Instuctor: Prof. Weinstock
Every point in the plane can be represented as a complex number x + iy. A Mobius transformation is a transformation of the plane of the form Tz = (az + b)/(cz + d) where a, b, c, d are complex numbers such that ad - bc is not equal to zero. These transformations have interesting algebraic and geometric properties. They include the rotations and translations. The purpose of this project is to discover some of these interesting properties.
Prerequisites: MATH 2164 and a willingness to play around with complex numbers.
Instuctor: Prof. Shafer
The student will review ideas of controllability and eigenvalue assignment from linear systems theory in order to construct an explicit homotopy between an arbitray real matrix A with positive determinant and the identity matrix.
Prerequisite: MATH 2164
Instuctor: Prof. Shafer
There is an entire system of trigonometric formulas for the non-euclidean geometry of Bolyai, Gauss, and Lobachevsky, now called hyperbolic geometry. In the eighteenth century J. H. Lambert speculated that this hyperbolic geometry corresponds to the geometry of a sphere of imaginary radius. Yhe student will derive some of the trigonometric formulas of hyperbolic geometry by inserting complex quantities into the usual formulas of spherical trigonometry, and using facts of complex analysis.
Prerequisites: MATH 3181; MATH 3146 might be helpful.
Instructor: Prof. Papadopoulos
This project uses statistical techniques to analyze some accident data in Mecklenburg County.
Instructor: Prof. Dai, Fret 390G
Frame wavelet sets (or Frame sets) are very special sets on number line. These sets provide simple examples for frame wavelets which has potential applications in signal processing and data compressing. However, it is an open question to characterize the sets. In this project, a student will use sufficient conditions to check whether certain give sets are frame sets or not.
Prerequisite : Math 2241 with a grade of C or better.
Instuctor: Prof. Nabors
See instructor for details.
Instuctor: Prof. Wihstutz
Why does the proportion of sequences such that heads come up exactly n/2 times go to zero as n goes to infinity? In this project, we study the Law of Large Numbers, which explains such phenomena as occured in the coin tossing experiments.
Literature: Introductory chapter of Breiman, Probability
Prerequisite: Stat1220 (or higher), Math 1241, Math 1242.
Instuctor: Prof. Reiter
See instructor for details.
Instructor: Prof. Biswas
See instructor for details.
Instuctor: Prof. Houston
Through a series of exercises (involving concepts from linear algebra), the student will prove the fundamental theorem of algebra: every polynomial with complex coefficients has a complex root.
Prerequisites: MATH 2164
Instuctor: Prof. Diao
The student will learn some basic knot theory, especially about the crossing number and the connected sum operation of knots. He or she will then apply the knowledge to find as many zero deficiency knots as they can. This could possibly a multi-student project. See Prof. Diao for details.
Prerequisite: Modern Algebra Math 3163
Instructor: Prof. Cifarelli
The student will use Geometer Sketchpad to investigate problems and theorems in geometry.
See Prof. Cifarelli for prerequisites and details.
Instructor: Prof. Sundaram
See Prof. Sundaram for details.
Prerequisites: Stat 1220/1222, Stat 3128 /Stat 2122. Willingness to learn software!
Instructor: Prof. Hetyei
Project description:
There are several matrices associated with graphs, such as the adjacency matrix and the incidence matrix, and there are problems in economics presented as a matrix problem which may be dealt with more easily when considering the structure of the underlying weighted graph.
This project will explore the interactions between the world of graphs and the world of matrices. Depending on the taste and background of the student, two options will be available. For the algebraically inclined, the study of the graph-theoretic meaning of eigenvalues and subdeterminants in matrices associated to graphs is offered. Those having a weaker algebra background and more interest in discrete mathematics, will focus on algorithms processing information on weighted graphs.
Prerequisite: Math 2164, Linear Algebra
Instructor: Prof. Johnson
Includes the history and the mathematical development.
Students need to have studied some non-Euclidean geometry.
Instructor: Prof. Johnson
How Newton and Leibniz came to understand the relation between differentiation and integration.
Prerequisite: The calculus sequence. Co-requisite: History of Mathematics.
Instructor: Prof. Lambert
All the problems listed below can be successfully completed using the prerequisites listed below.. Part of the project will be a review of the terminology and concepts from this prerequisite material. Let V be the n dimensional vector space consisting of all n-tuples of complex numbers, and let L(V) be the collection of all linear transformations from V to V.
Problem 1: How can one characterize the linear functionals on V; that is, the linear transformations from V to C ?
Problem 2: How can one characterize those members of L(V) whose rank (dimension of range) is less than or equal to k ?
Problem 3: How can one characterize the ideals within L(V) ?
Problem 4: How can one characterize those subrings R of L(V) which are transitive ?
Prerequisite: Math 2164
Instructor: Prof. Lambert
The first task in this project is to establish properties of a vector space of infinite dimension for which the infinite series corresponding to the finite sum in the inner product of a finite dimensional vector space converges. Call this vector space V. Then the inner product on V leads to a norm on V, which makes V into a metric space. The second goal of this project is to determine which linear transformations from V to V are continuous (with respect to this metric).
Prerequisite: Math 2164 and some knowledge about metric space.