In this paper we study the structure of standard Einstein solvmanifolds of arbitrary RANK. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

This is a survey on some recent progress in homogeneous Finsler geometry. Three topics are discussed, the classi, cation of positively curved homogeneous Finsler spaces, the geometric and topological properties of homogeneous Finsler spaces satisfying K ,0 and the (FP) condition, and the orbit number of prime closed geodesics in a compact homogeneous Finsler manifold. These topics share the same similarity that the same RANK inequality, i. e., RANKG ,RANKH +1 for G=H with compact G and H, plays an important role. In this survey, we discuss in each topic how the RANK inequality is proved, explain its importance, and summarize some relevant results.

We introduce the notions of algebraic and arithmetic derivation. As an application, we use the combinatorial decomposition of an ideal to provide a constructive method to find the algebraic invariants, as the arithmetical RANK, for a family of squarefree monomial ideals, the $k$--complete ideals $I_k^n, $ also known as squarefree Veronese ideals of degree $k$.

In this paper, we propose a nonparametric RANK-based alternative to the least-squares independent component analysis algorithm developed. The basic idea is to estimate the squared-loss mutual information, which used as the objective function of the algorithm, based on its copula density version. Therefore, no marginal densities have to be estimated. We provide empirical evaluation of the proposed algorithm through simulation and real data analysis. Since the proposed algorithm uses RANK values rather than the actual values of the observations, it is extremely robust to the outliers and suffers less from the presence of noise than the other algorithms.