Fall 2005 GSS Schedule

Organizers: Shannon Purvis and Alex Engau

Date Speaker Topic Abstract
September 5 Tim Krueger ,
Stacey Faulkenberg,
Jang-Woo Park ,
Brian Faulkner ,
Dave Szurley
Life in the Department of Mathematical Sciences In this talk, graduate students with experience in the M.S. and Ph.D. programs in the department of Mathematical Sciences at Clemson will discuss some of their experiences at different levels and how they differed from their expectations. Tim Krueger is a second-year master's student at Clemson. Stacey Faulkenberg recently completed the M.S. degree and is starting the doctoral program. Jang-Woo Park is a Ph.D. student who has a half research, half teaching assistantship. Brian Faulkner is also a Ph.D. student who recently completed the qualifying exams. Dave Szurley graduated in August with his doctorate.

The main purpose of this talk is to promote candid discussion among new and experienced graduate students about the realities of graduate school here.
September 12 Jeremy Lyle Fall Colorings of Graphs A fall coloring of a graph is a partition $\Pi = \{V_1, V_2,\dots V_k\}$ of the vertices of $G$ into independent sets $V_i$, which also dominate the vertices in $G-V_i$. This coloring partition is the intersection between proper colorings and domatic partitions. Unlike many other variants of proper colorings, not all graphs can be colored in this way. In this talk, the subclass of chordal graphs is examined in order to determine the circumstances under which a chordal graph is fall-colorable.
September 19 Lin Wei Testing the Equality of Two Single-Index Models Comparison of two nonparametric regression models has been extensively discussed in the literature when the covariate is one-dimensional. Comparison problem remains open for completely nonparametric models when the covariate is multi-dimensional. In this talk we address this issue under the assumption that both models are single-index models (SIMs). First we shall briefly introduce some popular methods used in the literautre for such kind of a testing problem and the difficulty in high-dimension settings. Then we introduce single-index models (SIMs) which serves as a dimension-reduction technique in regression analysis. And we propose a test for assessing the equality of the mean functions of two SIMs and show its asymptotic performance. Partial simulation results will also be given if it is time-permitted.
September 26 Alex Engau From radiation therapy in cancer treatment to a new class of network flow problems Radiation therapy is a major tool in fighting cancer. In order to be successful, it is of crucial importance to find optimal strategies to carefully design a cancer radiation treatment (CRT) which guarantees that tumor cells are destroyed while avoiding harm to neighboring organs at risk. One of the many aspects which need to be taken into consideration is that the treatment and radiation time for the patient should be as short as possible. In this talk, I first explain the concept of intensity matrices and show how they are used for mathematically modeling the CRT problem by a suitable integer programming formulation.

In a second part, we study a related integer program which is characterized by a binary constraint matrix that can be partitioned into submatrices that are consecutive one. Based on linear programming relaxation and duality, the associated constraints are transformed into flow conservation constraints for a collection of networks that are related through bijections on subsets of their respective arcs. The resulting problem turns out to be a generalization of the minimum cost flow problem and requires the detection of a collection of flows that have identical values on corresponding arcs in different networks. After investigation of this new problem as part of my diploma thesis for the University of Kaiserslautern, I plan to address remaining difficulties with emphasis on several interesting problem characteristics.
October 3 Shannon Purvis The Births of a Sequence Partition functions often come up in Combinatorics and other areas of math. In this talk, we will be talking about a restricted partition function and the sequence related to it, $b(n)$. Calkin and Wilf used this sequence to construct a binary tree that gives an enumeration of the positive rationals. We will talk about some properties of this tree as well as how we can use the tree to give a partial answer to a question about the births of $b(n)$. All definitions needed for this talk will be discussed.
October 10 Gilbert Eyabi Exploring L(2,1) coloring on Certain Classes of Graphs An \textit{L(2,1) coloring} of a graph $G=(V,E)$ is a vertex coloring $f:V(G) \rightarrow \{0,1,2, \dots ,k\} $ such that $|f(u) - f(v)|\geq 2$ for all $uv\in E(G)$ and $|f(u) - f(v)|\geq 1$ if $d(u,v)=2$. The \textit{span} $\lambda(G)$ is the smallest $k$ for which G has an \textit{L(2,1) coloring}. A \textit{span} coloring is a coloring whose greatest color is $\lambda(G)$. An \textit{L(2,1)-coloring} $f$ is a full-coloring if $f:V(G) \rightarrow \{0,1,2, \dots ,\lambda(G)\} $ is onto and f is an irreducible no-hole coloring ($inh$-coloring) if $f:V(G) \rightarrow \{0,1,2, \dots ,k\} $ is onto for some $k$ and there do not exists a coloring $g$ such that $g(u)\leq f(u)$ for all $u\in V(G)$ and $g(v)< f(v)$ for some $v\in V(G)$. In this talk, we present some basic concepts in \textit{L(2,1) coloring} and investigate some of its properties as related to certain classes of graphs. The talk shall be accessible to all even those who never took any Graph Theory class before. We shall present some past results, a few results from my present work and some open problems. Masters students still wondering what to work on may find this talk useful. Come ready to participate as we shall color graphs together.

