Word Length Perturbations in Certain Symmetric Presentations of Dihedral Groups

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities called $\lambda_{1}$ and $\lambda_{2}$ have been defined that allow for a precise measure of how stable a group is under certain types of small perturbations in the generating expressions for the elements of the group. These quantities provide a means to measure differences among all possible paths in a Cayley graph for a group, establish a group theoretic analog for the notion of stability in nonlinear dynamical systems, and play an important role in the application of groups to computational genomics. In this paper, we further expose the fundamental properties of $\lambda_{1}$ and $\lambda_{2}$ by establishing their bounds when the underlying group is a dihedral group. An essential step in our approach is to completely characterize so-called symmetric presentations of the dihedral groups, providing insight into the manner in which $\lambda_{1}$ and $\lambda_{2}$ interact with finite group presentations. This is of interest independent of the study of the quantities $\lambda_{1},\; \lambda_{2}$. Finally, we discuss several conjectures and open questions for future consideration

Universal deformation rings of modules for algebras of dihedral type of polynomial growth

Let k be an algebraically closed field, and let \Lambda\ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowro\'{n}ski. We describe all finitely generated \Lambda-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(\Lambda,V). We prove that only three isomorphism types occur for R(\Lambda,V): k, k[[t]]/(t^2) and k[[t]].Comment: 11 pages, 2 figure

A multisite study to examine the efficacy of the otoacousticemission/automated auditory brainstem response newborn hearing screening protocol: Results of visualreinforcement audiometry

Purpose: This 3rd of 4 articles on a study of the efficacy of the 2-stage otoacoustic emission/automated auditory brainstem response (OAE/A-ABR) newborn hearing screening protocol describes (a) the behavioral audiometric protocol used to validate hearing status at 8-12 months of age, (b) the hearing status of the sample, and (c) the success of the visual reinforcement audiometry (VRA) protocol across 7 sites. Method: A total of 973 infants who failed OAE but passed A-ABR, in one or both ears, during newborn screening were tested with a VRA protocol, supplemented by tympanometry and OAE screening at age 8-12 months. Results: VRA audiograms (1.0, 2.0, and 4.0 kHz) were obtained for 1,184 (82.7%) of the 1,432 study ears. Hearing loss was ruled out in another 100 ears by VRA in combination with OAE, for a total of 88.7% of the study sample. Permanent hearing loss was identified in 30 ears of 21 infants. Sites differed in their success with the VRA protocol. Conclusions: Continued monitoring of hearing beyond the newborn period is an important component of early detection of hearing loss. Using a structured protocol, VRA is an appropriate test method for most, but not all, infants. A battery of test procedures is often needed to adequately delineate hearing loss in infants. Examiner experience appears to be a factor in successful VRA