# Virtual Math Circle: Summer 2023 Research Projects

## LSU Math Circle Math Circle is a virtual summer camp for high school students. You can read more about how the program works.

Project posters, papers, and presentations appear in the public archive of Math Circle projects.

## Session 1: Mon, Jun 12, 2023 – Sat, Jul 1, 2023

### Lagrange Multipliers in Portfolio Optimization with Mixed Constraints

#### Virtual Math Circle Research Proposal

Session
Session 1 (Jun 12 – Jul 1, 2023)
Mentor
Minh Vu
PhD Student
Department of Mathematics
Louisiana State University

Project title
Lagrange Multipliers in Portfolio Optimization with Mixed Constraints
Topic area
Optimization

High school calculus
Yes.
Skills and background
This research project only requires a basic understanding of calculus, specifically derivatives of polynomials, to determine the maximum and minimum values of a function. Some additional calculus concepts will be introduced and taught as needed. However, high school algebra is a prerequisite.

Abstract
Portfolio optimization is a fundamental problem in finance, which entails identifying the optimal combination of assets to hold in a portfolio, given the investor’s goals and constraints. This research project focuses on the method of Lagrange multipliers as an effective tool for building an optimal investment portfolio with two stocks, A and B, characterized by known expected returns and standard deviations. Specifically, we concentrate on the scenario where these two stocks have different expected returns and non-zero standard deviations, while being uncorrelated.

We will start by formulating the problem as a nonlinear constrained optimization problem. Our primary objective is to determine the proportion of each stock in the portfolio that maximizes the expected return while minimizing the portfolio risk, subject to budget and risk constraints. We will then investigate the impact of a modified portfolio risk limit on the optimal portfolio allocation to further enhance our analysis. By the end of this research, we aim to provide a comprehensive understanding of the method of Lagrange multipliers and its practical application in optimizing investment portfolios.

Possible extension
This problem can be extended in several ways to explore various aspects of portfolio optimization. For instance, we can expand the portfolio by adding more stocks and examining the impact of diversification on the risk and return. We can also include more intricate conditions, such as sector constraint, to reflect practical investment scenarios.

Outline/timeline
Week 1: In the first week, we will provide an overview of calculus topics, including derivatives of polynomials, maximum and minimum values of a function, and the method of Lagrange multipliers . We will also cover the application of the method of Lagrange multipliers in solving nonlinear optimization problems subject to equality constraints. Since the optimization problem used in our research project consists of both equality and inequality constraints, we will introduce the concept of slack variables that can be added to an inequality constraint to transform it into an equivalent equality constraint . Several examples and assignments will be given to help students better understand the method of Lagrange multipliers and nonlinear optimization problems subject to mixed constraints [2, 3, 4].

Week 2: The second week will begin with an introduction to portfolio optimization and its role in finance. We will subsequently delve into statistical concepts and formulas that are essential in solving our portfolio optimization problem, including the expected return and risk of the two-asset portfolio . Next, we will formulate our research problem as a nonlinear optimization problem subject to budget and risk constraints. The risk inequality constraint will be converted to an equality constraint by introducing a slack variable. Then, the method of Lagrange multipliers will be utilized to determine the proportion of each stock in the portfolio that maximizes the expected return while minimizing the portfolio risk. After obtaining the general solution to our optimization problem, we will select fixed values for the expected return and standard deviation of the two stocks in the portfolio while considering two different portfolio risk constraints. By comparing the optimal portfolio allocation under different constraints, we will develop practical skills in analyzing investment options and designing an investment strategy that meets our goals.

Week 3: In the final week, we will finalize our results, write the report, and prepare the final presentation using LaTeX.

References
References are given below.
1. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, In: 10th ed. McGraw-Hill Education, 2013. Chap. 7.
2. Felix Fischer, The Method of Lagrange Multipliers,url: https://webspace.maths.qmul.ac. uk/felix.fischer/teaching/opt/notes/notes2.pdf. (accessed: 03.03.2023).
3. James Stewart, Calculus: Early Transcendentals ,In: 8th ed. Cengage Learning, 2015. Chap. 3.1, 4.1, 14.8.
4. Robert J. Vanderbei, Linear Programming: Foundations and Extensions ,3rd ed. Springer, 2008. Chap. 2.

