Random discrete groups of Möbius transformations : probabilities and limit set dimensions : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand

This thesis addresses two areas related to the quantification of discrete groups. We study "random" groups of Möbius transformations and in particular random two-generator groups; that is, groups where the generators are selected randomly. Our intention is to estimate the likelihood that such groups are discrete and to calculate the expectation of their associated geometric and topological parameters. Computational results of the author [55] that indicate a low probability of a random group being discrete are extended and we also assess the expected Hausdorff dimension of the limit set of a discrete group. In both areas of research analytic determinations are correlated with computational results. Our results depend on the precise notion of randomness and we introduce geometrically natural probability measures on the groups of all Möbius transformations of the circle and the Riemann sphere

A randomized single blind crossover trial comparing leather and commercial wrist splints for treating chronic wrist pain in adults

<p>Abstract</p> <p>Background</p> <p>To compare the effectiveness of a custom-made leather wrist splint (LS) with a commercially available fabric splint (FS) in adults with chronic wrist pain.</p> <p>Methods</p> <p>Participants (N = 25, mean age = 54) were randomly assigned to treatment order in a 2-phase crossover trial. Splints were worn for 2 weeks, separated by a one-week washout period. Outcomes were assessed at baseline and after each splint phase using the Australian/Canadian Osteoarthritis Hand Index (AUSCAN), the Canadian Occupational Performance Measure (COPM) and Jamar dynamometer by an observer blinded to treatment allocation.</p> <p>Results</p> <p>Both styles of wrist splint significantly reduced pain (effect size LS 0.79, FS 0.43), improved hand function and increased grip strength compared to baseline (all p < 0.05) with no increase in wrist stiffness. There was a consistent trend for the LS to be superior to the FS but this was statistically significant only for patient perceived occupational performance (p = 0.008) and satisfaction (p = 0.015). Lastly, 72% of patients preferred the custom-made leather splint compared to the commercially available splint.</p> <p>Conclusion</p> <p>Leather wrist splints were superior to a commercially available fabric splint for the short-term relief of pain and dysfunction.</p

Structural brain abnormalities in the common epilepsies assessed in a worldwide ENIGMA study

Progressive functional decline in the epilepsies is largely unexplained. We formed the ENIGMA-Epilepsy consortium to understand factors that influence brain measures in epilepsy, pooling data from 24 research centres in 14 countries across Europe, North and South America, Asia, and Australia. Structural brain measures were extracted from MRI brain scans across 2149 individuals with epilepsy, divided into four epilepsy subgroups including idiopathic generalized epilepsies (n =367), mesial temporal lobe epilepsies with hippocampal sclerosis (MTLE; left, n = 415; right, n = 339), and all other epilepsies in aggregate (n = 1026), and compared to 1727 matched healthy controls. We ranked brain structures in order of greatest differences between patients and controls, by meta-Analysing effect sizes across 16 subcortical and 68 cortical brain regions. We also tested effects of duration of disease, age at onset, and age-by-diagnosis interactions on structural measures. We observed widespread patterns of altered subcortical volume and reduced cortical grey matter thickness. Compared to controls, all epilepsy groups showed lower volume in the right thalamus (Cohen's d = \uc3\ua2 '0.24 to \uc3\ua2 '0.73; P < 1.49 \uc3\u97 10 \uc3\ua2 '4), and lower thickness in the precentral gyri bilaterally (d = \uc3\ua2 '0.34 to \uc3\ua2 '0.52; P < 4.31 \uc3\u97 10 \uc3\ua2 '6). Both MTLE subgroups showed profound volume reduction in the ipsilateral hippocampus (d = \uc3\ua2 '1.73 to \uc3\ua2 '1.91, P < 1.4 \uc3\u97 10 \uc3\ua2 '19), and lower thickness in extrahippocampal cortical regions, including the precentral and paracentral gyri, compared to controls (d = \uc3\ua2 '0.36 to \uc3\ua2 '0.52; P < 1.49 \uc3\u97 10 \uc3\ua2 '4). Thickness differences of the ipsilateral temporopolar, parahippocampal, entorhinal, and fusiform gyri, contralateral pars triangularis, and bilateral precuneus, superior frontal and caudal middle frontal gyri were observed in left, but not right, MTLE (d = \uc3\ua2 '0.29 to \uc3\ua2 '0.54; P < 1.49 \uc3\u97 10 \uc3\ua2 '4). Contrastingly, thickness differences of the ipsilateral pars opercularis, and contralateral transverse temporal gyrus, were observed in right, but not left, MTLE (d = \uc3\ua2 '0.27 to \uc3\ua2 '0.51; P < 1.49 \uc3\u97 10 \uc3\ua2 '4). Lower subcortical volume and cortical thickness associated with a longer duration of epilepsy in the all-epilepsies, all-other-epilepsies, and right MTLE groups (beta, b < \uc3\ua2 '0.0018; P < 1.49 \uc3\u97 10 \uc3\ua2 '4). In the largest neuroimaging study of epilepsy to date, we provide information on the common epilepsies that could not be realistically acquired in any other way. Our study provides a robust ranking of brain measures that can be further targeted for study in genetic and neuropathological studies. This worldwide initiative identifies patterns of shared grey matter reduction across epilepsy syndromes, and distinctive abnormalities between epilepsy syndromes, which inform our understanding of epilepsy as a network disorder, and indicate that certain epilepsy syndromes involve more widespread structural compromise than previously assumed

Genetic mechanisms of critical illness in COVID-19.

