MATH 498 Brief Description of Some Possible Topics for the Senior Assignment Fall 2007

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1 MATH 498 Brief Description of Some Possible Topics for the Senior Assignment Fall 2007 Dr. Marcus Agustin Office: SL Simulation Study of Software Reliability Models Prerequisites: STAT 480a&b or a solid background in STAT 380; programming background a plus 2. Estimating the Reliability of a Series-Parallel System Prerequisites: STAT 480a&b or a solid background in STAT 380; STAT 484 a plus 3. Parametric Models Under Censoring Prerequisites: STAT 480a&b or solid background in STAT 380; STAT 484 a big plus 4. Homogenous and Nonhomogeneous Poisson Processes Prerequisites: STAT 480a&b Dr. Zenia Agustin zagusti@siue.edu Office: SL Goodness-of-fit Testing with Applications in Reliability and Survival Analysis Prerequisites: STAT 480a&b or strong background in STAT Topics in Actuarial Science Prerequisites: STAT Advanced Topics in Theory of Interest Prerequisites: MATH Topics in Regression Prerequisite: Stat 380 or Stat 482 Dr. Song Foh Chew schew@siue.edu Office: SL Decision making under uncertainty: The use of utility functions This project looks into properties of utility functions, and studies the use of utility functions in the process of decision making under uncertainty. Prerequisites: OR 441 and adequate knowledge of STAT 480a&b 2. Queuing Theory: How long do I have to wait in line? The project investigates various queuing systems and their applications. Prerequisites: OR 441 and adequate knowledge of STAT 480a&b 3. What are random graphs? The project looks into why graphs are random and their applications. Prerequisites: MATH 423 and adequate knowledge of STAT 480a&b 1

2 4. Combinatorics: The counting problem The project studies how to count the number of ways of choosing r of n objects under various circumstances, and investigates how polynomials can be used for the counting problem. Prerequisite: MATH 423 Dr. Krzysztof Jarosz kjarosz@siue.edu Office: SL Set Theory: cardinals and ordinals, basic paradoxes and some ZF Prerequisites: Math 223 with at least solid B, strong interest in foundation of math 2. The Banach-Tarski paradox Prerequisites: Math 223 and Math 320 with at least B, Math 350; strong interest in foundation of math 3. Applications of quaternions Prerequisites: Math 223 and Math 321 with at least B 4. Fixed point theorems and applications Prerequisites: Mah Conformal mappings Prerequisites: Math 223 with at least B, Math 350, and Math 451 Dr. KoungHee Leem kleem@siue.edu Office: SL Iterative methods for solving linear systems. Prerequisite : MATH 321, MATH 465, MATH Eigenvalue problems. Prerequisite : MATH 321, MATH 465, MATH Preconditioning Techniques for Iterative Methods. Prerequisite : MATH 321, MATH 465, MATH 466. Through these projects, students will learn various numerical methods and how to implement these using MATLAB. Also, students will learn how to analyze the corresponding numerical results. Dr. Urszula Ledzewicz uledzew@siue.edu Office: SL Simulation studies of mathematical models for cancer treatments There are several models that have been proposed. This will be a nice interdisciplinary topic and is not as difficult as it may sound. Prerequisites: Knowledge of differential equations, MATH 305, and MATLAB or Mathematica are required 2. Stability of linear differential equations Prerequisites: Knowledge of differential equations, MATH 305, and linear algebra, MATH 321, are required. 2

3 Dr. Chunqing Lu Office: SL ODE related: (a) Population growth, Harvesting, Rockets (b) Mechanical vibrations, electrical networks (c) The pendulum motions, Population of interacting species (d) Why does the inverse Laplace Transform work? (e) Use the Phase Plane to solve some second order ODEs. Prerequisite: MATH Linear Algebra related: (a) LU factorization and its application (b) Eigenvalues and eigenvectors applied in ODE (c) About the determinant of a matrix (d) Apply Linear Transformation in Calculus Prerequisite: MATH PDE related: (a) Fourier Theorem with some applications in Calculus (b) Use Mathematica or Matlab solve some PDE and plot the solutions (c) Uniqueness of solutions for the three basic types of 2nd order linear PDEs. Prerequisite: Math Computer related: (a) Instability of numerical differentiation and stability of numerical integration (b) Instability of numerical solutions for some ODEs (c) Instability of computing eigenvalues of some matrices (d) Lagrange and Spline interpolations: compare their convergence (e) Chaos of some simple mappings Prerequisite: Math 465 or Math Some topics related to Nun-Euclidean geometry: The Pythagorean Theorem in Hyperbolic geometry, the half plane model and the disk model Prerequisite: Math Other applied topics in Mathematics 3

