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

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

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.
    \[f'(x) = \ln(x+1)-\ln x – \frac{1}{x+1} = \int_x^{x+1} \big(\frac{1}{t} – \frac{1}{x+1}\big)\dt > 0.\]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s