Organizers: Bogdan Suceavă (CSUF) and Peter Jipsen (Chapman University)
2:00 pm John Simanyi (Cal State Fullerton) - Complex Series at the Radius of Convergence
Abstract: In complex analysis, as in the real case, we can find a radius of convergence for a given series, dividing the complex plane into separate regions of certain convergence and divergence. What happens at the border, on the circle of precisely that radius? We consider a few examples to investigate.
2:30 pm Dr. Mark Filowitz (Associate Dean) - Welcoming Address
2:35 pm Alex Barrett (Chapman University) - A Two-Light Version of the Classical Hundred Prisoners and a Light Bulb Problem: Optimizing Experimental Design through Simulations
Abstract: We propose five original strategies of successively increasing complexity and efficiency that address a novel version of a classical mathematical problem that, in essence, focuses on the determination of an optimal protocol for exchanging limited amounts of information among a group of subjects with various prerogatives. The inherent intricacy of the problem–solving protocols eliminates the possibility to attain an analytical solution. Therefore, we implemented a large-scale simulation study to exhaustively search through an extensive list of competing algorithms associated with the above-mentioned 5 generally defined protocols. Our results show that the consecutive improvements in the average amount of time necessary for the strategy-specific problem-solving completion over the previous simpler and less advantageously structured designs were 18, 30, 12, and 9% respectively. The optimal multi-stage information exchange strategy allows for a successful execution of the task of interest in 1722 days (4.7 years) on average with standard deviation of 385 days. The execution of this protocol took as few as 1004 and as many as 4965 with median of 1616 days.
3:00 pm Heng Sok (Chapman University) - On the Performance of Exact Testing Procedures with Respect to Comparisons of Several Multinomial Distributions in Small Samples
Abstract: This research project focuses on the validity of the exact p-value method under the fully specified and non-specified null hypotheses with respect to comparison of multiple multinomial distributions in small samples, where classical chi-square testing procedure is inappropriate. Here by validity, we mean the type I error rates at several significance levels and the power of the test under various alternatives. The two sample comparison problem arises often in two treatment randomized trials and case-control studies where comparison of background characteristics is an important step of the study analysis. Small samples are an inherent trait of pilot studies and studies with recruitment difficulties due to rare conditions, lack of interest and consent or budget and time restrictions. Further, this analysis could be extended to encompass the comparison of more than two multinomial distributions that would be applicable to multiple treatment randomized trials and as done in the classical large sample theory. Our results show that under the non-specified null hypothesis, the exact p-value method is severely conservative due to the absence of adjustment structure in these tests due the estimation of the common parameters and predictably, as the number of sample increases, the type 1 error rate increases but still remains below its nominal level. Consequently, the power of the test is low when the underlying multinomial probability distributions are relatively close to each other and the number of categories is large compared to the sample size.
3:30 pm Nicholas Blackford, Daniel Lenders, and Danny Orton - An Inverse-Based Analogue of the Probability that Two Elements of a Finite Group Commute
Abstract: Given a finite group G, the probability that two randomly chosen elements of G commute has long been viewed in the literature as a natural measure of the degree of commutativity enjoyed by the group. Many variants on this probabilistic question have arisen in the literature recently, and our research introduces yet another such variation that provides somewhat different information. In particular, given a product of elements of G, we investigate the likelihood of being able to permute the order of the elements in the product and obtain the inverse of the original element. With the help of the software program Groups, Algorithms and Programming (GAP) , we have discovered patterns leading to interesting results about this variant. In our paper, we will examine these patterns for such familiar finite groups as cyclic, dihedral, and symmetric groups, and describe the general results we have obtained. Some of these results are at odds with analogous ones known for the commutativity measure studied in recent articles, thereby adding further interest into our investigation.