4 Dr. George Pelekanos Office: SL Discrete Fourier Transform in data interpolation We will derive the discrete Fourier Transform, we will use it to interpolate a data set and we will examine few of its applications. 2. Edge detection using linear spatial filtering Edges are one of the gray-level discontinuities in a digital image, an element of image segmentation. Segmentation breaks an image into its basic parts and should stop when the objects that we are analyzing have been isolated. 3. JPEG image compression We will demonstrate how JPEG compression works, why it is used, and how effective it is. In particular, we will use an actual digital camera picture as a demonstration and we will observe that the image is compressed with barely any visual difference. The key tool here will be the Discrete Cosine Transform. 4. Differentiation matrices We will derive second and fourth order differentiation matrices. In a situation where we have a discrete set of points rather than a continuous function we can use differentiation matrices to obtain the derivative directly. Otherwise we will have to construct an interpolating polynomial and then find its derivative. 5. Image Metamorphosis Among the many ways to manipulate an image is a technique known as morphing. Image morphing is a special effect that transforms or morphs one image into another. We will achieve that by using affine transformations and MATLAB. 6. Location of a known object using GPS from N sensors We will explore the accuracy achieved by using GPS sensors to determine the location of an object of interest. Prerequisites: MATH 250, MATH Computer Graphics and Animation Computer graphics is the generation and transformation of pictures on the computer. This is a hot topic with important applications in science and business as well as in Hollywood (computer special effects and animated films). In this project we will show how matrices can be used to perform certain types of geometric operations on objects. In our illustrations objects will be two-dimensional. 8. The Conjugate Gradient Method We will explore the Conjugate Gradient Method (CG) originally developed by Hestenes and Stiefel as a direct method to solve an nxn positive definite linear system. 4

5 9. Using Vector Spaces for Information Retrieval As the internet grows, massive amounts of data need to be efficiently indexed. One way to achieve this is via the construction and utilization of a vector space model. Singular Value Decomposition will be extensively used) 10. Cubic Spline Interpolation (QR decomposition and Real world data is usually difficult to analyze. Any function which would effectively correlate the data would be difficult to obtain. Hence, the idea of the cubic spline is developed. Using this approach, a series of unique cubic polynomials are fitted between each of the data points, with the condition that the curve obtained be continuous and appear smooth. These cubic splines can then be used to determine rates of change over an interval. 11. Circulant Matrices and Polynomials We will provide a simple and unified approach to the solutions of quadratic and cubic polynomial equations. Prerequisites: MATH 250, MATH 321, MATH Cryptography We will discuss a method of encoding and decoding messages using matrix systems and modular arithmetic. We will also show how Gaussian elimination can sometimes brake down an opponents code. Prerequisites: MATH 250, MATH 321. Possibly some MATLAB programming will take place. In addition the student working on this project should be willing to learn some elementary number theory. 13. Solving Dirichlet Problems with Conformal Mapping Very often the difficulty in solving a Dirichlet problem is due to geometry of the region on which the problem is stated. Conformal mappings can be used to transform a region to one on which the ensuing boundary value problem is easier to solve. Prerequisites: MATH 250, MATH 321, MATH 464, MATH Numerical Solution of Fredholm Integral Equations: Dr. Steven Rigdon srigdon@siue.edu Office: SL Using Mathematica to compute power and sample sizes Prerequisites: STAT 480a&b and a knowledge of (or willingness to learn) Mathematica 2. Using the Kalman Filter to analyze successive sample surveys, for example, polls leading up to an election. Prerequisites: STAT 480a&b and the ability to do scientific programming 3. Using dynamic graphics, animations, and interactivity to present topics in calculus / statistics / differential equations / geometry Prerequisites: Advanced course work in some area (either MATH or STAT) and the ability to write programs in an object oriented programming (OOP) language 5