Keywords: L(2,1) colorings; inh-coloring; Holes; Channel Assignment Problems.
October 17 Fall Break
October 24 Yuko Palesch Application of Biostatistics to Clinical Trials All scientific research requires sound statistical foundation to make valid inferences from their results. Biomedical research, including clinical trials, provides fertile grounds where statistical theories are applied and where new statistical methods are developed. The presentation covers a brief introduction of clinical trials and highlights two statistical issues that arise in clinical trials - randomization and treatment of interaction effects in interim analyses. These issues are discussed in the context of two large clinical trials of ischemic stroke. The presentation will conclude with a description of three NIH-funded training grants in biostatistics available in the Department of Biostatistics, Bioinformatics and Epidemiology at MUSC.
October 31 John Chrispell Time Stepping Techniques to Find Numerical Solutions for PDEs There are many methods in which the solution to a differential equation can be advanced in time. Specifically in this talk we aim to discuss explicit, and implicit methods as well as the ability to use a weighted average of these methods to create a Crank-Nicolson type $\theta$-scheme. Also discussed will be a fractional step $\theta$-method that allows for operator splitting. Thus, by using a fractional step $\theta$-method convection can be decoupled from diffusion, or pressure from stress when solving various partial differential equations.
November 7 Shaina Race Configurations of Non-Attacking Kings on a Chessboard How many different ways can you place an arbitrary number of non-attacking kings on an N x M chessboard? Although easily stated, this problem has proven quite difficult to conquer. With applications in statistical mechanics and in physics, The Kings Problem is an intriguing combinatorial puzzle that is still largely open. This talk will focus on Fibonacci type recurrences, the approach of adjacency matrices, dominant eigenvalues and computational successes and boundaries.
November 14 Bryan Faulkner A Graphical Approach to Computing Selmer Groups of Congruent Number Curves A positive integer $n$ is called a congruent number if there exist a right triangle with rational length sides and area $n$. It can be shown that the elliptic curve defined by, $E_n : y^2 = x^3 - n^2x$, has infinitely many rational points if and only if $n$ is a congruent number. One common way of bounding the number of rational points on such a curve is to study its corresponding "Selmer group". In this talk I will give a description of some Selmer groups, $S_n$, in terms of certain graphs. A formula for the size of the Selmer group is found by finding the dimension of certain subspaces of the null space of the Laplace matrix.
November 21 Meng Zhao Consistent Linear Model Selection We examine the penalty term in linear model selection using penalized least squares. The rate of divergence of the penalty term for consistent model selection is discussed under a general error structure.

Key Words: Design variables; Linear Model; Squared Error Loss
November 28 Vijay Singh Equitable Efficiency in Multiple Criteria Optimization Multiple criteria optimization is an integral part of multiple criteria decision making. Multiple criteria optimization problems are mostly concerned with finding/approximating efficient solutions. Standard approach starts with the assumption of incomparability of criteria. However, in many applications, the criteria are uniform in the sense of scale; they represent the same physical aspect of the problem under study. Equitale efficiency was developed in the early 1990's (Kostreva and Ogryczak) to deal with such problems.

We discuss the concept of equitable efficiency, look at some standard solution procedures and apply them to compare/discuss solutions of certain problems.
December 5 Kanoktip Nimitkiatklai An Analytical Approach for a Lifetime Distribution The lifetime distribution in a thin homogeneous system is modeled as a Markov process on a thin lattice. The system evolves by adding edges at random times. When there is a path connecting the top and bottom layers, the system fails. The lifetime is then the first passage time to a particular subset of the state space. For large system, calculation of the lifetime is intractable. We show via extreme value theory that the lifetime is approximately Weibull distributed.