The purpose of this paper is to introduce the concept of a cone Symmetric space and to investigate relationship between (cone) metric spaces and (cone) Symmetric spaces. Among other things, we shall also extend some fixed point results from metric spaces to cone metric spaces (Theorem 3.3), and to Symmetric spaces (Theorems 3.2 and 3.5) under some new contraction conditions.

A series of Symmetric and aSymmetric binuclear α-diimine nickel(II) complexes toward ethylene polymerization were successfully synthesized and characterized by 1HNMR. All the catalysts were typically activated with MAO and displayed good activity at room temperature under 1atm ethylene pressure. The Symmetric catalyst containing isopropyl on ligand activated with 500 equiv MAO produced high molecular weight polyethylene with a PDI=1. 75, while the Symmetric catalyst containing methyl on ligand produced relatively low molecular weight polyethylene with a PDI=1. 74. By contrast, the molecular weight, branching density of resultant polymer produced by aSymmetric catalyst decrease roughly somewhere between that of polymers prepared by two Symmetric catalysts.

Recently, we have used the Symmetric bracket of vector fields, and developed the notion of the Symmetric derivation. Using this machinery, we have defined the concept of Symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of Symmetric forms which has been discussed by the authors. We introduce a new class of geometric vector fields and prove some basic facts about them. We call these vector fields affinewise. By contraction of the Symmetric curvature, we define two new curvatures which have direct relations to the notions of divergence, Laplacian, and the Ricci tensor.

We discuss  some potential advantages of the  orthogonal Symmetric-diagonal reduction in  two main versions of the Schur-QR method  for Symmetric positive definite  generalized eigenvalue problems. We also advise and use the appropriate reductions  as preprocessing on  the solvers, mainly  the Cholesky-QR method, of the  considered  problems. We discuss numerical stability of the  methods via providing upper bound for backward error of the computed eigenpairs and via investigating two kinds of  scaled residual errors. We also propose  and apply  two kinds of symmetrizing  which  improve  the stability and the performance  of the methods. Numerical experiments show that the  implemented versions of the Schur-QR method and the preprocessed versions of the Cholesky-QR  method are  usually more stable than the Cholesky-QR method.

A ring $R$ is called a left $\mathcal{Z}$-Symmetric ring if $ab \in \mathcal{Z}_l(R)$ implies $ba \in \mathcal{Z}_l(R)$, where $\mathcal{Z}_l(R)$ is the set of left zero-divisors. A right $\mathcal{Z}$-Symmetric ring is defined similarly, and a $\mathcal{Z}$-Symmetric ring is one that is both left and right $\mathcal{Z}$-Symmetric. In this paper, we introduce the concept of $\mathcal{Z}$-Symmetric modules as a generalization of $\mathcal{Z}$-Symmetric ring. Additionally, we introduce the concept of an eversible module as an analogy to eversible rings and prove that every eversible module is also a $\mathcal{Z}$-Symmetric module, like in the case of rings.