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

Apolipoprotein E epsilon 4 (APOE-ε4) genotype is associated with decreased 6-month verbal memory performance after mild traumatic brain injury

Introduction: The apolipoprotein E (APOE) ε4 allele associates with memory impairment in neurodegenerative diseases. Its association with memory after mild traumatic brain injury (mTBI) is unclear. Methods: mTBI patients (Glasgow Coma Scale score 13–15, no neurosurgical intervention, extracranial Abbreviated Injury Scale score ≤1) aged ≥18 years with APOE genotyping results were extracted from the Transforming Research and Clinical Knowledge in Traumatic Brain Injury Pilot (TRACK-TBI Pilot) study. Cohorts determined by APOE-ε4(+/−) were assessed for associations with 6-month verbal memory, measured by California Verbal Learning Test, Second Edition (CVLT-II) subscales: Immediate Recall Trials 1–5 (IRT), Short-Delay Free Recall (SDFR), Short-Delay Cued Recall (SDCR), Long-Delay F

T1ρ and T2 -based characterization of regional variations in intervertebral discs to detect early degenerative changes.

Lower back pain is one of the main contributors to morbidity and chronic disability in the United States. Despite the significance of the problem, it is still not well understood. There is a clear need for objective, non-invasive biomarkers to localize specific pain generators and identify early stage changes to enable reliable diagnosis and treatment. In this study we focus on intervertebral disc degeneration as a source of lower back pain. Quantitative imaging markers T1ρ and T2 have been shown to be promising techniques for in vivo diagnosis of biochemical degeneration in discs due to their sensitivity to macromolecular changes in proteoglycan content and collagen integrity. We describe a semi-automated technique for quantifying T1ρ and T2 relaxation time maps in the nucleus pulposus (NP) and the annulus fibrosus (AF) of the lumbar intervertebral discs. Compositional changes within the NP and AF associated with degeneration occur much earlier than the visually observable structural changes. The proposed technique rigorously quantifies these biochemical changes taking into account subtle regional variations to allow interpretation of early degenerative changes that are difficult to interpret with traditional MRI techniques and clinical subjective grading scores. T1ρ and T2 relaxation times in the NP decrease with degenerative severity in the disc. Moreover, standard deviation and texture measurements of these values show sharper and more significant changes during early degeneration compared to later degenerative stages. Our results suggest that future prospective studies should include automated T1ρ and T2 metrics as early biomarkers for disc degeneration-induced lower back pain. © 2016 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 34:1373-1381, 2016