A Fast Algorithm for Computing the p-Curvature

We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is \softO (p d r^\omega) operations in the ground field (where $\omega$ denotes the exponent of matrix multiplication), whereas the size of the output is about $p d r^2$. Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} $\omega = 2$). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdo

On the reduction of the degree of linear differential operators

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type

The effect of a sport-based intervention to prevent juvenile delinquency in at-risk adolescents

Despite the wide implementation of sport-based crime prevention programs, there is a lack of empirical knowledge on the effectiveness of these interventions. This study evaluated a Dutch sport-based program in N = 368 youth at risk for juvenile delinquency. Intervention effects were tested in a quasi-experimental study, comparing the intervention group with a comparison group using multiple sources of information. The study was conducted under conditions that resemble real-life implementation, thereby enhancing the relevance of this contribution to practitioners. The primary outcome was juvenile delinquency, measured by official police data. The secondary outcomes were risk and protective factors for delinquency, assessed with self- and teacher reports. A significant effect was found on one delinquency measure. The intervention group consisted of fewer youth with police registrations as a suspect than the comparison group (d = −0.34). We did not find an intervention effect on the number of registrations as a suspect in each group. In addition, no significant intervention effects were found on the secondary outcomes. Implications for theory and practice concerning the use of sport-based crime prevention programs are discussed