Host-mediated lung inflammation is present1, and drives mortality2, in the critical illness caused by coronavirus disease 2019 (COVID-19). Host genetic variants associated with critical illness may identify mechanistic targets for therapeutic development3. Here we report the results of the GenOMICC (Genetics Of Mortality In Critical Care) genome-wide association study in 2,244 critically ill patients with COVID-19 from 208 UK intensive care units. We have identified and replicated the following new genome-wide significant associations: on chromosome 12q24.13 (rs10735079, P = 1.65 × 10-8) in a gene cluster that encodes antiviral restriction enzyme activators (OAS1, OAS2 and OAS3); on chromosome 19p13.2 (rs74956615, P = 2.3 × 10-8) near the gene that encodes tyrosine kinase 2 (TYK2); on chromosome 19p13.3 (rs2109069, P = 3.98 ×  10-12) within the gene that encodes dipeptidyl peptidase 9 (DPP9); and on chromosome 21q22.1 (rs2236757, P = 4.99 × 10-8) in the interferon receptor gene IFNAR2. We identified potential targets for repurposing of licensed medications: using Mendelian randomization, we found evidence that low expression of IFNAR2, or high expression of TYK2, are associated with life-threatening disease; and transcriptome-wide association in lung tissue revealed that high expression of the monocyte-macrophage chemotactic receptor CCR2 is associated with severe COVID-19. Our results identify robust genetic signals relating to key host antiviral defence mechanisms and mediators of inflammatory organ damage in COVID-19. Both mechanisms may be amenable to targeted treatment with existing drugs. However, large-scale randomized clinical trials will be essential before any change to clinical practice

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

Case Reports1. A Late Presentation of Loeys-Dietz Syndrome: Beware of TGFβ Receptor Mutations in Benign Joint Hypermobility

Background: Thoracic aortic aneurysms (TAA) and dissections are not uncommon causes of sudden death in young adults. Loeys-Dietz syndrome (LDS) is a rare, recently described, autosomal dominant, connective tissue disease characterized by aggressive arterial aneurysms, resulting from mutations in the transforming growth factor beta (TGFβ) receptor genes TGFBR1 and TGFBR2. Mean age at death is 26.1 years, most often due to aortic dissection. We report an unusually late presentation of LDS, diagnosed following elective surgery in a female with a long history of joint hypermobility. Methods: A 51-year-old Caucasian lady complained of chest pain and headache following a dural leak from spinal anaesthesia for an elective ankle arthroscopy. CT scan and echocardiography demonstrated a dilated aortic root and significant aortic regurgitation. MRA demonstrated aortic tortuosity, an infrarenal aortic aneurysm and aneurysms in the left renal and right internal mammary arteries. She underwent aortic root repair and aortic valve replacement. She had a background of long-standing joint pains secondary to hypermobility, easy bruising, unusual fracture susceptibility and mild bronchiectasis. She had one healthy child age 32, after which she suffered a uterine prolapse. Examination revealed mild Marfanoid features. Uvula, skin and ophthalmological examination was normal. Results: Fibrillin-1 testing for Marfan syndrome (MFS) was negative. Detection of a c.1270G > C (p.Gly424Arg) TGFBR2 mutation confirmed the diagnosis of LDS. Losartan was started for vascular protection. Conclusions: LDS is a severe inherited vasculopathy that usually presents in childhood. It is characterized by aortic root dilatation and ascending aneurysms. There is a higher risk of aortic dissection compared with MFS. Clinical features overlap with MFS and Ehlers Danlos syndrome Type IV, but differentiating dysmorphogenic features include ocular hypertelorism, bifid uvula and cleft palate. Echocardiography and MRA or CT scanning from head to pelvis is recommended to establish the extent of vascular involvement. Management involves early surgical intervention, including early valve-sparing aortic root replacement, genetic counselling and close monitoring in pregnancy. Despite being caused by loss of function mutations in either TGFβ receptor, paradoxical activation of TGFβ signalling is seen, suggesting that TGFβ antagonism may confer disease modifying effects similar to those observed in MFS. TGFβ antagonism can be achieved with angiotensin antagonists, such as Losartan, which is able to delay aortic aneurysm development in preclinical models and in patients with MFS. Our case emphasizes the importance of timely recognition of vasculopathy syndromes in patients with hypermobility and the need for early surgical intervention. It also highlights their heterogeneity and the potential for late presentation. Disclosures: The authors have declared no conflicts of interes

