Quantitative Singularity Theory for Random Polynomials

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-W singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-W singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset W of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-W singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function p we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center p⁠, in which the W-singularity structure is constant

Intimate Partner Violence in Urban, Rural, and Remote Areas: An Investigation of Offense Severity and Risk Factors

This study compared the severity of intimate partner violence (IPV) and the relationship between risk factors for IPV and overall risk judgements of future IPV in urban, rural and remote areas. IPV risk assessments conducted by the Swedish police between 2010 and 2014 in urban (n = 564), rural (n = 456), and remote (n = 196) areas were examined. Rurality was associated with the severity of IPV reported, as well as the presence of risk factors and their relationship to overall risk judgements. Cases in remote areas included more severe IPV as well as more risk factors

Race/ethnicity and potential suicide misclassification: window on a minority suicide paradox?

<p>Abstract</p> <p>Background</p> <p>Suicide officially kills approximately 30,000 annually in the United States. Analysis of this leading public health problem is complicated by undercounting. Despite persisting socioeconomic and health disparities, non-Hispanic Blacks and Hispanics register suicide rates less than half that of non-Hispanic Whites.</p> <p>Methods</p> <p>This cross-sectional study uses multiple cause-of-death data from the US National Center for Health Statistics to assess whether race/ethnicity, psychiatric comorbidity documentation, and other decedent characteristics were associated with differential potential for suicide misclassification. Subjects were 105,946 White, Black, and Hispanic residents aged 15 years and older, dying in the US between 2003 and 2005, whose manner of death was recorded as suicide or injury of undetermined intent. The main outcome measure was the relative odds of potential suicide misclassification, a binary measure of manner of death: injury of undetermined intent (includes misclassified suicides) versus suicide.</p> <p>Results</p> <p>Blacks (adjusted odds ratio [AOR], 2.38; 95% confidence interval [CI], 2.22-2.57) and Hispanics (1.17, 1.07-1.28) manifested excess potential suicide misclassification relative to Whites. Decedents aged 35-54 (AOR, 0.88; 95% CI, 0.84-0.93), 55-74 (0.52, 0.49-0.57), and 75+ years (0.51, 0.46-0.57) showed diminished misclassification potential relative to decedents aged 15-34, while decedents with 0-8 years (1.82, 1.75-1.90) and 9-12 years of education (1.43, 1.40-1.46) showed excess potential relative to the most educated (13+ years). Excess potential suicide misclassification was also apparent for decedents without (AOR, 3.12; 95% CI, 2.78-3.51) versus those with psychiatric comorbidity documented on their death certificates, and for decedents whose mode of injury was "less active" (46.33; 43.32-49.55) versus "more active."</p> <p>Conclusions</p> <p>Data disparities might explain much of the Black-White suicide rate gap, if not the Hispanic-White gap. Ameliorative action would extend from training in death certification to routine use of psychological autopsies in equivocal-manner-of-death cases.</p

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Household illness, poverty and physical and emotional child abuse victimisation:Findings from South Africa’s first prospective cohort study

Physical and emotional abuse of children is a large scale problem in South Africa, with severe negative outcomes for survivors. Although chronic household illness has shown to be a predictor for physical and emotional abuse, no research has thus far investigated the different pathways from household chronic illness to child abuse victimisation in South Africa.Confidential self-report questionnaires using internationally utilised measures were completed by children aged 10-17 (n = 3515, 56.7% female) using door-to-door sampling in randomly selected areas in rural and urban locations of South Africa. Follow-up surveys were conducted a year later (96.7% retention rate). Using multiple mediation analyses, this study investigated direct and indirect effects of chronic household illness (AIDS or other illness) on frequent (monthly) physical and emotional abuse victimisation with poverty and extent of the ill person's disability as hypothesised mediators.For children in AIDS-ill families, a positive direct effect on physical abuse was obtained. In addition, positive indirect effects through poverty and disability were established. For boys, a positive direct and indirect effect of AIDS-illness on emotional abuse through poverty were detected. For girls, a positive indirect effect through poverty was observed. For children in households with other chronic illness, a negative indirect effect on physical abuse was obtained. In addition, a negative indirect effect through poverty and positive indirect effect through disability was established. For boys, positive and negative indirect effects through poverty and disability were found respectively. For girls, a negative indirect effect through poverty was observed.These results indicate that children in families affected by AIDS-illness are at higher risk of child abuse victimisation, and this risk is mediated by higher levels of poverty and disability. Children affected by other chronic illness are at lower risk for abuse victimisation unless they are subject to higher levels of household disability. Interventions aiming to reduce poverty and increase family support may help prevent child abuse in families experiencing illness in South Africa