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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…

Image courtesy of geralt via Pixabay

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…


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. …


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


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…

Picture courtesy of geralt via Pixabay


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…


Python is without question the programming language I am most comfortable with, and this is largely because it’s in such high demand. I have been in internship positions where the programming language of choice was Python, and so I had to learn how to use it. I also took two programming courses at Carnegie Mellon that were largely about programming in Python. Despite this, it took until my second year of graduate school before I needed the capabilities of solving algebraic and differential equations in a programming language.

While the main focus of my research has been in nonlocal…

While I am currently doing research in analysis and partial differential equations as a Ph.D. student at the University of Tennessee-Knoxville, my love of mathematics was focused in discrete mathematics, particularly combinatorics, during the earlier years of my undergraduate study. This was in part due to a research topics course I took during my freshmen year of college at Carnegie Mellon with visiting professor Steven Miller, and in part due to my experience competing in competitions such as ARML and the Putnam. Nonetheless, my math textbook collection grew to include various discrete math textbooks, particularly those on combinatorics. …

Joshua Siktar

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

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