_k}$ 307 | 308 | \item 309 | Every finite Abelian group $\mathbf{G}$ is isomorphic to \emph{some} 310 | direct sum of the form: \\ 311 | $\mathbb{Z}_{(p_1)^{k_1}} \times \mathbb{Z}_{(p_2)^{k_2}} \times \dotsm \times \mathbb{Z}_{(p_n)^{k_n}}$ \\ 312 | where $p_i$ are not necessarily distinct, and $k_i$ are not necessarily 313 | distinct. These are the \emph{elementary divisors} of $\mathbf{G}$. 314 | 315 | \item 316 | Alternatively, we can write it another way:\\ 317 | $\mathbb{Z}_{m_1} \times \mathbb{Z}_{m_2} \times \dotsm \times \mathbb{Z}_{m_n}$\\ 318 | Where $m_1 \geq 2$, $m_1 | m_2$, \dotso, $m_{n-1} | m_n$ \\ 319 | Note that, $m_1, \dots, m_n$ are not distinct, but is a \emph{unique} list. 320 | These are the \emph{invariant factors} of the group. 321 | \end{itemize} 322 | 323 | \subsubsection{An Example} 324 | 325 | Show how many distinct (up to isomorphism) Abelian groups of order 600 exist: 326 | 327 | $600 = 2^3 \cdot 3 \cdot 5^2$ \\ 328 | \begin{align*} 329 | 600 & = 2 \cdot 2 \cdot 2\cdot 3 \cdot 5 \cdot 5 \\ 330 | & = 2 \cdot 2^2 \cdot 3 \cdot 5 \cdot 5 \\ 331 | & = 2^3 \cdot 3 \cdot 5 \cdot 5 \\ 332 | & = 2 \cdot 2 \cdot 2\cdot 3 \cdot 5^2 \\ 333 | & = 2 \cdot 2^2 \cdot 3 \cdot 5^2 \\ 334 | & = 2^3 \cdot 3 \cdot 5^2 \\ 335 | \end{align*} 336 | 337 | Or rather $3 \cdot 1 \cdot 2 = 6$ 338 | 339 | % 340 | % 341 | % 342 | \pagebreak 343 | \subsection{Group Homomorphisms} 344 | \label{sub-group-homomorphisms} 345 | 346 | $\phi$ is a homomorphism if: \\ 347 | $\phi(a \cdot b) = \phi(a)\ast\phi(b)$ 348 | 349 | \subsubsection{Properties of Homomorphisms} 350 | 351 | \begin{itemize} 352 | \item If $e$ is the identity element of $\mathbf{G}$, then $\phi(e)$ is the identity element of $\mathbf{G}'$ 353 | \item If $g \in \mathbf{G}$ has finite order $m$, then $\phi(g)$ has order $m$ 354 | \item If $a^{-1}$ is the inverse of $a \in \mathbf{G}$, then $\phi{a^{-1}}$ is the inverse of $\phi{a}\in\mathbf{G}'$ 355 | \item If $\mathbf{H}$ is a subgroup of $\mathbf{G}$, then $\phi(\mathbf{H})$ is a subgroupf of $\mathbf{G}'$ 356 | \item If $\mathbf{G}$ is finite, then the order of $\phi(\mathbf{G})$ divides the order of $\mathbf{G}$ 357 | \item If $\mathbf{G}'$ is finite, then the order of $\phi(\mathbf{G})$ divides the order of $\mathbf{G}'$ 358 | \item 359 | If $\mathbf{H}'$ is a subgroup of $\mathbf{G}'$, then $\phi^{-1}(\mathbf{H}')$ is a subgroup of $\mathbf{G}$, where: \\ 360 | $\phi^{-1}(\mathbf{H}') = \set{h \in \mathbf{G}'}{\phi(h) \in \mathbf{H}'}$ 361 | \end{itemize} 362 | 363 | {\noindent}The kernel of $\phi$ is defined:\\ 364 | {\indent}$\ker \phi \defeq \set{g\in\mathbf{G}}{\phi(g) = e'}$ 365 | And is a group, naturally. 366 | 367 | 368 | % RINGS 369 | % RINGS 370 | % RINGS 371 | \subsection{Rings} 372 | 373 | {\noindent}{\bf Definition.} A ring is a set with two binary operators on, such that: \\ 374 | \begin{itemize} 375 | \item $(\mathbf{R}, +)$ is an \emph{Abelian Group} 376 | \item $(\mathbf{R}, \cdot)$ is a \emph{semigroup} (closed, assosciateive) 377 | \item 378 | Has the distributive law defined on it: 379 | \begin{itemize} 380 | \item $a \cdot (b + c) = a \cdot b + a \cdot c$ 381 | \item $(a + b) \cdot c = a \cdot b + a \cdot c$ 382 | \end{itemize} 383 | \end{itemize} 384 | 385 | % RING H-MORPHISMS 386 | % RING H-MORPHISMS 387 | % RING H-MORPHISMS 388 | \subsection{Ring Homomorphisms} 389 | 390 | {\noindent}$(\mathbf{R}, +, \times) \cong (\mathbf{R}', \oplus, \otimes)$ iff:\\ 391 | \begin{itemize} 392 | \item 393 | $\phi(a + b) = \phi(a) \oplus \phi(b),\ \forall a, b \in \mathbf{R}$ 394 | \item 395 | $\phi(a \times b) = \phi(a) \otimes \phi(b),\ \forall a, b \in \mathbf{R}$ 396 | \end{itemize} 397 | 398 | \subsubsection{Properties of Rings} 399 | 400 | \begin{itemize} 401 | 402 | \item 403 | The \emph{kernel} of a ring homomorphism is the set 404 | $\ker \phi \defeq \set{a\in\mathbf{R}}{\phi(a)=0'}$, where $0'$ is the 405 | additive identity in $\mathbf{R}$ 406 | 407 | \item 408 | The image of $\mathbf{R}$, $\phi(\mathbf{R}) = \set{\phi(r)}{r \in \mathbf{R}}$ 409 | is a subring in $\mathbf{R}'$ 410 | 411 | \item 412 | $\phi(0) = 0'$ and $\phi(-r) = -\phi(r), \forall r \in \mathbf{R}$ 413 | 414 | \end{itemize} 415 | 416 | 417 | 418 | % INTEGRAL DOMAINS 419 | % INTEGRAL DOMAINS 420 | % INTEGRAL DOMAINS 421 | \subsection{Integral Domains} 422 | 423 | % --- 424 | % @@@ 425 | % @@@ 426 | % @@@ 427 | % --- 428 | 429 | % FIELDS 430 | % FIELDS 431 | % FIELDS 432 | \subsection{Fields} 433 | 434 | 435 | \end{document} 436 | --------------------------------------------------------------------------------