Introduction to the Hermite Functions

As I teach partial differential equations and compose sections for a textbook on the subject, I often run across special families of functions that I rarely see in other contexts. Recently this involved the Hermite functions which are related to the Hermite polynomials. The linked notes assume the reader is familiar with the Hermite polynomials and points out some of the connections between the Hermite polynomials and the Hermite functions. To me the most amazing feature of the Hermite functions is that they are eigenfunctions of the Fourier Transform.

The Laplacian in Spherical Coordinates

The Laplacian operator is frequently used in multivariable calculus and partial differential equations. The calculation of the operator in Cartesian coordinates is straightforward and versions of the operator for polar, cylindrical, spherical, and other coordinate systems exist, though often just the result is present without the justifying steps. As an instructor of mathematics, I believe everyone (myself included) should derive the Laplacian in these other coordinate systems at least once in their mathematical career. The linked notes provide an outline of one method of deriving the Laplacian in spherical coordinates from the Laplacian in Cartesian coordinates.

Solving Non-Homogeneous nth Order Linear Boundary Value Problems

With no disrespect to my professors in undergraduate and graduate school, I do not seem to recall spending much time on the topic of solving non-homogeneous, linear boundary value problems. Even as a professor of mathematics myself, I do not usually have the time during the semester to discuss boundary value problems in our ODE course. I feel plenty of time is spent on the method of undetermined coefficients and variation of parameters for solving second order non-homogeneous differential equations. I recently had the need to organize some notes on using Green’s functions to solve non-homogeneous initial and boundary value ordinary differential equations and initial, boundary value partial differential equations. If you are interested in seeing the method developed and applied to nth order, linear, non-homogeneous boundary value problems on intervals of the form [a, b], check out these notes. I have attempted to concisely motivate the method, so that it can be immediately employed in examples. Two examples are included.

Modes of Convergence of Random Variables

While correcting some typographical errors and making other clarifications to my textbook An Undergraduate Introduction to Financial Mathematics, 4th edition, I began to think that a set of notes on the various modes of convergence of random variables associated with probability spaces would be useful. Hence I have put together these notes on convergence of random variables. The notes define several modes of convergence, give elementary examples, and attempt to compare and contrast the modes of convergence to one another.

Airy Functions and Modified Bessel Functions

The Airy differential equation is often one of the first nontrivial differential equations used to demonstrate the infinite series method for determining a solution. In my own classes I use it to illustrate series techniques near an ordinary point. However, I usually do not emphasize that special functions are often defined as the solutions to specific classes of differential equations. In the brief note attached below, the Airy functions Ai(x) and Bi(x) are explored in more detail. The Bessel differential equation is also an important source of examples in ODE courses, usually to illustrate the procedure for finding Frobenius-type solutions. In fact, the Airy functions can be expressed in terms of the solutions to the Bessel differential equation of order 1/3 and the modified Bessel differential equation of order 1/3. This relationship is also verified in the note.

Airy and Bessel Functions

A Detailed Derivation of the Euler-Mascheroni Constant

The Euler-Mascheroni constant (or Euler gamma) is found throughout analytic number theory and applied mathematics. Any student working with the Bessel functions will have encountered this constant. The attached article goes through a fairly rigorous derivation of this constant.

Euler-Mascheroni Constant

A Brief Introduction to Principal Component Analysis

As a continuation of my earlier exposition on the Singular Value Decomposition, I have a brief article on the statistical technique known as Principal Component Analysis (PCA). The article contains a linear algebra-based derivation of PCA, several examples of its application, and Mathematica code for people willing to try it out.

Principal Component Analysis

Singular Value Decomposition and an Application

For the past four fall semesters I have taught a course in matrix algebra and applications. One of my favorite parts of the semester is the discussion surrounding the topic of the Singular Value Decomposition (SVD). In this blog entry I will outline the SVD, how to calculate it, what the SVD can do with data, and use it as a kind of image compression algorithm.

Singular Value Decomposition

Random Harmonic Oscillator

The simple harmonic oscillator is frequently encountered in ordinary differential equations courses and in classical mechanics. This short post explores the harmonic oscillator driven by white noise.

Random Harmonic Oscillator

One more file on Hilbert Spaces

The final lecture introducing concepts related to Itô Calculus was given by Dr. Mine Çaglar of Koç University. I have summarized it in the attached notes.

A Brief Introduction to Itô Calculus

During 2022 the SIAM student chapter at Sabanci University produced a series of videos on Itô calculus presented by Dr. Ali Süleyman Üstünel of Bilkent University. I have attempted to summarize and re-organize some of the material and to flesh out the definitions, lemmas, theorems, and other propositions. I hope readers find this helpful in studying Itô calculus.

• Probability Spaces, Random Variables, and Expectation

• Convergence of Sequences of Random Variables

• Additional Convergence Results

• Homework Exercises and Additional Definitions