# Expository Writings, Notes, Problem Solutions, etc.

The following are (expository) papers I wrote as an undergrad to satisfy various course and degree requirements.

• Fourier Analysis on the Torus – An introduction to Fourier analysis on the Torus, which includes Kolmogorov’s construction of an $L^{1}$ function whose Fourier series diverges almost everywhere and basic results on the Dirichlet and Fejer kernels. The paper was my final project for Math 212br (Spring ’12) with Professor A. Knowles.
• D.J. Newman PNT Proof – A detailed exposition of D.J. Newman’s extremely elegant proof of the prime number theorem. This was an expansion on my talk to my Math 229x class (Spring ’12) with Professor B. Mazur. It was also my submission to satisfy the Harvard math department’s expository paper requirement.
• Mathematical Methods in Wind Power Modeling – A survey of some mathematical methods in wind power estimation and modeling, with a focus on the use of the two-parameter Weibull distribution. This was my final project for the SPU 25 with Professor M. McElroy.
• Generalized Martingale ABRACADABRA Problem – Some basic results in the theory of discrete-time martingales, plus an application to the problem of determining the expected time until a fixed pattern occurs (informally, how long until a monkey types ABRACADABRA on a typewriter). The exposition is adapted from one of Professor S. Sternberg’s Math 212a problem sets.

I like to have proved at least once every non-trivial result I need to use in mathematical doings. What follows below is partial documentation of this idiosyncracy.

Classical Analysis

Functional Analysis

Harmonic Analysis

• Hormander’s Staircase – Construction of Hörmander’s staircase used in the proof of the Malgrange-Ehrenpreis theorem.
• A Useful Fourier Transform – Computation of the Fourier transform of  $\left|x\right|^{-n}$ for $0 < \alpha < n$, where $n$ denotes the dimension of the underlying Euclidean space.
• Generalization of HL Maximal Inequality – Generalization of the Hardy-Littlewood Maximal Inequality to sigma-finite measure spaces where the underlying set is also a separable metric space.
• DJ Newman’s Proof of Fourier Uniqueness – A short, elegant proof of the injectivity of the Fourier transform as a map $L^{1}(\mathbb{R}) \rightarrow C(\mathbb{R},\mathbb{C})$.

PDE

Complex Analysis

Measure Theory and Integration

Probability Theory

Analytic Number Theory

Linear Algebra

• Trace and Determinant – Coordinate-free approach to the trace and determinant of a linear operator on a finite-dimensional vector space.

General Topology

• Chernoff’s Proof of Tychonoff Theorem
• Moore-Smith theorems – Proofs of the Moore-Smith theorem and the Arzela-Ascoli theorem for families of equicontinuous functions on complete, separable metric spaces into compact metric spaces.
• Compactness Equivalence and Application to Proper Maps – We define a topological space $K$ to be compact if and only if every open cover of $K$ contains a finite subcover. We prove a theorem which gives an equivalent criterion for a space to be compact in terms of the closedness of the projection map $\pi_{Z}: K \times Z \rightarrow Z$. We then use this theorem to prove that a proper map is closed and has compact fibers.

Applied Mathematics

### 3 Responses to Expository Writings, Notes, Problem Solutions, etc.

1. Hansen says:

Hi, Matt:

I am reading your writeup on Hardy’s Inequality. On page 2, line 7, says $\int_n^{n+1} (1-\frac{n}{x})dx>\frac{1}{n+1}$. That is not right, the inequality should be reversed. Of course, this blocks the way to proving the discrete version of the Hardy’s Inequality.

2. Hansen says:

Hi, Matt:

On another note, regarding the compound interest note, here is a more straightforward way of proving $f(x) = x\ln(1+\frac{1}{x})$ is nondecreasing, and indeed, strictly increasing.
$f'(x) = \ln(x+1)-\ln x – \frac{1}{x+1} = \int_x^{x+1} \big(\frac{1}{t} – \frac{1}{x+1}\big)\dt > 0.$