An update on statistical boosting in biomedicine

Statistical boosting algorithms have triggered a lot of research during the last decade. They combine a powerful machine-learning approach with classical statistical modelling, offering various practical advantages like automated variable selection and implicit regularization of effect estimates. They are extremely flexible, as the underlying base-learners (regression functions defining the type of effect for the explanatory variables) can be combined with any kind of loss function (target function to be optimized, defining the type of regression setting). In this review article, we highlight the most recent methodological developments on statistical boosting regarding variable selection, functional regression and advanced time-to-event modelling. Additionally, we provide a short overview on relevant applications of statistical boosting in biomedicine

Rate of Convergence Towards Hartree Dynamics

We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the effective nonlinear Hartree dynamics is at most of the order 1/N, for any fixed time. The N-dependence of the bound is optimal.Comment: 26 page

Derivation of the Cubic Non-linear Schr\"odinger Equation from Quantum Dynamics of Many-Body Systems

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.Comment: 72 pages, 17 figures. Final versio

Dynamical Collapse of Boson Stars

We study the time evolution in system of $N$ bosons with a relativistic dispersion law interacting through an attractive Coulomb potential with coupling constant $G$. We consider the mean field scaling where $N$ tends to infinity, $G$ tends to zero and $\lambda = G N$ remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schr\"odinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large $\lambda$ (the sub-critical case has been studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the Coulomb interaction in the many body Hamiltonian with an $N$ dependent cutoff that vanishes in the limit $N\to \infty$. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval $[-T,T]$, then the many body Schr\"odinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time $T$ (in the sense that the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then also the solution of the linear Schr\"odinger equation collapses (in the sense that the kinetic energy per particle diverges) if $t \to T$ and, simultaneously, $N \to \infty$ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.Comment: 40 page

Short and oral antimicrobial therapy for diabetic foot infection: a narrative review of current knowledge

Diabetic foot infection is a frequent complication in long-standing diabetes mellitus. For antimicrobial therapy of this infection, both the optimal duration and the route of administration are often based more on expert opinion than on published evidence. We reviewed the scientific literature, specifically seeking prospective trials, and aimed at addressing two clinical issues: (1) shortening the currently recommended antibiotic duration and (2) using oral (rather than parenteral) therapy, especially after the patient has undergone debridement and revascularization. We also reviewed some older key articles that are critical to our understanding of the treatment of these infections, particularly with respect to diabetic foot osteomyelitis. Our conclusion is that the maximum duration of antibiotic therapy for osteomyelitis should be no more than to 4-6 weeks and might even be shorter in selected cases. In the future, in addition to conducting randomized trials and propagating national and international guidance, we should also explore innovative strategies, such as intraosseous antibiotic agents and bacteriophages