We adress the problem of qualitative properties of multipeakons, particular solutions of the Camassa-Holm equation. Our approach makes use of the well-known fact that the evolution of multipeakons is governed by the geodesic motion of a particle on an N -dimensional surface whose metric tensor is given via the inverse matrix to the one defining the Hamiltonian. Our approach yields some properties of twopeakons in a very simple way. We classify initial shapes of twopeakons according to the occurrence of collision. Moreover we extend the class of matrices that are invertible for similar reasons to the one occurring in the Hamiltonian. We get exact formulas for the inverses.2010 Mathematics Subject Classification: 53A17, 15A09.

We consider selection of random predictors for high-dimensional regression problem with binary response for a general loss function. Important special case is when the binary model is semiparametric and the response function is misspecified under parametric model fit. Selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a two-step selection procedure which consists of screening and ordering predictors by Lasso method and then selecting a subset of predictors which minimizes Generalized Information Criterion on the corresponding nested family of models. We prove consistency of the selection method under conditions which allow for much larger number of predictors than number of observations. For the semiparametric case when distribution of random predictors satisfies linear regression conditions the true and the estimated parameters are collinear and their common support can be consistently identified.