# Undergraduate Research Projects

The Department of Mathematics and Computer Sciences offers undergraduate majors an opportunity to work closely with a professor on a project. The project might be publishable research, but it could also just be an interesting project that introduces students to some new mathematics. In any case, the project would be worthy of a student presentation at a conference.

The duration of the project might be short or long: a student could work for more than one year on a long-term project, or less than a year on a short-term one.

This is not clerical work. The student will not receive payment or course credit for the project. This will be interesting mathematics that leads to something that a student can write or present. A student could end up with a finished project to hand to a prospective employer or graduate school admissions officer; or just to keep and say, "I did this."

If you are interested in working on a project, please contact the professor listed. You can also discuss project ideas you have with any faculty member, and that faculty member can suggest someone who will work with you.

## Potential Projects

### Simple Differential Equations and the Growth and Decay of Ice Sheets, Dr. Rick Adkins

In this project, we will re-visit and expand upon a project of John Imbrie of the University of Virginia and his daughter Katherine matching periodic earth temperatures reflected in ice cores to when the earth's axis tilt wobbled and the planet's relative annual position to the sun. We will investigate how key aspects of the ice-age record (such as shifts in dominant periodicities) follow from simple ordinary differential equations capturing the essential physics of the growth and decay of ice sheets.

** Prerequisite:** A student working on this project should have successfully completed MATH 241 Differential Equations.

### 'Area' and 'Length' Application Using Green's Theorem and More, Dr. John Chrispell

The idea of this project is to create an app for an Android or Apple device that allows the user to calculate the area of a selected region using a touch interface. For instance, an application user could calculate the approximate size of the state of New York. To calculate New York's area a map interface would be used to collect user-specified points around the boundary outlining the state and the application would compute the area using a line integral.

The application would be even more useful on a smaller scale. Computing areas of plots of land or towns by again selecting points around the region. As an added feature computing the length of a path selected on an image would also be a neat mathematical feature to add.

** Prerequisite:** The mathematics needed would be based on MATH 225 Calculus III with the use of Green's Theorem and arclength being the core concepts.

### The Gamma Function, Dr. Alfy Dahma

The theory of the gamma function was developed in connection with the problem of generalizing the factorial function of the natural numbers. The gamma function is defined as a definite, improper integral, and the notion of factorials is extended to complex and real arguments. This function crops up in many unexpected places in mathematical analysis, such as finding the volume of an n-dimensional "ball." In this project, we develop and explore the basic properties of this function.

** Prerequisite:** MATH 225 Calculus III preferably, but you can get by with a good understanding of infinite series and integration techniques used in MATH 126 Calculus II.

### Problems In Mathematical Analysis, Dr. Alfy Dahma

Are you interested in learning skills and techniques required to do research in mathematical analysis? If so, I have just the project for you. I've compiled a list of interesting problems in many areas of mathematics: calculus, set theory, linear algebra, topology, advanced calculus, general elementary analysis, etc. These problems range in difficulty level, and many of the solutions will require you to do some of your own independent research. Solutions to some of these problems would be appropriate for presentation at an IUP colloquium or possibly a conference sponsored by a professional mathematical organization.

** Prerequisite:** Most problems require experience with writing proofs, but there's no harm in trying.

### Managing the Grading Process of Open-ended Questions on Nationwide Test, Dr. Yu-Ju Kuo

Have you ever taken a nationwide standardized test, such as an AP exam? Do you wonder how and where they are graded in a limited amount of time? This is a type of logistic problem occurring in business settings. Through this project, students will build a simplified *Arena* (simulation software) model for the above system, conduct sensitivity analysis for various resources and cost factors, propose reasonable ways to improve the system, and verify the proposed methods.

** Prerequisite**: No previous knowledge of *Arena* is required. Students should be comfortable playing with the software and will investigate the model in the directions of their choice.

### Least is the Best, Dr. Yu-Ju Kuo

A common concern in industry is optimization: minimizing the cost, maximizing the profit, optimizing resource utilization, and so on. Students learn basic optimization techniques in calculus courses. But to what is it applied? What if the objective function is non-differentiable? What if variables are discrete? In this project, students can choose their preferred "no-so-nice" application and explore heuristic approaches to estimate the optimum and the optimizer.

** Prerequisite**: A student working on this project should have successfully completed MATH 225 Calculus III, MATH 171 Linear Algebra, and COSC 110.

### Outdoor Mobile Robot Navigation, Dr. Terry Fries

Many shortest path algorithms exist, but most assume a relatively uniform surface such as a floor in a building. In this project, we address the effects of varying terrain conditions on selecting the best path. We will represent terrain conditions such as sloped, sandy, and rocky in varying degrees with fuzzy sets. These conditions will influence the path an autonomous mobile robot may take. Applications for this research include planetary exploration, investigating a hazardous environment such as volcanoes, and search and rescue operations.

**Prerequisite**: COSC 405 Artificial Intelligence preferably, but you can quickly learn the aspects of AI you will need for this project. Also, you will need good programming skills from COSC 310 Data Structures.

### Network Intrusion Detection using Big Data, Dr. Terry Fries

The amount of data available in network traffic is so large that big data methodologies must be used to study it. In this project, we will use a variety of big data techniques to evaluate network traffic to determine signatures of intruders. Polymorphic intruders are even harder to identify because their signature is constantly changing. We will use artificial intelligence techniques such as fuzzy clustering to categorize intruder signatures and identify network intrusion as they occur.

**Prerequisite**: Good programming skills from COSC 310 Data Structures. Knowledge of artificial intelligence techniques from COSC 405 are beneficial but not necessary.

### Anomaly Detection, Dr. Waleed Farag

Dr. Waleed Farag uses student researchers in his Anomaly Detection research project.