An **ordinary differential equation** (ODE) is an equation involving a scalar-valued function from the real line to the real line, and some of its derivatives. A **partial differential equation** (PDE) is a generalization of the ordinary differential equation to the event where the argument is in multiple dimensions. In this case we have **partial derivatives** in each direction. For example, if the input is an ordered pair, then the input belongs to R², so we may have partial derivatives in two different directions. …

**Introduction: What is Calculus of Variations?**

The **calculus of variations **refers to the science (or the art) of solving optimization problems containing integrals. In other words, out of a collection of functions, choose the one that makes the integral of those functions as small as possible. Such a function will be called a **minimizer.** Sometimes the integrand will depend on the function itself alone; sometimes dependence on derivatives of the function will be allowed. Here is an abstract example of what one of these problems looks like:

If you aren’t familiar with what the “W” means, don’t worry about it…

I’ve just finished my fourth semester in the University of Tennessee’s Mathematics Ph.D. program! That means I’m nearly at the halfway point of the program, and I am getting ever closer to the oral exam. All that being said, this seemed like a good time to reflect on how I’ve developed an answer to the question, “what do I want to do after I graduate?” Moreover, I want to evaluate if I have been spending my time and energy on suitable activities to reach this goal.

At this point I’ve finished my preliminary exams and selected a research topic and…

As of August 2020 I had passed my preliminary examinations at the University of Tennessee-Knoxville, in analysis and partial differential equations. In the mindset of many potential advisors, this was the green light to begin working on research with them. Of course, I had to first identify an advisor and a project. While I detail this process more in a different blog post, I want to jump to what happened from a logistical standpoint after I made these important decisions.

In the Fall 2020 semester, I had extra work to do each week as my project initiated, along with one…

**Introduction and outline**

In this article I’m going to describe some mechanisms for carrying out an election that are mathematical and deterministic in nature (i.e. noting happens randomly). These mechanisms are called **voting methods**, because they are essentially ways of running an election to determine a winner. The data used in these elections come in the form of **preference tables**. While there are many voting methods studied in research papers and utilized as examples in recreational mathematics classes, this article will one particular voting method as a case study:

**Borda Count: **each person voting in the election ranks all of…

**Introduction**

A **voting method** is a mechanism for determining the winner in an election, with a series of steps explicitly listed. These methods generally have some means of scoring candidates. Each voting method is used on a **preference table**, which is a chart that indicates people’s relative preferences between two or more options.

Voting methods are often found as a topic in the curriculum of a recreational math or math modeling course at the high school or collegiate level. …

**Introduction**

Consider a sequence of numbers with a distinct pattern, such as

1, 4, 7, 10, 13, 16, 19, …

For this thought exercise, the sequence needs to be chosen with some purpose, rather than at complete random. When this is achieved, the sequence can be described by two types of formulas. For this specific example, those formulas take the following two forms:

S_n = 3n + 1

AND

S_n = S_{n-1} + 3 (with S_1 = 1)

Both formulas above help to describe the sequence we started with, but there is one significant difference between them. The first formula…

**Introduction and Outline**

It’s just the greatest number of votes wins the election, right? Not so fast.

While a simple counting of votes can be an effective method in some situations, the theory of designing voting methods is a lot more complicated (and intriguing) than that. When one asks, “which voting method is the best,” this is both a mathematical and a philosophical question, not to mention a question that lends itself to interesting pedagogical opportunities in a math classroom. In this article I’m going to break this question into many pieces, but let’s start with defining the phrase “voting…

**Introduction**

In this article, I’m going to give a brief introduction as to what recursion is, how it can be used to generate sequences of numbers, and how to perform some of these basic operations in Python. Two sequences of real numbers that will be pertinent to our discussion are the Lucas Numbers and the Fibonacci numbers.

Ultimately,** recursion** refers to the buildup of a sequence of numbers based on an explicit pattern that is given. This patten comes in the form of an equation known as the **recurrence relation**. Moreover, any sequence has to have a starting point, known…

**Introduction: what are Zeckendorf Decompositions?**

In this article I’m going to talk about Zeckendorf Decompositions and how to manipulate and construct these sums in Python. Of course, I should probably start by defining what these mysterious sums are. A **Zeckendorf decomposition** is a way of writing a positive integer as a sum of non-consecutive Fibonacci numbers. For clarify, the Fibonacci numbers are the sequence

1, 2, 3, 5, 8, 13, …

defined with the recursive relation

F_n = F_{n-1} + F_{n-2}

Sometimes an additional 1 is stuck at the front of the Fibonacci number sequence, but when dealing with Zeckendorf…

Math PhD Student University of Tennessee | Academic Sales Engineer | Writer, Educator, Researcher