### Mathematical Modeling of the Transmission of Multi-strain Infectious Diseases

#### Virtual Math Circle Research Proposal

Session
Session 1 (Jun 12 – Jul 1, 2023)
Mentor
Dr. Olusegun Michael Otunuga
Assistant Professor of Mathematics
Department of Mathematics
Augusta University

Project title
Mathematical Modeling of the Transmission of Multi-strain Infectious Diseases
Topic area
Mathematical Biology; Epidemiology

High school calculus
A knowledge of High School Calculus and Differential Equations will help. These will also be taught/revised as needed.
Skills and background
High School Calculus

Abstract
A vaccine breakthrough infection case is when an individual vaccinated against infection with a virus, bacterium, or any germ still gets infected with such diseases. A rebound infection happens when someone recovering from a virus or bacterium begins to get symptoms after recovery. This work develops an innovative multi-strain susceptible-vaccinated-exposed asymptomatic-symptomatic-recovered epidemic model to analyze how such transmission occurs.

We assume the population is completely susceptible to $n$ different variants of the disease, and those who are vaccinated and recovered from a specific strain of the disease are immune to the present strain and its predecessors but can still be infected by newer emerging strains. The visualization of the outbreak transmission is enabled using a block diagram environment in Simulink MATLAB. The model is used to estimate epidemiological parameters, namely, the latent period, the transmission rates, vaccination rates, and recovery rates of the variants and lineages of the COVID-19 virus in the United States. The reproduction number of the disease is also calculated and reported. This is used to determine if the population is in an endemic state with one or more other strains or if it is free of specific variants of the disease.

Possible extension
Possibility of extending the model to a case where the transmission rate of the disease is assumed to be fluctuating due to environmental perturbations.

Outline/timeline
Week Mentor’s Duty Student’s Duty
Week 1 Meet students, provide reading research materials, read provided materials, and research a similar list of needed statistical and mathematical software. Read provided materials, research similar materials.
Introduction and revision of necessary topics needed (Differential Equation (DE), Calculus). Revise material, write summarized reports on materials.
Introduction to numerical methods for systems of differential equations, introduction to MATLAB, Estimation scheme (statistical and mathematical). Complete assigned exercises, solve DE numerically by writing code(s) in MATLAB or using inbuilt MATLAB ODE45 code.

Week 2 Data collection relating to research work, data cleaning, and sorting. Development of mathematical and statistical models to describe data trajectory. Students will participate in data collection, sorting, and modeling.
Analyze biological data, design epidemiological model, analyze model, estimate epidemiological parameters. Students will participate in the research work by assisting in the analysis, designing, and estimation process.

Week 3 Analyze biological data, design epidemiological model, analyze model, estimate epidemiological parameters. Student will participate in the research work by assisting in the analysis, designing, and estimation process.
Introduction to using LaTeX on Overleaf, manuscript writing. Students will learn to write complex mathematical equations using LaTeX.

References
References are given below.
1. S. Nagle, E. Saff, A. Snider, Fundamentals of Differential Equations, 9th ed., Pearson Education, Inc., 2018.
2. O.M. Otunuga, Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19, PLoS ONE, 17(7): e0271446, 2022.
3. O.M. Otunuga, Stochastic modeling and forecasting of Covid-19 deaths: Analysis for the fifty states in the United States, Acta Biotheoretica 70:25, 2022.
4. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci 180:29-48, 2002.

### Walking to Infinity Along Primes

#### Virtual Math Circle Research Proposal

Session
Session 1 (Jun 12 – Jul 1, 2023)
Mentor
Bencheng Li
Department of Mathematics
Georgia Institute of Technology

Project title
Walking to Infinity Along Primes
Topic area
Elementary Number Theory and Probability Theory

High school calculus
Not a prerequisite.
Skills and background
This research project requires only a little elementary knowledge of arithmetic and algebra, such as functions and sequences. Familiarity with number theory and probability theory is encouraged, but the mentor will teach some related basic concepts, such as primes and sequences. Also, we will implement either Python or R to simulate some random walks, which will be taught. These computational and simulation skills will be important for students' future undergraduate studies. When preparing for the final presentation, we will use Beamer, ShareLaTeX, or Powerpoint, depending on students' preferred tools. If necessary, the mentor will also teach basic LaTeX typesetting skills.

