# Assorted Academic Writing

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:

- Integrability of \( f\) on \(\mathbb{R}\) does not necessarily imply the convergence of \(f(x)\) to \(0\) as \(x\to\infty\)
- If \( f\) is integrable on \(\mathbb{R}\), then \(F(x) = \displaystyle\int_{-\infty}^x f(t)\ dt\) is uniformly continuous.
- Chebyshev’s Inequality
- \(f\) 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.
- A function can be integrable yet unbounded in any interval.

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