Current and Past Undergraduate Research Projects

High-Impact Practices (HIPs) in The Mathematics Classroom: What Difference Do They Make?

Researched by: Makaila Valley and Circe Gedeon

Mentor: Dr. Mercedes Franco

Abstract: The purpose of this research project is to collect, digitalize, organize, and analyze data from course sections I have had the opportunity to teach with and without HIPs and to attempt to capture and compare the student experience in these varied settings. Aspects of the student experience that will be examined relate to student participation and satisfaction with the course (e.g. attendance, withdraw rates, student evaluations) and student learning outcomes (e.g. final grades, proficiency on final exam questions/topics covered on both traditional and HIP courses).

Mellin Integral Transform Method for solving Fractional Differential Equations

Researched by: Yahao Chen

Mentor: Dr. Lyubomir Boyadzhiev

Abstract: The project aims at acknowledging undergraduate students with the basic properties of the Mellin Integral Transform and its application to derivatives of arbitrary (not necessarily integer) order (fractional derivatives). The students will learn how to apply the Mellin Integral Transform for solving Fractional Differential Equations. The focus of this project will be on solving the fractional telegraph and the fractional diffusion-wave equations.

Minimal Log Discrepancies of Kawamata Log Terminal Singularities

Researched by: Siying Li

Mentor: Dr. Fei Ye

Abstract: Minimal log discrepancy is an very important birational invariant in algebraic geometry. Despite its
crucial role in minimal model program, it also proves to be very useful in studying global generations of adjoint line bundles. It was known that for a surface rational singularity of multiplicity m, the minimal log discrepancy is always less than 2/m. However, this upper bound in no means is optimal. In this project, we will study minimal log discrepancy of a special type of rational singularities, Kawamata log terminal singularities (klt for short). The goal is to find optimal upper bounds for this type of singularities in terms of the multiplicity and dual graphs of their minimal resolutions.

High-Impact Practices (HIPs) in The Mathematics Classroom: What Difference Do They Make?

Researched by: Makaila Valley and Circe Gedeon

Mentor: Dr. Mercedes Franco

Abstract: The purpose of this research project is to collect, digitalize, organize, and analyze data from course sections I have had the opportunity to teach with and without HIPs and to attempt to capture and compare the student experience in these varied settings. Aspects of the student experience that will be examined relate to student participation and satisfaction with the course (e.g. attendance, withdraw rates, student evaluations) and student learning outcomes (e.g. final grades, proficiency on final exam questions/topics covered on both traditional and HIP courses).

Predicting stock prices

Researched by: Nazif Alam

Mentor: Dr. Yusuf Danisman

Abstract: For this project, different machine learning techniques will be used to predict stock prices of S&P
500 Index and the NYSE Composite. The data from the webpage https://www.marketwatch.com/investing/index/spx
will be used to train and test the models. Based on the results a portfolio will be created.

Laplace Integral Transform Method for solving Fractional Differential Equations

Researched by: Rupakshi Aggarwal 

Mentor: Dr. Lyubomir Boyadzhiev

Abstract: The project aims to make undergraduates familiar with the basic properties of the Laplace Integral Transform, its application to derivatives of arbitrary (not necessarily integer) order (fractional derivatives) and as an immediate consequence of this, to get proficient on solving Fractional Differential Equations. The topic is one of the most intensively developing calculus areas due to a numerous applications of the derivatives and integrals of arbitrary order (FRACTIONAL CALCULUS) in medicine, chemistry, physics, engineering, finance, astronomy, fluid mechanics etc.

High-Impact Practices (HIPs) in The Mathematics Classroom: What Difference Do They Make?

Researched by: Marcos Navarro and Rhojay Brown

Mentor: Dr. Mercedes Franco

Abstract: The purpose of this research project is to collect, digitalize, organize, and analyze data from course sections I have had the opportunity to teach with and without HIPs and to attempt to capture and compare the student experience in these varied settings. Aspects of the student experience that will be examined relate to student participation and satisfaction with the course (e.g. attendance, withdraw rates, student evaluations) and student learning outcomes (e.g. final grades, proficiency on final exam questions/topics covered on both traditional and HIP courses).

