The Crossed Product by a Partial Endomorphism and the Covariance Algebra

Given a local homeomorphism \sigma:U -> X where U is a clopen subset of an compact and Hausdorff topological space X, we obtain the possible transfer operators L_\rho which may occur for \al:C(X) -> C(U) given by \al(f)=f\sigma. We obtain examples of partial dynamical systems (X_A,\sigma_A) such that the construction of the covariance algebra C^*(X_A,\sigma_A) and the crossed product by partial endomorphism O(X_A,\al,L) associated to this system are not equivalent, in the sense that there does not exists invertible function \rho in C(U) such that O(X_A,\al,L_\rho)=C^*(X_A,\sigma).Comment: 13 pages, no figure

On the mass formula and Wigner and curvature energy terms

The efficiency of different mass formulas derived from the liquid drop model including or not the curvature energy, the Wigner term and different powers of the relative neutron excess $I$ has been determined by a least square fitting procedure to the experimental atomic masses assuming a constant R$_{0,charge}$/A$^{1/3}$ ratio. The Wigner term and the curvature energy can be used independently to improve the accuracy of the mass formula. The different fits lead to a surface energy coefficient of around 17-18 MeV, a relative sharp charge radius r$_0$ of 1.22-1.23 fm and a proton form-factor correction to the Coulomb energy of around 0.9 MeV

Adaptive Clustering through Semidefinite Programming

We analyze the clustering problem through a flexible probabilistic model that aims to identify an optimal partition on the sample X 1 , ..., X n. We perform exact clustering with high probability using a convex semidefinite estimator that interprets as a corrected, relaxed version of K-means. The estimator is analyzed through a non-asymptotic framework and showed to be optimal or near-optimal in recovering the partition. Furthermore, its performances are shown to be adaptive to the problem's effective dimension, as well as to K the unknown number of groups in this partition. We illustrate the method's performances in comparison to other classical clustering algorithms with numerical experiments on simulated data