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.