Test of Austin’s Custom LaTeX Style

This document tests various custom commands and environments from austin.sty.

Custom Math Commands

Here are some custom math symbols:

  • Complex numbers: $\mathbb{C}$

  • Natural numbers: $\mathbb{N}$

  • Rational numbers: $\mathbb{Q}$

  • Real numbers: $\mathbb{R}$

  • Integers: $\mathbb{Z}$

Some custom operators: $\mathrm{cis}(\theta) = \cos(\theta) + i\sin(\theta)$

The least common multiple: $\mathrm{lcm}(12, 18) = 36$

Automorphism group: $\mathrm{Aut}(G)$

Custom Math Shortcuts

Cube root: $\sqrt[3]{27} = 3$

Floor and ceiling: $\left\lfloor 3.7 \right\rfloor = 3$ and $\left\lceil 3.2 \right\rceil = 4$

Boxed result: $\boxed{E = mc^2}$

Fraction shortcut: $\frac{a}{b} = \frac{a}{b}$

Vector notation: $\mathbf{v} = \begin{bmatrix}x \ y \ z\end{bmatrix}$

Theorem-Like Environments

Theorem:

If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then \(\int_a^b{f(x) \, dx} = F(b) - F(a)\)

Lemma:

Every partially ordered set in which every chain has an upper bound contains at least one maximal element.

Corollary:

Every vector space has a basis.

Proposition:

For any vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$: \(|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \|\mathbf{v}\|\)

Definition:

A group is a set $G$ together with a binary operation $\cdot: G \times G \to G$ satisfying:

  1. Associativity: $(a \cdot b) \cdot c = a \cdot(b \cdot c)$

  2. Identity: $\exists e \in G \ \text{s.t.} \ a \cdot e = e \cdot a = a$ for all $a \in G$

  3. Inverses: For each $a \in G$, $\exists a^{-1} \in G \ \text{s.t.} \ a \cdot a^{-1} = a^{-1} \cdot a = e$

Example:

The symmetric group $S_n$ consists of all permutations of $n$ elements. It has order $n!$.

Remark:

The notation $\mathrm{Hom}(G, H)$ denotes the set of all homomorphisms from $G$ to $H$.

Probability and Statistics

Expected value: $\mathbb{E}{\left[X\right]} = \sum_{x} x \cdot P(X = x)$

Variance: $\text{Var}\left(X\right) = \mathbb{E}{\left[X^2\right]} - \mathbb{E}{\left[X\right]}^2$

Covariance: $\text{Cov}\left(X, Y\right) = \mathbb{E}{\left[XY\right]} - \mathbb{E}{\left[X\right]}\mathbb{E}{\left[Y\right]}$

Bernoulli distribution: $X \sim \text{Bern}\left(p\right)$

Binomial distribution: $Y \sim \text{Binom}\left(n, p\right)$

Code Listings

def fibonacci(n):
    if n <= 1:
        return n
    return fibonacci(n-1) + fibonacci(n-2)

Problem Environment

Problem:

Prove that $\sqrt{2}$ is irrational.

Subproblem:

Assume $\sqrt{2} = \frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $\gcd(p, q) = 1$.

Subproblem:

Derive a contradiction.