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Note: Full report is only available under request when not indicated by an asterisk. (*)

##### Full List

In order of Personal Importance and Recognition:

- 2020 Korea Science and Engineering Fair (KSEF) Physics Bronze Award

- 2020 Korea Science and Engineering Fair International (KSEFI) Earth Science Bronze Award

- 2020 Youth International Science Fair (YISF) National Representative

- Published on the International Journal of STEAM

* Who is the Greatest Athlete of All Times: A Probabilistic Rating Model (2021/03), S Yoon. et al.

- 2021 International Mathematical Modelling Challenge (IMMC) Honorable mention (3rd Place)

- 2021 International Mathematical Modelling Challenge (IMMC) National Winner, Representative

* Derivation and Generalization of the Law of Cosines onto the nth Dimension (2021/02), S Yoon.

- International Baccalaureate (IB) Extended Essay

- Submission to the International Journal of STEM Education

- 2020 International Mathematical Modelling Challenge (IMMC) Honorable Mention (3rd Place)

- 2020 International Mathematical Modelling Challenge (IMMC) National Representative

- Revised during Internship at the French National Institute for Research in Digital Science and Technology (INRIA)

* A Mathematically Constructed Simulation of Korean Traditional Painting Minhwa (2021/05), S Yoon.

- International Baccalaureate (IB) Mathematics Internal Assessment (IA)

- 2020 High School Mathematical Contest in Modeling (HiMCM) National Representative

- HarvardX Data Science Professional Certificate Capstone Project

- Published on the NLCS Journal of Pure and Applied Mathematics

Reports below are in chronological order; the list above is in order of recognition.

##### 2021

# Using Local Indicators to Unitary and Balanced Transport Problems with Concave Costs

# (Applied Mathematics)

J. Delon., J. Salomon., A. Sobolevskii.

Revised by S. Yoon. under supervision of Dr. J. Salomon.

Optimal transport problems consist of transferring mass from a series of supplies to demands. Within this paper, we focus on a specific instance of such problems, which is the unitary and balanced transport problem with concave cost. Unitary means that there is no mass left after the transportation is finishes; balanced means that there is an equal number of supplies and demands. On a one dimensional case of such problems, that is, such problem on a line, we define structures called chains. Using concavity of the cost function, we seek to define a local indicator on a chain, which will then be used to deduce the global optimum transport plan.

The algorithm allows a deeper understanding of optimal transport problems on a line. Although the time complexity of the algorithm has not been so well-understood, the prospect for local matching indicators to be used as a solution for the unsolved optimal transport problem in two dimensions or 1.5 dimensions (i.e. a graph) still remains.

# A Mathematically Constructed Simulation of Korean Traditional Painting Minhwa (Applied Mathematics)

S. Yoon.

International Baccalaureate (IB) Mathematics Internal Assessment (IA)

Within this research, I will reflect upon my personal experience of being raised by my grandmother, a traditional folk painter, and animate one of her paintings with a butterfly. The aim of the research is to create a mathematical model that is computationally light (that is, without use of the Navier-Stokes equations) and is aesthetically pleasing at the same time. Within the research, we use ellipsoids and Catmull-Rom Splines to generate the body and wings of the butterfly. Then, rotational motion using rotational matrices and translation using vectors will be applied to model the movement of the butterfly.

Although the model would be less effective in an academic context due to the many assumptions we made regarding flight and motion, it would still be of much use in the realm of media art, where mathematical or physical accuracy can be compromised for beauty.

# Who is the Greatest Athlete of All Times:

A Probabilistic Rating Model (Applied Mathematics)

S. Yoon., M. Chang., Y. Oh., S. Yoo.

2021 International Mathematical Modeling Competition South Korea Winner

To achieve this objective, we first identified factors that would affect determining the greatness of an athlete. These included: the number of matches, aggregated number of points, or strength of the opponent. Building the foundation to finding the GOAT of our chosen individual sport, we first examined the results of the 2018 Grand Slam tournaments. Giving each of the participants an individual rating, we were able to conclude that the great-est women’s tennis player of 2018 was Simona Halep, which was a natural result looking at the provided dataset.

