ON P-ADIC FIELDS AND P-GROUPS

The dissertation is divided into two parts. The first part mainly treats a conjecture of Emil Artin from the 1930s. Namely, if f = a_1x_1^d + a_2x_2^d +...+ a_{d^2+1}x^d where the coefficients a_i lie in a finite unramified extension of a rational p-adic field, where p is an odd prime, then f is isotropic. We also deal with systems of quadratic forms over finite fields and study the isotropicity of the system relative to the number of variables. We also study a variant of the classical Davenport constant of finite abelian groups and relate it to the isotropicity of diagonal forms. The second part deals with the theory of finite groups. We treat computations of Chermak-Delgado lattices of p-groups. We compute the Chermak-Delgado lattices for all p-groups of order p^3 and p^4 and give results on p-groups of order p^5

Are circles isoperimetric in the plane with density er?

We prove that an isoperimetric region in R2 with density er must be convex and contain the origin, and provide numerical evidence that circles about the origin are isoperimetric, as predicted by the Log-Convex Density Conjecture

Self-reporting of psychiatric illness in an online patient registry is a good indicator of the existence of psychiatric illness

Online registries offer many advantages for research, including the ability to efficiently assess large numbers of individuals and identify potential participants for clinical trials and genetic studies. Of particular interest is the validity and utility of self-endorsement of psychiatric disorders in online registries, which, while increasingly more common, remain understudied. We thus assessed the comparability of prevalence estimated from self-endorsement of psychiatric disorders in one such registry, the Brain Health Registry (BHR) to prevalence computed from large US-based epidemiological studies and the degree to which BHR participants report psychiatric disorders consistently. We also examined the concordance between self-report and clinically determined diagnoses of various DSM-5 psychiatric disorders in a subset of participants who underwent direct assessments and identified possible reasons for discordance. Rates of self-reported psychiatric disorders were closest to previously reported population prevalence rates when endorsed at multiple timepoints, and accuracy was at least 70% for all except Hoarding Disorder as compared to the clinical diagnoses. Clinical data suggested that self-endorsement of a given psychiatric diagnosis was indicative of the presence of a closely related condition, although not necessarily for the specific disorder, with the exception of major depressive disorder, panic disorder, and hoarding disorder, which had high positive predictive values (85%, 73%, 100%, respectively). We conclude that self-reporting of psychiatric conditions in an online setting is a fair indicator of psychopathology, but should be accompanied by more in-depth interviews if using data from a participant for a specific disorder

Identifying psychiatric and neurological comorbidities associated with hoarding disorder through network analysis.

BACKGROUND: The relationships between hoarding disorder (HD) and other neurological and psychiatric disorders remain largely unknown. Although psychiatric burden in those with HD is high, less is known about neurological disorders. Furthermore, which disorders are primarily associated with HD vs which can be better explained via a relationship with another disorder has not been determined. To address these questions, we examined comorbidity patterns of psychiatric and neurological disorders in a large online registry of adults using network analyses. METHODS: We first examined psychiatric comorbidity among 252 participants completing clinician administered psychiatric assessments. Using the Brain Health Registry (BHR) (N = 15,978), we next analyzed prevalence of self-reported neurological and psychiatric disorders among participants with no/minimal hoarding, subclinical hoarding, and clinically significant hoarding and used network analyses to identify direct and indirect relationships between HD and the assessed psychiatric and neurological disorders. RESULTS: The most prevalent comorbidity in clinically assessed participants with HD was major depressive disorder (MDD, 62%), followed by generalized anxiety disorder (GAD, 32%). Network analyses in the BHR indicated that the strongest direct relationships with HD were attention-deficit hyperactivity disorder (ADHD), major depressive disorder (MDD), and obsessive-compulsive disorder (OCD). The relationships between HD and neurological disorders, including mild cognitive impairment, were weak or non-existent after controlling for other disorders. CONCLUSIONS: ADHD, MDD, and OCD form a triad of psychiatric disorders directly associated with HD. Despite their high comorbidity rates, the associations among anxiety disorders and HD were weak or indirect

Internet-based hoarding assessment: The reliability and predictive validity of the internet-based Hoarding Rating Scale, Self-Report.

The Hoarding Rating Scale, Self Report (HRS-SR) is a 5-item assessment developed to ascertain the presence and severity of hoarding symptoms. This study aimed to evaluate the validity of an online adaptation of the HRS-SR in a remote, unsupervised internet sample of 23,214 members of the Brain Health Registry (BHR), an online research registry that evaluates and longitudinally monitors cognition, medical and psychiatric health status. Convergent validity was assessed among a sub-sample of 1,183 participants who completed additional, remote measures of self-reported hoarding behaviors. Structured clinical interviews conducted in-clinic and via video conferencing tools were conducted among 230 BHR participants; ROC curves were plotted to assess the diagnostic performance of the internet-based HRS-SR using best estimate hoarding disorder (HD) diagnoses as the gold standard. The area under the curve indicated near-perfect model accuracy, and was confirmed with 10-fold cross validation. Sensitivity and specificity for distinguishing clinically relevant hoarding were optimized using an HRS-SR total score cut-off of 5. Longitudinal analyses indicated stability of HRS-SR scores over time. Findings indicate that the internet-based HRS-SR is a useful and valid assessment of hoarding symptoms, though additional research using samples with more diverse hoarding behavior is needed to validate optimal cut-off values