The eleventh meeting of the Carolina Mathematics Seminar took place at Benedict College, Columbia, SC. This is a list of our speakers, titles and abstracts.
Gurcan Comert and Anton Bezuglov (Benedict College)
Title: Predicting System States in Transportation Networks.
Abstract: Traffic systems often exhibit nonlinearities and sometimes abrupt changes due to various factors such as traffic accidents, inclement weather conditions, and demand surges. Developing robust adaptive models for real-time system state prediction is a significant challenge. This research develops strategies and models to predict traffic conditions under abrupt changes using Hidden Markov and Time Series Models. The data set used has incident databases that reveal the causes of sudden changes in traffic parameters (e.g., speed, occupancy, and flow). In the models, all of the significant characteristics of the changes (e.g., duration, amplitude) are statistically detected and analyzed.
Eva Czabarka (USC Columbia)
Title: Orthogonal arrays and transversals.
Abstract: With H. Aydinian, K. Engel, P.L. Erdos and L.A. Szekely we investigated a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in axis-parallel directions (i.e. in a 1-dimensional subgrid). This concept is motivated by error correcting codes and more-part Sperner theory, and it is closely connected to orthogonal arrays. We extend this concept from 1 to d-dimension: the bounds are given on the number of allowed entries in a d-dimensional subgrid. We prove that there are packing arrays that always reach the natural upper bound for their size, and prove some related extremal results.
László Zsilinszky (University of North Carolina)
Title: Infinities – Stuff That Make People Go Nuts.
Abstract: A gentle introduction to some questions involving the notion of infinity in Mathematics, and Set-Theory.
Danny Rorabaugh (Ph.D. Student at USC Columbia)
Title: Collatz Generalized: An Expansion of the 3x+1 Problem
Abstract: The focus of this 2010 undergraduate research project is a generalization of the Collatz conjecture – an unsolved number theory problem involving the “3x+1 function” on the positive integers: if x is odd then C(x) = 3x + 1; if x is even then C(x) = x/2. The Collatz conjecture is that, given any positive integer x, the infinite sequence {x,C(x),C(C(x)),...} – the trajectory of x - contains the number 1. Although the conjecture has been proven for x up to at least 10^17, it remains unproven for all positive integers.
This project investigates the problem within a broader context of the following “Ax+B function”: if x is odd then C(x) = Ax + B; if x is even then C(x) = x/2. Under this wider scope, the project explores the relationships between A, B and x and whether a trajectory contains 1, loops without reaching 1, or is unbounded with no positive integer occurring twice. Understanding these relationships may help to shed light on why the trajectory of every positive integer under the 3x+1 function contains one, if such is in fact the case.