6 Dr. Edward Sewell Office: SL Deterministic Operations Research Prerequisite: OR Discrete Optimization Prerequisite: OR 440 or MATH Graph Theory Prerequisite: MATH 423. Dr. Myung-Sin Song Office: SL Wavelet image compression Wavelets are used in JPEG2000, FBI fingerprint compression and in many other compressions in industry. Demonstration of wavelet image compression using different wavelets. The mechanism of wavelet image compression, and how the performances differ depending on the properties of different wavelets will be studied. The actual digital image will be used to be compressed. Prerequisites: MATH 321, MATH 450, MATH 451, MATH 466, CS 330 (some computer programming background needed) 2. Edge detection in images Wavelets are good at detecting certain types of edges in images, but curvelets which came out to overcome the problems wavelets have do better job in certain cases. The reason behind such differences will be studied with different types of digital images. Prerequisites: MATH 321, MATH 450, MATH 451, MATH 466, CS 330 (some computer programming background needed) 3. Wavelet noise removal Some digital images have noises that worsens the quality of the images. Wavelets enables us to detect those noises and we then can remove most of the noises. Then reconstruction of the digital images can be done yielding better quality images. Noise removal from the noisy images will be demostrated using wavelets. Prerequisites: MATH 321, MATH 450, MATH 451, MATH 466, CS 330 (some computer programming background needed) 4. Google Matrix Google is one of the most used search engines. It uses matrices such as PageRank matrix to effective search for an input. The network searched on World Wide Web is represented as graphs and the records of it as matrices. From these PageRank matrix is computed. It involves various theories from linear algebra. The study of this PageRank matrix on how it works will be done using various theories learnt in MATH 421, and MATH423 especially. Prerequisites: MATH 421, MATH 350, MATH423, MATH451 6

7 Dr. Adam Weyhaupt Office: SL 1316 Dr. Weyhaupt s web page, aweyhau/teaching, has a little bit of additional information on these topics. 1. Mathematics of Sudoku An interesting new paper relates Sudoku and graph theory in an interesting way. They obtain some nice results about the clues in a Sudoku puzzle and prove some interesting facts that have not been obtainable using other methods. Prerequisite: No course would be an explicit prerequisite. 2. Geodesics on polyhedra A geodesic on a surface is, more or less, the shortest connection between two points. (That s a lie, but it s true for small pieces of a geodesic, and we can sort out the details when we meet). Geodesics in the plane are lines, and geodesics on the sphere are great circles. What are geodesics on a 3-dimensional polyhedron? What properties do they have? Prerequisite: It would be preferable that students would have had differential geometry (437), but almost no one will meet this preference, so we can develop the necessary machinery. Math 350 is required. 3. Discrete geometry I m interested in learning more about discrete geometry. In differential geometry, you usually study smooth (or at least continuous and differentiable) curves and surfaces. What if you decided to approximate a smooth surface by points and triangles? Do any of the usual geometry theorems have analogs here? Prerequisite: It would be preferable that students would have had differential geometry (437), but almost no one will meet this preference, so we can develop the necessary machinery. Math 350 is required. 4. Calculus of Variations Calculus is the study of change. The calculus of variations is, roughly, the study of optimal solutions to certain problems that involve variations of some object. For example, one could cast the problem of determining the shape of a planar region with fixed perimeter and largest possible area (a circle) as a problem in the calculus of variations. Prerequisite: Math Minimal surfaces A minimal surface M is a mathematical model of a soap film. Small pieces of these surfaces have the smallest possible area of any other surface with the same boundary. An introduction to the subject can be accessible without too much technical knowledge necessary. Prerequisite: Math 350 is required, Math 305 and 437 and 451 are preferred but not necessary. 6. Point-set topology Point set topology is the study of the closeness of points in a set. In many ways, it generalizes the notions from the real line of open intervals and limits. In Math 350 you learn some point-set topology, but there s lots more to consider. Prerequisite: Math The mathematics of alien tiles Alien tiles is an amusing game one can play on the web that is kind of like a combination of Minesweeper and Sudoku. Prerequisites: Math 320 and 321 7