The seventh meeting of the Carolina Mathematics Seminar took place at Benedict College, Columbia, SC. This is a list of our speakers, titles and abstracts.
Keyona James (Benedict College)
Title: African-Americans in Mathematics.
Abstract: In this talk we will discuss contributions made by African-American mathematicians and their involvement in the mathematical sciences. Not much is known, taught, or written about African-American mathematicians. Information lacks on their past and present contributions and on the qualitative and quantitative nature of their existence throughout mathematics. In this talk we will provide some details about great mathematicians such as Benjamin Banneker, Elbert Frank Cox, Evelyn Boyd Granville, J. Ernest Wilkins, Jr., etc.
Peter Nyikos (USC Columbia)
Title: Discontinuities and smooth curves in n-space.
Abstract: The following theorem, with various wordings, can be found in a number of calculus texts.
Theorem 1. If the limit of a real-valued function on R 2 exists at a point p, then it will also be the limit along any smooth curve through p.
This theorem clearly extends to all n ≥ 2. The converse is true, but seems to be missing from all the standard calculus textbooks. In fact, something stronger is true:
Theorem 2. If f is a real-valued function defined in a deleted neighborhood of p in Rm, and the limit of f at p does not exist, then either:
(1) there is a smooth curve through p on which the limit does not exist, or
(2) there are two straight lines through p on which the limits exist, but are unequal.
László Zsilinszky (UNC at Pembroke)
Title: On some topological games.
Abstract: The so-called Banach-Mazur game was first introduced and studied in the 1930's by Stanislaw Mazur and Stefan Banach,, who played it on the unit interval. Oxtoby generalized and studied the game in topological spaces, later Choquet rediscovered it, and introduced a modification, which is now termed the strong Choquet game, to characterize complete metrizability in a metrizable setting. The talk will be an introduction to these games with a review of some interesting related results and examples.
Gurcan Comert (Benedict College)
Title: Traffic parameter estimation from probe vehicles at signalized intersections.
Abstract: Instrumented vehicle data (i.e., probe data) is becoming more important for real-time system parameter estimation in transportation networks. Probe data can be tracked anonymously and can report data on their locations, speeds, travel times, and arrival times as they perform their regular trips. This research develops analytical models for the real-time estimation of key traffic parameters (e.g., queue length, delay) at signalized intersections using the fundamental information (i.e., location, count, and time). For a single queue with Poisson arrivals, analytical models are developed to evaluate how error changes in estimation as percentage of probe vehicles in the traffic stream varies. The formulations presented assess the error in estimation for various scenarios of probe vehicle market penetration rates and congestion levels.
Ralph Howard (USC Columbia)
Title: The Geometry of Mirrors.
Abstract: We use the principle of least action to show why light bounces off a mirror so that the angle of incidence equals the angle of reflection. One geometric consequence of this is that when looking a mirror, one sees a left right reversed version of one's self. Another is that light bouncing off of a corner will return parallel to its original path. We also discuss how to construct the angles of a kaleidoscope so as to have non-overlaping images. There will be mirrors and kaleidoscopes for hands on experimenting.
Pictures