To extend our model from the previous approach, we selected Men’s individual Epee Fencing for our Individual sport, and data was collected from the Olympics since 1932. We made substantial improvements upon the previous Tennis model, firstly developing a Performance Probability Distribution Function (PPDF) to represent an individual’s skill at that particular time. We then developed a Performance Indicator to quantify the performance of an athlete at each match. We used the Maximum Likelihood Estimate (MLE) method to estimate each PPDF that fitted its corresponding performance indicators. Suitable modifications were made to guarantee accuracy of the PPDFs, such as modifying the Rating Deviation of athletes during a period of non-observation by the model. (i.e. the 4 years between each Olympic)

# *Derivation and Generalization of the Law of Cosines onto the nth Dimension (Pure Mathematics)

S. Yoon.

International Baccalaureate (IB) Extended Essay

While the Law of Cosines is one of the most well-known and straightforward theorems in mathematics, would it also hold true in higher dimensions? While much research has been conducted on the generalization of Cosine or operators such as the cross product to any arbitrary dimension, there is a lack of research on bringing the Law of Cosines to a higher dimension. It must be noted that a couple of research has successfully generalized the theorem, but this report will prove the theorem without the use of the Divergence theorem yet with greater mathematical rigour. By using a basic property of determinants and the exterior product on a n dimensional simplex, we were able to successfully prove the Law of Cosines in any finite dimension. Finally, we were able to confirm that the law is true by applying it to a 4 dimensional simplex.

An implication of this theorem is the multidimensional Pythagorean rule for special cases of the simplex. While the theorem might not have practical applicability due to the vast amount of information about the simplex needed to apply the theorem to a simplex, introducing the exterior product is new in research on this theorem. Further research may be conducted to generalize the theorem to a hyperbolic simplex, combining the hyperbolic law of cosines and the multidimensional law of cosines investigated in this report.

Keywords: Simplex, Inner product, Exterior product, Law of Cosines, Determinant

# *The 2008 Netflix Challenge: Development of a Machine Learning Algorithm in Movie Rating Prediction (Applied Mathematics)

S. Yoon.

HarvardX Data Science Professional Certificate Capstone Project 2021

Inspired by the 2008 Netflix challenge, this report is an implementation of machine learning algorithms to tackle the same task that was designated to participants in the Netflix challenge. The primary aim of this project will be developing a model that predicts the rating of a given movie using the dataset Movielens. To assess our model, we will calculate the root mean squared error (RMSE) to calculate error. Through this report, we will construct multiple models to predict the ratings, and the model with the lowest RMSE will be used.

# *Application of the Central Limit Theorem: Predicting the 2016 Brexit Referendum Results (Applied Mathematics)

S. Yoon.

The aim of this research is to obtain a prediction of the 2016 Brexit referendum results using 127 datasets of pre-voting polls obtained from UK pollsters. After gathering the dataset, we filtered the data that would be within a time frame that makes the polls valid, and constructed a 95% confidence interval of our estimation using the Central Limit Theorem, which states that the sample mean of sufficiently many random, large, and independent samples created with replacement can be approximated by the normal distribution.

Overall, we were able to obtain a lower interval of 0.4521997 and an upper interval of 0.4693453, which did not include the real proportion 0.481, but did correctly predict the side that would win the referendum. The interval also did not include 0.5, meaning that its prediction is not sheer guessing.