The Construction of a Class Schedule Website via Django and GraphQL

Researched by: Brian Ryu

Mentor: Dr. Kwang Kim

Abstract: This project is a continuation of previous project in MA905.  In MA905, we developed some front side codes related to a class scheduling website for faculties using React Framework.  Now we are developing sever side codes using the Django framework which is used by Instagram.  One of most challenging parts of project is connecting Front-side React framework and Server-side Django framework efficiently.  We will achieve the goal by using GraphQL.  This approach is a recent development direction, so  the student needs to find how to implement it with GraphQL.  Eventually, we will provide GraphQL api of a class schedule server for a future mobile development.

Variations on the Cantor set

Researched by: Nikola Baci

Mentor: Dr. Danial Garbin

Abstract: The Cantor set is a fascinating example of a set of real numbers obtained by removing the open middle third part of a compact interval and then repeating this process ad infinitum on all the remaining compact parts. As a set of real numbers, the Cantor set is a perfect measure zero set, yet it is uncountable. Furthermore, all its numbers have a power series representation in base 3. We construct several generalizations of the Cantor set, including higher dimension Cantor variants. While all of the variants have measure zero, are uncountable and perfect sets, not all of them seem to possess a natural representation via power series.

Reconstruction Threshold for the Multi-State Hard Core Model

Researched by: Ziyan Lin

Mentor: Dr. Wenjian Liu

Abstract: Ramanan et al. proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the multi-state hard core model, the capacity C is allowed to be a positive integer, and a configuration in the model is an assignment of states from {0, . . . ,C} subject to the constraint that the states of adjacent nodes may not sum to more than C. This project will focus on the reconstruction bound of the multi-state hard core model on regular d-ary trees. The purpose of this project is to establish the distributional recursion and moment recursion by analyzing the recursive relation between nth and (n+1)th generation's structure of the tree. Then we will display the spectrum analysis of the Markov transition matrix and high degree discussion by applying the Central Limit Theorem and Gaussian approximation to approximate the moment recursion obtained from the first stage. Therefore, we expect to figure out the critical reconstruction threshold of the multi-state hard core model.

Integer Values of Generating Functions

Researched by: Danial Mohktari Sharghi

Mentor: Dr. Andrew Bulawa

Abstract: The generating function F(x) of a sequence a(n) is the power series having that sequence as its coefficients.  Let I denote the interval of convergence of F(x).  The generating function for the Fibonacci sequence has been shown to exhibit the following property: any rational number x in I is of the form x=a(n)/a(n+1) if and only if F(x) is an integer.  In my project I investigate other sequences that have this property, focusing in particular on the case where a(n) is an integer valued polynomial.

Using the Eiegenvalues of Transition Matrix of Markov Chains

Researched by: Jiawei Ren

Mentor: Dr. Haishen Yao

Abstract: The Markov chain is a type of random process, of which that the future states only depend on current state rather than the history of the process. Therefore the transition probability matrix is the essential part of the markov chain. According to the property of the markov chain, it can be classified to ergodic,   irreducible chain, and periodic chain. The states can be classified to transient and persistent.  We use linear algebra to find the eigenvalues and eigenvectors, so that we can classify the type Markov chains fast.

Implementation of N*N Tic Tac Toe Using REACT and Styled Component

Researched by: Brian Ryu

Mentor: Dr. Kwang Kim

Abstract: The final goal of our project is to create a class schedule website for the Math and CS department here at QCC. In order to do so, we need to test which web frame (Angular, React, Vue) is rich enough to make our website. We started with React by using a 3x3 Tic Tac Toe example in a React tutorial and further expanded on this concept into a nxn game. We used a dynamic layout system and used styled components. To create the dynamic win condition, we optimized by creating a winning verification algorithm. Came to the conclusion that React is very flexible and has many third party libraries for various uses. Decided on React as the web frame to use. The next concept we will be exploring is a calendar library and a Drag and Drop system to incorporate into our web development.

