This is an archive of expository papers I have written. I have attempted to organize them by most relevant subject area.

Algebra

The existence of an injective homomorphism between $\text{Hom}\left(\dfrac{\mathbb{Q}\lvert\zeta\rvert}{\mathbb{Q}}\right)$ and the multiplicative group $R_n$ where $\{x\in R_n\colon \gcd(x, n) = 1\}$
The group of rigid motions of the cube is isomorphic to $S_4$, and some group actions from a group to itself.

Linear Algebra

A series of notes I typed up for the benefit of a student over the course of a spring linear algebra class.
Part 1: Definitions, Zorn’s Lemma
Part 2: Representation, Products, Duals
Part 3: Infinite Product/Sum Isomorphism, Riesz
Part 4: Tensor Products
Part 5: Exterior Products
Part 6: Determinants

Numerical Analysis

Crank-Nicolson notes

A series of notes I typed up for the benefit of a student over the course of a fall numerical analysis class.
Part 1: Function Spaces
Part 2: Multi-Index, Fourier Transform
Part 3: Stability, DFT, Lax Equivalence
Part 4: Lax-Richtmyer Theorem
Part 5: Elliptical PDEs, FEM

Real Analysis/Measure Theory

Proofs of the following:

1. Integrability of $f$ on $\mathbb{R}$ does not necessarily imply the convergence of $f(x)$ to $0$ as $x\to\infty$
   
2. If $f$ is integrable on $\mathbb{R}$, then $F(x) = \displaystyle\int_{-\infty}^x f(t)\ dt$ is uniformly continuous.
   
3. Chebyshev’s Inequality
   
4. $f$ real-valued and integrable on $\mathbb{R}^d$ and $\displaystyle\int_E f(x)\ dx\geq 0$ for every measurable $E$ implies $f(x) \geq 0$ a.e.
   
5. A function can be integrable yet unbounded in any interval.
   PDF

Quantum Mechanics

Bell’s Inequality discussion.