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 study aimed to evaluate the receptor activator of nuclear factor-κB (RANK) expression statuses of esophageal squamous cell carcinoma (ESCC) patients, high-grade intraepithelial neoplasia (HGIN), low-grade intraepithelial neoplasia (LGIN), and normal esophageal tissues (NETs) in Chinese Han and Kazakh ethnic, as well as the correlation of RANK expression with clinicopathological characteris...

In this paper, we introduce a new notion of quadratic Killing normal Jacobi operator (simply, operator) and its geometric meaning for real hypersurfaces in the complex Grassmannians rank two G2m+2(c), c≠0. addition, give classification theorems Hopf with two-plane compact type non-compact type.

We study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗ Hk. The rank of u is the minimal number p of pure states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C) and give normal forms for tens...

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solu...

Given a rank-two sub-Riemannian structure ( M , Δ) and point x 0 ∈ singular curve is critical of the endpoint map F : γ ↦ (1) defined on space horizontal curves starting at . The typical least degenerate these structures are called regular ; they nice if their not conjugate along main goal this paper to show that locally around once we choose suitable topology control can find normal form for m...

We present a preconditioned Monte Carlo method for computing high-dimensional multivariate normal and Student-t probabilities arising in spatial statistics. The approach combines tile-low-rank representation of covariance matrices with block-reordering scheme efficient quasi-Monte simulation. decomposes the problem into many diagonal-block-size problems low-rank connections. reorders between wi...