Undergraduate Research Projects
These projects were done at the University of St. Thomas, St. Paul,
MN in their summer research program.
- Stability of Linear Structures - Summer 2000
Advisor: C.
Shakiban
Abstract: In this paper we will study the vibrations of structures made of bars and determine when such structures are stable under moderate stress. Linear systems of equations represent these structures. The structure's stability is then discussed using linear algebra. We will investigate the stability of several structures, including those made in the shape of the platonic solids. Some applications of these calculations are then considered. Applications are found in areas such as architecture, robotics, art, and toy making.
Paper
Power Point Presentation
- Stability of Nonlinear Structures (cont. of previous summer) - Summer 2001
Advisor: C. Shakiban
Abstract: In any field of study idealizations make the world easier to deal with and easier to understand. It is common to use generalizations and simplifications to understand basic principles and get a fuzzy image of what is being dealt with. This is a necessary first step in attaining a sharper picture of the world, but there comes a point when we have to leave utopia and come back to the real world. This research project deals with determining the stability of structures, which are by nature nonlinear. Nonlinear systems, however, are messier and more difficult to work with. This paper approaches the problem in a couple of different ways.
Paper (note: some of the imported
data in the file has become corrupted and hence unfortunately illegible)
Power Point Presentation
(note: some of the imported data in the file has become corrupted and hence unfortunately illegible)
- Symmetrical Musical Pieces: frieze patterns in music - Summer 2001
Advisor: C. Shakiban
Abstract: Symmetry in decorative arts has played an important role throughout the ages. Many cultures have used various symmetric groups to decorate vases, bracelets, borders, etc. Modern artists such as M.C. Escher have also utilized the beauty of transformations. These patterns are pleasing to the eye and mind because of their symmetric and infinite properties. There are seven such frieze groups (borders, strips) and seventeen wallpaper patterns. The next area to explore within the realm of symmetric groups is their application to a different form of art. Music is naturally intimate with and enhances math. This paper will explore the application of frieze patterns in composing music.
Paper
Power Point Presentation
- Exploration into Mathematical Logic (history and foundations) - Summer 2002
Advisor: J. McLean
Abstract: Logic is the science of reasoning. It is a wide and varied field of study that is intimately connected with a variety of areas in mathematics, science, and philosophy including, but not limited to, algebra, epistemology, foundations of mathematics, and symbolic logic. This paper will discuss some of these connections in the manner of the above quote. First, it is worthwhile to start with the ontology of numbers, which leads to one possible definition of numbers. Transfinite and infinitesimal numbers will then be considered along with some questions that arise from their existence. How we know the nature and existence of numbers relates to how we can know scientific truths. The purpose of this paper is to present some basic, important topics in logic and philosophical thought.
Paper (for my Summa examination)
Power Point Presentation