Math Awareness Month Symposium
Friday April 5, 2013, 9:00am - 12:30pm, 256 DSC
Mathematics Awareness Month is held each year in April. To celebrate Math Awareness Month, the UNO Math department holds a symposium in which students from the Kerrigan Research Minigrant Program present the results of their work.
The topic for Mathematics Awareness Month 2013 is Mathematics of Sustainability.
Schedule of Events
9:00am - 9:05am,
Dr. Mahboub Baccouch, Chair, MAM Committee
Welcome
9:05am - 9:30am,
Amanda Ludes, (Advisor: D. Matache)
Average Influence of Threshold Boolean Functions and Heterogeneous Kauffman Networks
Abstract
+
For this project we wish to find a formula for the average influence of threshold Boolean functions and then use it to model the influence and find the critical values for combinations of different types of Boolean functions in a heterogeneous random Boolean network. Besides threshold functions which are the main focus, other functions considered in the heterogeneous model fall in the categories of: canalizing functions with one canalizing input, canalizing functions with two canalizing inputs, and biased functions. While recent research has focused on finding phase transitions for homogeneous network ensembles with just one type of underlying Boolean function, real biological networks are comprised of a number of different types of Boolean functions, potentially with links between nodes of different classes. Here, we find the average influence of functions for the heterogeneous network in terms of the parameters for the network. This will allow us to find the critical condition, which divides the phase space into ordered and chaotic regions. We plot this influence and use it to find the critical condition in phase diagrams for both networks consisting only of threshold functions, and networks of ensembles of the four different types of functions.
9:30am - 9:55am,
Celeste Mott, (Advisor: D. Matache)
Simulation of a heterogeneous boolean network
Abstract
+
A Boolean network is a system of nodes linked to one another, each node capable of two states, active or inactive. As time passes, each node examines the state of its input nodes and determines its own state according to a specific rule, or Boolean function. As the network evolves, it can either regress into a cycle of repeating states or erupt into chaos under certain conditions. This paper examines a computer simulation that creates a Boolean network of n nodes where each node has a connectivity (number of input nodes), k, drawn from a uniform distribution, and one of four general function types-canalyzing with one canalyzing input, canalyzing with two canalyzing inputs, threshold, or bias functions. The goal is to estimate the probability that the network returns to its original behavior after a small disturbance. The main purpose of this computer simulation is a means of reference. Since the program has a great number of editable parameters, one can use the simulation to estimate how modifying a parameter will affect the stability of a network or compare that with mathematical models or real data against the simulation.
9:55am - 10:20am,
Balagbo Lawson, (Advisor: M. Baccouch)
A Discussion on the St. Petersburg Two-Envelope Paradox
Abstract
+
The discontinuous Galerkin (DG) finite element method provides an appealing approach to address problems having discontinuities, such as those that arise in hyperbolic conservation laws. In this talk, we study the global convergence of the implicit residual-based a posteriori error estimates for a DG method applied to one-dimensional linear hyperbolic conservation laws. We use a recent superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+3/2, when p-degree piecewise polynomials are used. Finally, we prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity indices are proved to be p+5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.
10:20am - 10:45am,
Joonhee Lee, (Advisor: M. Baccouch)
Superconvergence and a posteriori error estimation for the local discontinuous Galerkin method for solving convection-diffusion problems in one space dimension
Abstract
+
The discontinuous Galerkin (DG) method is a class of finite element methods using discontinuous piecewise polynomials as the solution and the test spaces. DG method is a powerful tool for approximating some partial differential equations which model problems in physics, especially in fluid dynamics or electrodynamics. DG combines many attractive features of the classical finite element, finite volume, and finite difference methods. The local discontinuous Galerkin (LDG) finite element method is an extension of the DG method aimed at solving ordinary and partial differential equations containing higher than first-order spatial derivatives. In this talk we study the superconvergence properties of the LDG method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local spatial discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the local discretization errors for the p-degree LDG solution and its spatial derivative achieve (p+2)th order superconvergent at the roots of the right and left Radau polynomials of degree p+1, respectively. These results are used to construct asymptotically correct a posteriori error estimates. Several numerical simulations are performed to validate the theory.
10:45am - 11:10am,
Teng Li, (Advisor: V. Matache)
Complex Dynamics, Geometric Function Theory, and a Fixed Point Theorem
Abstract
+
Geometric function theory is a body of knowledge that covers the geometric properties of analytic functions. Conformal transforms and their properties are among the most popular topics belonging to it. We will present aspects less known to the general public, such as the behavior of the sequence of iterates of analytic selfmaps of the unit disc, which leads to the famous Denjoy-Wolff theorem, and based on that theorem, to a recent fixed point theorem, whose message is that families of analytic selfmaps of the disc simultaneously have a fixe point in that set (not the same for all of them), if they interpolate between select interpolation data sets. The study of iterates of analytic selfmaps is designated by the term complex dynamics.
11:10am - 11:35am,
Nguyen Nguyen, (Advisor: V. Matache)
Hyperbolic composition operators and the invariant subspace problem
Abstract
+
About 20 years ago, three mathematicians proved that the Invariant Subspace Problem, which is still open for Hilbert space operators, can be reduced to the study of the invariant subspace lattice of a single operator: a hyperbolic composition operator acting on the Hilbert Hardy space H2 of all functions analytic in the open unit disc, having square sumable Maclaurin coefficients. We will show how this approach leads to the problem of deciding if the only minimal invariant subspaces of a hyperbolic composition operator are the 1-dimensional eigenspaces or not, and report on several results obtained in this direction.
11:35am - 12:00pm,
Li Westman, (Advisor: Z. Wang)
Ranking Fuzzy Numbers by Their Left and Right Wingspans
Abstract
+
Based on the area between the curve of the membership function and the horizontal real axis, concepts of left and right wingspans are introduced. By them, a new index, called the w-center for fuzzy numbers is proposed. It is continuous with respect to the convergence of fuzzy number sequence. An intuitive and reasonable ranking method for fuzzy numbers based on their w-center is also established. This new ranking method is useful in fuzzy decision making and fuzzy data mining. The paper also points out that, in literature, some ranking methods based on the centroid of fuzzy numbers are not reasonable.
12:00pm - 12:25pm,
Bo Guo, (Advisor: Z. Wang)
On Pseudo Gradient Search for Solving Nonlinear Multiregression with the Choquet Integral
Abstract
+
The objective function in some real optimization problems may not be differentiable with respect to the unknown parameters at some points such that the gradient does not exist at those points. Replacing the classical gradient search, the method of pseudo gradient search has been proposed and used for solving nonlinear optimization problems, such as nonlinear multiregression based on the Choquet integral with a linear core. It is a local search with rapid search speed. To improve the search tactics, a random angle search in randomly selected dimensions is also involved. Our experiments show that the proposed pseudo gradient search is effective and efficient. It can be widely used for solving nonlinear optimization problems with continuous objective function.
