Applied Mathematics
Graduates from our mathematics program make an immediate impact in a variety of careers. Our alumni are working as actuaries, computer programmers, engineers, hospital administrators, and secondary teachers.
For those choosing to continue their education, the choices are just as varied. Recent graduate school opportunities for our mathematics majors include:
- applied mathematics (University of Colorado)
- computer science (Arizona State University)
- dental school (Case-Western Reserve University)
- hospital management (University of Washington)
- industrial engineering (University of Washington)
- law school (Gonzaga University)
- mathematics (University of Iowa)
- Water Resources (University of Colorado)
Honors Thesis
Traveling Wave Excitation of Soft X-Ray Lasers
- Steffan Francischetti, Class of 2008, Mathematics.
This honors thesis covers the physics of lasers from simple operation to mode locking lasers that are a key part of the research done in this thesis. The thesis then details the design and use of a key optical component called a “Step Mirror” which governs the timing of plasma excitation. Finally the thesis explains how mode locked lasers are used to align the step mirror using second harmonic generation in order to perform the ultra precise alignment necessary for the step mirror to be used in the soft x-ray system.
Heuristics for the Knapsack Problem
- Claire Gilbertson, Class of 2008, Mathematics/Computer Science/Spanish.
We explore the knapsack problem where the goal is to maximize the value of packed objects for a certain container. The knapsack problem is NP-complete which means the time needed to solve it exactly grows exponentially as the size of the data set increases. Due to the infeasibility of using algorithms which find all possible solutions, we can use heuristics which are methods used to estimate a solution rapidly. We create a java program to test seven heuristics on 21 data sets. The seven picking algorithms include four basic greedy algorithms, two backtracking methods, and one random picker. We use data sets from a standard library, and create our own data sets with specific characteristics. The greedy algorithm which sorts by largest utility and the one of the backtracking methods perform the best.
Using Circuitry and Computer Analysis for Modern Cryptology
- Pete Lavallee, Class of 2008, Mathematics/Civil Engineering
Modern cryptology has allowed business to expand at an incredible rate by allowing people to transact business securely online. A common encryption process for encoding and decoding information utilizes a linear feedback shift register (LFSR) to generate a quasi-random string of binary digits. This “key stream” is then combined with the digits of the original or “plain text” using the exclusive-or logical operation, to create the encrypted or “cipher text.” Using modern circuit components, a circuit to implement this type of encryption was constructed and evaluated. The circuit’s operation was then simulated by the specialized computer program Cryptographic Analysis Program (CAP). Using CAP allowed for multiple methods of generating quasi-random bit strings: both the LFSR and the method of cellular automata were examined here. After these two key stream generators were analyzed, some observations were made concerning the general security of each.
Inbreeding Curvature: A Geometrical Approach For Describing A Population’s Tendency To Inbreed Or Outbreed
– John Riggs, Class of 2008, Mathematics/Biology.
Isolated populations of organisms face an imposing challenge: the challenge of inbreeding. For some species, inbreeding leads to the depression of vital population statistics--birth and survival rates—but others overcome this obstacle to form well-adapted groups. But how can researchers or conservationists know which path a population will take? To answer this question, I approached the problem of inbreeding from a geometrically oriented, mating-pair based perspective. I develop a perspective to discuss populations of organisms with regard to their mate choice patterns, and link these patterns to determine how populations will respond to inbreeding. I tested my hypotheses on two data sets from a Costa Rican population of the Ocellated Antbird, Phaenostictus mcleannani, and discuss how best to analyze the results of my calculations.
Pulverized Pavement Performance Evaluation
– Tiffany Rochelle, Class of 2008, Mathematics.
In the past 20 years, highway agencies began rehabilitating pavements using newer technologies such as pavement pulverization. Pulverization can be more cost effective and more environmentally friendly than pavement reconstruction. Pavement pulverization consists of pulverizing and blending the full thickness of the plant mix surfacing and a predetermined portion of the crushed base course to provide a homogeneous base material. The pulverized base course is then overlaid with new plant mix surfacing resulting in a new roadway. The purpose of this study is to analyze the actual performance of pulverized projects on Montana’s roadways. The evaluation determined how well pulverized pavements are performing and compared the performance of these pavements to conventional flexible pavements. The pavement age, traffic level, subgrade type, project location, annual precipitation, plant mix surfacing, and the Recycled Asphalt Pavement to crushed based course ratio were analyzed to identify factors which contribute to the performance. Multiple linear regression analyses were conducted to describe the relationship between these factors and the pavement. The pulverized pavements were also compared to the performance of adjacent non-pulverized pavement and the performance of Montana’s overall road system. It was found that among the characteristics in this study, the RAP/CAC ratio, age, annual precipitation, and project location were the most important factors affecting the performance. It can be concluded that pulverized roads in Montana are performing satisfactorily and equivalent to non-pulverized pavements. Based on cost and performance it is recommended that the MDT continues to build pulverized pavements.
A Solid Background
At Carroll, the Computer Science, Civil Engineering, and Mathematics majors focus on the skills you need to succeed at the next level.
Our emphasis on real-world problems, through group and independent projects, and the effective communication of results helps our graduates when looking for graduate school opportunities and jobs.
Independent Explorations with Faculty Members
One of Carroll's requirements to graduate with Honors is the writing and presentation of a thesis during the student's senior year. Within the Mathematics, Engineering, and Computer Science Department, students have the opportunity to work with faculty of varied backgrounds and interests. At left are thesis abstracts of our 2008 Honors graduates, and here is a list of recent thesis titles.