Abstract
An interesting problem in number theory is whether, given a prime number, we can construct a sequence of primes by appending a digit to the right of the previous primes. For example, if we start at $3$, which is a prime, the next possible prime we can get is $31$ by appending $1$ to the right of $3$. We can also add another $1$ to the right of $31$ to get a new prime, $311$. Now the question is whether we can continue to add digits one at a time to get a sequence of primes. If not, what is the longest prime walk we can get starting at $3$? How about starting at another prime instead of $3$? Can we write some programs to simulate this process?

In this project, we will try to devise an algorithm to compute the longest prime walk starting at a certain prime $x$ and compare the cases with different $x$. If this can be achieved, a further question is how can we add two, three, or a bounded number of digits at a time instead of only one? Will that give us a walk to infinity? This project is based on my research in 2020 Polymath REU .

Possible extension
We can also investigate the walk along squares or other interesting sequences instead of primes. Are we able to walk to infinity along the squares? Can we modify our codes to simulate the squares walk?

Outline/timeline
Week 1: The first week will be all about introducing the project-related concepts, including primes, sequences, and some significant results in number theory, such as Dirichlet's Theorem. In addition to that, some significant results in the paper  will be presented. After that, we will set up the coding environment and introduce the basic Python or R programming skills necessary for this project.

Week 2: The second week will be based heavily on coding. The objective is to use what we've learned in the first week to develop an algorithm that uses a specific prime $x$ as input and outputs the shortest prime walk we can get by appending only one digit at a time. This whole process will be guided by the mentor, and necessary hints will be provided. Once an algorithm is found, we can use that to compare the results with different input primes $x$.

Week 3: The first half of the last week will be spent finalizing our results to see if we can modify our codes to get an algorithm for the longest prime walk by appending a bounded number of digits at a time. The second half will be spent on creating the final presentation using Beamer, ShareLaTeX, or PowerPoint to present our algorithms.

References
References are given below.
1. S. J. Miller, F. Peng, T. Popescu, J. M. Siktar, N. Wattanawanichkul, The Polymath REU Program, Walking to Infinity Along Some Number Theory Sequences, arXiv:2010.14932, https://doi.org/10.48550/arXiv.2010.14932

## Session 2: Mon, Jul 17, 2023 – Sat, Aug 5, 2023

### Stochastics in Physics

#### Virtual Math Circle Research Proposal

Session
Session 2 (Jul 17 – Aug 5, 2023)
Mentor
Moises Gomez Solis
PhD Student
Department of Mathematics
Louisiana State University

Project title
An Introduction to Stochastics in Physics
Topic area
Applied mathematics, Stochastic processes, Mathematical physics.

High school calculus
Not required. It is going to be taught as needed.
Skills and background
For this project, it will be expected to have knowledge of high school algebra and some very basics of coding. This project represents a nice introduction to some applications of stochastic processes and how it is often encountered in other fields of science, in this case, statistical mechanics. By the end of this project, the students will develop a good sense of mathematical intuition and how to code their own Markov Chain Monte Carlo (MCMC) simulations in Python. During the preparation of the final presentation, students will use tools such as Powerpoint and TeX.

Abstract
When trying to model a real problem via identically distributed random variables, we often find that they are not always interesting by themselves, mostly due to them behaving "essentially" the same. So, in order to introduce some variability we can always introduce some dependency to previous states; this is, to follow the same route as in the deterministic difference equations, and define a stochastic difference equation via: \begin{align*} X_{n+1} = f(X_{n}, Z_{n+1}) \end{align*} To this goal, we are going to study some effects of thermodynamics; in particular, consider a system with $N$ particles; these particles can be found in a compartment labeled A or in a compartment labeled B. Consider a time $n\geq 0$, with $X_n = i$ (the number of particles in A). By choosing a particle at random, such that it has moved to the other compartment at time $n+1$, we see that the state $X_{n+1}$ is either $i+1$ or $i-1$. Meaning that for all times $n\geq0$ \begin{equation*} X_{n+1}= X_n + Z_{n+1}, \end{equation*} where $Z_n$ is either $1$ or $-1$. What can we infer from this model? Do all particles end up in one compartment? Is there a time $T$ such that we have gone through all possible states? What is the connection between this simple model and statistical mechanics?

Possible extension
This project can be extended further by examining some simulations of diffusion processes.

