Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Prevalence of inflicted and neglectful femur shaft fractures in young children in national level I trauma centers

BACKGROUND: The prevalence of inflicted femur fractures in young children varies (1.5-35.2%), but these data are based on small retrospective studies with high heterogeneity. Age and mobility of the child seem to be indicators of inflicted trauma. OBJECTIVE: This study describes other factors associated with inflicted and neglectful trauma that can be used to distinguish inflicted and neglectful from accidental femur fractures. MATERIALS AND METHODS: This retrospective study included children (0-6 years) who presented with an isolated femur fracture at 1 of the 11 level I trauma centers in the Netherlands between January 2010 and January 2016. Outcomes were classified based on the conclusions of the Child Abuse and Neglect teams or the court. Cases in which conclusions were unavailable and there was no clear accidental cause were reviewed by an expert panel. RESULTS: The study included 328 children; 295 (89.9%) cases were classified as accidental trauma. Inflicted trauma was found in 14 (4.3%), while 19 (5.8%) were cases of neglect. Indicators of inflicted trauma were age 0-5 months (29%, positive likelihood ratio [LR +] 8.35), 6-12 months (18%, LR + 5.98) and 18-23 months (14%, LR + 3.74). Indicators of neglect were age 6-11 months (18%, LR + 4.41) and age 18-23 months (8%, LR + 1.65). There was no difference in fracture morphology among groups. CONCLUSION: It is unlikely that an isolated femur fracture in ambulatory children age > 24 months is caused by inflicted trauma/neglect. Caution is advised in children younger than 24 months because that age is the main factor associated with inflicted trauma/neglect and inflicted femur fractures

The prevalence of non-accidental trauma among children with polytrauma: A nationwide level-I trauma centre study

OBJECTIVE: We aimed to investigate the prevalence and characteristics of non-accidental trauma (NAT) in children with polytrauma treated at level-I trauma centres (TC). SUMMARY OF BACKGROUND: Data 6-10% Of children who present at the emergency department with injuries, sustain polytrauma. Polytrauma may result from either accidental (AT) or NAT, i.e. inflicted or neglect. The prevalence of NAT among children with polytrauma is currently unclear. METHODS: This is a retrospective study that included children (0-18 years) with an Injury Severity Score >15, who presented at one of the 11 Level-I trauma centers (TC) in the Netherlands between January 1, 2010 and January 1, 2016. Outcomes were classified based on the conclusions of the Child Abuse and Neglect-team. Cases in which conclusions were unavailable and there was no clear accidental cause of injuries were reviewed by an expert panel. RESULTS: The study included 1623 children, 1452 (89%) were classified as AT, 171 (11%) as NAT; 39 (2,4%) inflicted and 132 (8,1%) neglect. Of pre-school aged children (<5 years) 41% sustained NAT (OR26.73, 95%CI 17.70-40.35), 35/342 (10%) inflicted and 104/342 (31%) neglect. Admission due to 'cardiopulmonary arrest' was the result of inflicted trauma (30% vs 0%,p < 0.001). NAT had a higher mortality rate (16% vs 10%, p = 0.006). Indicators of NAT were: (near-)drowning (OR10.74, 95%CI 5.94-19.41), burn (OR8.62, 95%CI 4.08-18.19) and fall from height (OR2.18, 95%CI 1.56-3.02). CONCLUSIONS: NAT was the cause of polytrauma in 11% of children in our nationwide level-I TC study; 41% of these polytrauma were the result of NAT experienced by preschool-aged children. Our data show the importance of awareness for NAT