The eleventh meeting of the Carolina Mathematics Seminar took place at USC Salkehatchie, Salkehatchie, SC. This is a list of our speakers, titles and abstracts.
Joshua Smoak (USC Salkehatchie)
Title: An Identity for Exradii.
Abstract: In our talk, we will start with some basic properties of exradii of a triangle. We then will use a theorem introduced by Johnson (Modern Geometry, Dover, 2007) to prove an identity proposed in The American Mathematical Monthly Volume 118, No.8.
Kuo-Wei Yao (The Citadel)
Title: Low Rank Matrix Approximation and Its Applications in Image Processing.
Abstract: The aim of this paper is to find efficient methods to estimate approximation errors of low rank matrix approximations of a m × n matrix A Consider A ≈Ak formed by the first k singular values and their corresponding singular vectors where k is much smaller than the rank of A. We study the relation of the ratio of the kth largest and the largest singular values, and the change of the approximation error of Ak. We have applied our finding to applications in image processing to determine the rank of an approximation matrix that gives an acceptable image with the least storage possible. Results will be reported in the paper.
Virginia Johnson (USC Columbia)
Title: Counting gene trees.
Abstract: Gene trees used in biology to describe the evolution of genetic material throughout different species. Internal nodes of the tree correspond to speciation or duplication events, and the leaves are labeled with the name of the species the gene comes from. Consequently, gene trees are leaf-labeled trees which ideally but not necessarily are rooted, the root is the only vertex that may have degree 2, and labels in the label set may be used multiple times or not at all (the latter corresponding to deletion events). Otter in 1949 has proved a formula on unlabeled trees that connects counts of rooted trees to corresponding counts of unrooted trees. We generalize this formula for semi-labeled graphs, and use this to provide ordinary generating functions for gene trees (binary or non-binary, rooted or unrooted) and leaf-labeled trees (where internal nodes may have degree 2 even if they are not the root). This is joint work with Éva Czabarka, Peter L Erdös, Virginia Johnson and Vincent Moulton.
Upasana Kashyap (The Citadel)
Title: Picard group of dual operator algebras.
Abstract: We discuss the Picard group of dual (weak*-closed) operator algebras. We prove that for a weak*-closed function algebra A, the weak Picard group Picw(A) is a semidirect product of the automorphism group of A, and subgroup of Picw(A) consisting of symmetric equivalence bimodules. In particular we show that the weak Picard group of space of bounded analytical functions is isomorphic to the group of conformal automorphisms of the disk.
Paul Nietert (Medical University of South Carolina)
Title: Use of novel biostatistics methods by researchers publishing in general/internal medicine journals.
Abstract: Novel statistical methods are constantly being developed within the context of biomedical research; however, the characteristics of biostatistics methods that have been adopted into the field of general / internal medicine (GIM) is unclear. This study highlights the statistical journal articles, the statistical journals, and the types of statistical methods that appear to be having the most direct impact on GIM research. Descriptive techniques, including analyses of articles’ keywords and controlled vocabulary terms, were used to characterize the articles published in statistics and probability journals that were subsequently referenced within GIM journal articles during a recent 10-year period (2000-2009). From the 45 statistics and probability journals of interest, a total of 597 unique articles were identified as being cited by 900 (out of a total of about 10,501) unique GIM journal articles. The most frequently cited statistical topics included general/other statistical methods, followed by epidemiologic methods, randomized trials, generalized linear models, meta-analysis, and missing data. I will briefly summarize these methods and what their uses. As statisticians continue to develop and refine techniques, the promotion and adoption of these methods should also be addressed so that their efforts spent in developing the methods are not done in vain.
Fidele Ngwane (USC Salkehatchie)
Title: Towers and Type I curves.
Abstract: Type I curves and Towers of functions are very useful in coding theory. We will discuss Type I curves, Towers of function and we will show how to put Towers of functions into Type I curves. All the Towers of functions considered here are asymptotically good towers.