Ultra-rare genetic variation in common epilepsies: a case-control sequencing study

BACKGROUND:Despite progress in understanding the genetics of rare epilepsies, the more common epilepsies have proven less amenable to traditional gene-discovery analyses. We aimed to assess the contribution of ultra-rare genetic variation to common epilepsies. METHODS:We did a case-control sequencing study with exome sequence data from unrelated individuals clinically evaluated for one of the two most common epilepsy syndromes: familial genetic generalised epilepsy, or familial or sporadic non-acquired focal epilepsy. Individuals of any age were recruited between Nov 26, 2007, and Aug 2, 2013, through the multicentre Epilepsy Phenome/Genome Project and Epi4K collaborations, and samples were sequenced at the Institute for Genomic Medicine (New York, USA) between Feb 6, 2013, and Aug 18, 2015. To identify epilepsy risk signals, we tested all protein-coding genes for an excess of ultra-rare genetic variation among the cases, compared with control samples with no known epilepsy or epilepsy comorbidity sequenced through unrelated studies. FINDINGS:We separately compared the sequence data from 640 individuals with familial genetic generalised epilepsy and 525 individuals with familial non-acquired focal epilepsy to the same group of 3877 controls, and found significantly higher rates of ultra-rare deleterious variation in genes established as causative for dominant epilepsy disorders (familial genetic generalised epilepsy: odd ratio [OR] 2·3, 95% CI 1·7-3·2, p=9·1 × 10-8; familial non-acquired focal epilepsy 3·6, 2·7-4·9, p=1·1 × 10-17). Comparison of an additional cohort of 662 individuals with sporadic non-acquired focal epilepsy to controls did not identify study-wide significant signals. For the individuals with familial non-acquired focal epilepsy, we found that five known epilepsy genes ranked as the top five genes enriched for ultra-rare deleterious variation. After accounting for the control carrier rate, we estimate that these five genes contribute to the risk of epilepsy in approximately 8% of individuals with familial non-acquired focal epilepsy. Our analyses showed that no individual gene was significantly associated with familial genetic generalised epilepsy; however, known epilepsy genes had lower p values relative to the rest of the protein-coding genes (p=5·8 × 10-8) that were lower than expected from a random sampling of genes. INTERPRETATION:We identified excess ultra-rare variation in known epilepsy genes, which establishes a clear connection between the genetics of common and rare, severe epilepsies, and shows that the variants responsible for epilepsy risk are exceptionally rare in the general population. Our results suggest that the emerging paradigm of targeting of treatments to the genetic cause in rare devastating epilepsies might also extend to a proportion of common epilepsies. These findings might allow clinicians to broadly explain the cause of these syndromes to patients, and lay the foundation for possible precision treatments in the future. FUNDING:National Institute of Neurological Disorders and Stroke (NINDS), and Epilepsy Research UK

Topographic divergence of atypical cortical asymmetry and atrophy patterns in temporal lobe epilepsy

Temporal lobe epilepsy, a common drug-resistant epilepsy in adults, is primarily a limbic network disorder associated with predominant unilateral hippocampal pathology. Structural MRI has provided an in vivo window into whole-brain grey matter structural alterations in temporal lobe epilepsy relative to controls, by either mapping (i) atypical inter-hemispheric asymmetry; or (ii) regional atrophy. However, similarities and differences of both atypical asymmetry and regional atrophy measures have not been systematically investigated. Here, we addressed this gap using the multisite ENIGMA-Epilepsy dataset comprising MRI brain morphological measures in 732 temporal lobe epilepsy patients and 1418 healthy controls. We compared spatial distributions of grey matter asymmetry and atrophy in temporal lobe epilepsy, contextualized their topographies relative to spatial gradients in cortical microstructure and functional connectivity calculated using 207 healthy controls obtained from Human Connectome Project and an independent dataset containing 23 temporal lobe epilepsy patients and 53 healthy controls and examined clinical associations using machine learning. We identified a marked divergence in the spatial distribution of atypical inter-hemispheric asymmetry and regional atrophy mapping. The former revealed a temporo-limbic disease signature while the latter showed diffuse and bilateral patterns. Our findings were robust across individual sites and patients. Cortical atrophy was significantly correlated with disease duration and age at seizure onset, while degrees of asymmetry did not show a significant relationship to these clinical variables. Our findings highlight that the mapping of atypical inter-hemispheric asymmetry and regional atrophy tap into two complementary aspects of temporal lobe epilepsy-related pathology, with the former revealing primary substrates in ipsilateral limbic circuits and the latter capturing bilateral disease effects. These findings refine our notion of the neuropathology of temporal lobe epilepsy and may inform future discovery and validation of complementary MRI biomarkers in temporal lobe epilepsy.11Nsciescopu