Big Data Information Inference on the Infinite Communication Tree Network of DNA Evolution

Researched by: Jiayao Sun

Mentor: Dr. Wenjian Liu

Abstract: The big data information reconstruction problem on the infinite communication tree network, is to collect and analyze massive samples at the n-th level of the phylogenetic tree to identify whether there is non-vanishing information of the root, as n goes to infinity. Although it has been studied in numerous contexts such as information theory, genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In this project, we focus on the form of signals' probability transition matrix corresponding to a classical DNA evolution model, the Felsenstein 1981 (F81) model, while further allow the existence of a guanine-cytosine content bias. The corresponding information reconstruction problem in molecular phylogenetics will be explored, by means of the refined analysis of moment recursion on a weighted version of the magnetization, concentration investigation and in-depth investigation on the resulting nonlinear second order dynamical system. Our purpose is to figure out under what condition of the base frequencies of adenine and thymine is the reconstruction solvable.

The Uniqueness of the Portfolio that Pays the Maximum Dividend Rate

Researched by: Zhenyi Wang

Mentor: Dr. Wenjian Liu

Abstract: Recently, Hansen and Scheinkman (2009) and Steve Ross (2013) have shown applications of principal eigenvalues and eigenfunctions and its appropriate generalizations to recovering market beliefs from option prices. The main purpose of this project is to show the uniqueness and calculate maximal dividend yields for financial portfolios under multiple driving state variables, by means of generalizing Ross' principal eigenvalue skills. Specifically, a financial portfolio typically pays dividend based on its value. We want to show that there is a unique portfolio that pays the maximum dividend rate while remaining solvent, under appropriate assumptions. The mentee will try to characterize the maximum dividend yield and the portfolio itself by the eigenfunction of a certain second order partial differential equation. Then in order to optimize the dividend yield, the mentee will analyze some basic properties of eigenvalues of this PDE with the knowledge of differential equations. Moreover, the mentee will develop an algorithm based on the preceding theory to calculate and simulate the portfolio paying the maximum dividend rate.

The Barrier Problems of Dice Sums

Researched by: Amy Zhang

Mentor: Dr. Haishen Yao

Abstract: In the real world, dice are polyhedra made of plastic, wood, ivory, or other hard material. Each face of the die is numbered, or marked in some way. Mathematically, we consider a die to be a random variable that takes on only finitely many distinct values. Usually, these values will constitute a set of positive integers {1, 2,..., n} in such cases, we will refer to the die as n-sided. A die is rolled repeatedly and summed. What can you say about the expected number of rolls until the sum is greater than or equal to some given number x? In other words, we are interested in the expected time of rolls to hit or pass a given number. The mentee is expected to apply the recursive method and generating function to give a delicate analysis of the sum barriers by estimating order of convergence, and obtain a general asymptotic result. Moreover, we will utilize MAPLE/MATLAB to simulate the rolling process.

Space Filling Curves

Researched by: Nikola Baci

Mentor: Dr. Daniel Garbin

Abstract: In this research project, we study space-filling curves. As an example of such curves, fractals are mathematical models that resemble various phenomena from nature. We see them every day in the tree branches, water bubbles, and even seashells such as the chambered nautilus. Understanding space filling curves gives us a better grasp of the world around us and allows us to apply this knowledge in other fields of science such as physics or biology. We start our research with the Cantor set from which we construct several variations. In this direction, we need to handle mathematical concepts such as measure of the set and the base n representation of numbers. The latter allows for generalizations on the Cantor set. Aside from the understanding the mathematics behind fractals, we also make use of computer graphics to graph such recursively defined structures.

Noncommutative Geometry of Finite Sets

Researched by: Gurpal Singh

Mentor: Dr. David Pham

Abstract: The notion of differentiation is central to calculus. An interesting problem is whether the idea of differentiation can be extended to functions on a finite set X. The subject of noncommutative geometry provides a solution to this question with the idea of a first order differential calculus (FODC). My research project is to study certain properties of FODCs as they pertain to group actions on X. Due to the severe time constraint, my talk will focus exclusively on introducing and motivating the idea of FODCs over F(X), where the latter is the algebra of functioins on X.

Click here for program information and requirements.