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.

The RANK-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For a noisy quantum channel, a quantum error correcting code of dimension k exists, if and only if the associated joint RANK-k numerical range is non-empty. In this paper, the no-tion of joint RANK-k numerical range is generalized, and some state-ments of [2011, Generalized numerical ranges and quantum error correction, J. Operator Theory, 66: 2, 335-351. ] are extended.

By reflection and thinking about the mood and RANK in mysticism and Sufism, one can realize that a large number of mystics have mingled the Sufism and mystical expressions and Kheikhs’ discourses with their poetic desires, and produced verses. In general, one can say that Sufism is the practical aspect of mysticism and Sufi uses spiritual journey and acquires RANKs and tendency toward the mood that can observe the Almighty in his heart. The Sufi’ s travel is the best means of knowing about the Almighty. The Sufis are the best class of people and their enthusiasm is the best kind of mood. The mood of a sensation that fills the breath of the Sufi’ s travelling, and that is the generosity of God; RANK is a station for praying and strife of creature standing before God and hopeless from other to join Him.

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.