Outline/timeline
Week 1: During the first week, students will be introduced to some very elementary probability and linear algebra. If required, we are going to cover all the algebra and calculus necessary to understand the computations involved, with a minimal focus put on the concept of proofs and their structure.

Week 2: Over the course of this week, we are going to work on some numerical experiments to visualize the results obtained.

Week 3: The first half is going to be dedicated to double-checking some of our models and their python implementation, and the latter half of the week is going to be used to prepare the presentation in either Powerpoint or LaTeX.

References
References are given below.
1. Pierre Bremaud, Markov chains: Gibbs fields, Monte Carlo simulation, and queues, Springer, 1999. isbn: 9780387985091; 0387985093. url: libgen.li/%20file.php?md5=07d6ec859d0%5C% 5C88102708d951aec9b391b.
2. Franklin Mendivil, Ronald W. Shonkwiler, Explorations in Monte Carlo Methods, 1st ed. Undergraduate Texts in Mathematics. Springer, 2009. isbn: 038787836X; 9780387878362; 9780387878379. url: libgen.li/file.php?md5=7b88c5ba42b9bb2f5e79e23fae052e85.

### Walk to Infinity Along Square-Free Numbers

#### Virtual Math Circle Research Proposal

Session
Session 2 (Jul 17 – Aug 5, 2023)
Mentor
Bencheng Li
Department of Mathematics
Georgia Institute of Technology

Project title
Walking to Infinity Along Squares
Topic area
Elementary Number Theory and Probability Theory

High school calculus
Not a prerequisite.
Skills and background
This research project requires only a little elementary knowledge of arithmetic and algebra, such as functions and sequences. Familiarity with number theory and probability theory is encouraged, but the mentor will teach some related basic concepts, such as squares and sequences. Also, we will implement either Python or R to simulate some random walks, which will be taught. These computational and simulation skills will be important for students' future undergraduate studies. When preparing for the final presentation, we will use Beamer, ShareLaTeX, or Powerpoint, depending on students' preferred tools. If necessary, the mentor will also teach basic LaTeX typesetting skills.

Abstract
An interesting problem in number theory is whether, given a square of the form $x^2$ for some integer $x$, we can construct a sequence of squares by appending a digit to the right of the previous squares. For example, if we start at $1$, which is a square, the next possible square we can get is $16$ by appending $6$ to the right of $1$. We can also add another $9$ to the right of $16$ to get a new prime, $169 = 132$. Now the question is whether we can continue to add digits one at a time to get a sequence of squares. If not, what is the longest square walk we can get starting at $1$? How about starting at another square instead of $1$? Can we write some programs to simulate this process?

In this project, we will try to devise an algorithm to compute the longest square walk starting at a given square $x^2$ and compare the cases with different $x$. If this can be achieved, a further question is how we can add two, three, or a bounded number of digits at a time instead of only one. Will that give us a walk to infinity? This project is based on my research in 2020 Polymath REU .

Possible extension
We can also investigate the walk along squares or other interesting sequences instead of primes. Are we able to walk to infinity along the squares? Can we modify our codes to simulate the squares walk?

Outline/timeline
Week 1: The first week will be all about introducing the project-related concepts, including primes, sequences, and some significant results in number theory, such as Dirichlet's Theorem. In addition to that, some important developments in the paper  will be presented. After that, we will set up the coding environment and introduce the basic Python or R programming skills necessary for this project.

Week 2: The second week will be based heavily on coding. The objective is to use what we've learned in the first week to develop an algorithm that uses a certain square number $x^2$ as input and outputs the longest possible square walk we can get by appending only one digit at a time. This whole process will be guided by the mentor, and a necessary hint will be provided. Once an algorithm is found, we can use that to compare the results with different inputs $x$.

Week 3: The first half of the last week will be spent finalizing our results to see if we can modify our codes to get an algorithm for the longest prime walk by appending a bounded number of digits at a time. The second half will be spent on creating the final presentation using Beamer, ShareLaTeX, or PowerPoint to present our algorithms.

References
References are given below.
1. S. J. Miller, F. Peng, T. Popescu, J. M. Siktar, N. Wattanawanichkul, The Polymath REU Program, Walking to Infinity Along Some Number Theory Sequences, arXiv:2010.14932, https://doi.org/10.48550/arXiv.2010.14932

## Questions?

Contact Isaac Michael <imichael@lsu.edu>.