##### 2020

# *†Research on Possibility of Ring Formation of 10199 Chariko by Partially Inelastic Collision (Physics)

S. Yoon.

Youth International Science Fair Bronze

Published on International Journal of STEAM

Korea Science and Engineering Fair Physics Bronze

KSEF International Earth Science Bronze

Discovered in 2014 to have two rings, the asteroid 10199 Chariklo became a counterexample to conventional theory that minor planets cannot have rings. However, how Chariklo obtained its two rings remained a point of speculation. The main goal of this research is to explore various possibilities of the mechanism of Chariklo’s rings’ creation. Two possibilities of particles outside Chariklo forming the ring were considered, yet rejected as external particles simply cannot drift into the rings’ orbits. Thus, we hypothesized that the ring formed through three steps: (1) an external object (e.g. a small asteroid) collided with Chariklo, making it spin faster. (2) Then, the high angular velocity ejected particles from the surface, and (3) these particles collided with one another to form a ring. We simulated perfectly and partially inelastic collisions.

In conclusion, perfectly inelastic collisions were not able to form a ring identical to those of Chariklo, but they were able to create a ring at a smaller radius. On the other hand, rings identical to those of Chariklo were able to be formed through partially inelastic collisions. It was further discovered that a very wide range of scenarios allowed ring creation through this mechanism, further validating the hypothesis.

# Use of the Greedy Algorithm and Reinforcement Learning to Devise Optimum Plant Conservation Plan (Applied Mathematics)

S. Yoon., M. Chang., Y. Oh., Y. Park.

High School Mathematical Contest in Modeling 2020

Biodiversity and conservation of different organisms is crucial to preserve the elaborate balance of the ecosystem. However, the current conservation endowments are not being distributed in the most efficient way due to the constraints imposed on the amount of funds. A new scheme is imperative for a more efficient conservation for the biodiversity of plants.

In this paper we aim to construct a model which takes into account all the factors the board may consider when selecting the plants to subsidize. To achieve this objective, we implemented the use of two different models. We inferred from the level of availability of funds that the current method was unsuccessful in conserving the most number of species while utilizing the minimum cost. Having a multi model approach was used to our benefit as we could compare the results from both approaches and calculate a value that was closest to the true optimal combination.

# Simulation to develop Optimum Store Layout during

# Black Friday to Minimize Store Damage

# (Applied Mathematics)

S. Yoon., M. Chang., Y. Oh., Y. Park.

International Mathematical Modeling Competition South Korea Representative, Honorable Mention

During Black Friday sales, consumers often rush to grab their desired products, causing much damage to the store in the process. In the worst case, such frenzied shopping practices even cause deaths or immense loss to the shopowners. Within this report, we aim to construct two models, that differ based on consumer knowledge of the store layout, to predict the damage that will occur during Black Friday. Upon the results, we will create an optimum store layout plan that will help to minimize such damage.

# Deriving Gravitational Acceleration from pendulum oscillation with rotational motion by the arbitrary amplitude method (Physics)

S. Yoon.

The pendulum is an object suspended from a pivot by a string that can be utilized to calculate gravitational acceleration. However, most works utilize the sine small angle approximation formula, hence producing much inaccuracy. Instead, energy conservation is utilized to derive an equation of the period for arbitrary amplitudes. Furthermore, the ideal model of a pendulum has a period only dependent upon length of the string and gravitational acceleration, one in the real world is affected by air resistance, rotation of the bob and other external factors. Hence, within this report, I aim to develop the arbitrary-amplitude method to take energy used for rotation into consideration, and derive the gravitational acceleration of Earth.

##### 2019

# The Formation of a Sierpinski Triangle by the Chaos Game and its Proof using Iterative Function System (Pure Mathematics)

S. Yoon.

Fractals, structures formed by iterations of identical shapes, can be found in numerous places including nature. The Chaos Game is an embodiment of this emergence of patterns from seemingly random actions. Using a dice and a piece of paper, one is able to generate the Sierpinski Triangle, a form of fractal structures. Within this report, I aim to empirically demonstrate the Chaos Game and explain why the Chaos game is able to produce such structures using the notion of iterative function systems.