Optimality of a Class of Entanglement Witnesses for 3⊗3 Systems

Let $\varPhi_{t,\pi}: M_{3}({\mathbb{C}}) \rightarrow M_{3}({\mathbb{C}})$ be a linear map defined by $\varPhi_{t,\pi}(A)=(3-t)\*\sum_{i=1}^{3}E_{ii}AE_{ii}+t\sum_{i=1}^{3}E_{i,\pi (i)}AE_{i,\pi(i)}^{\dag}-A$ , where 0≤t≤3 and π is a permutation of (1,2,3). We show that the Hermitian matrix $W_{\varPhi_{t,\pi}}$ induced by Φ _t,π is an optimal entanglement witness if and only if t=1 and π is cyclic.     

Global Regularity of the 3D Axi-Symmetric Navier–Stokes Equations with Anisotropic Data

In this paper, we study the 3D axisymmetric Navier–Stokes equations with swirl. We prove the global regularity of the 3D Navier–Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some L ^p norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.     

Permanence and periodic solutions for a class of delay Nicholson’s blowflies models

This paper covers the dynamic behaviors for a class of Nicholson’s blowflies models with a nonlinear density-dependent mortality term. By using inequality analyze technique and coincidence degree theory, some sufficient conditions are determined that guarantee the permanence of the model and the existence of positive periodic solutions. Moreover, we give an example to illustrate our main results.     

Distance unimodular equivalence of graphs

Let G be a connected graph and D(G) be its distance matrix. In this article, the Smith normal forms of the integer matrices D(G) are determined for trees, wheels, cycles, complements of cycles and are reduced for complete multipartite graphs.     

The Join of Split Graphs Whose Regular Endomorphisms Form a Monoid

In this paper, the regular endomorphisms of the join of split graphs are investigated. We give a condition under which the regular endomorphisms of the join of split graphs form a monoid.     

Observation of the growth of carbon nanotubes prepared at low temperature

Lack of information during the growth of carbon nanotubes has hindered efforts to understand the growth mechanism of these unique nanoscale structures. Here, we report that Multi‐walled carbon nanotubes with bamboo‐shaped structure may be synthesized by solvothermal method at 300°C using benzene as carbon source and Zn as catalyst. We have obtained direct experimental information on the growth under Transmission electron microscopy. It has been shown that all the compartments of bamboo‐like tube successively joined leading to the formation of regularly segmented nanobamboos. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

DBU-promoted tandem Michael-addition/cyclization for the synthesis of polysubstituted pyrroles

An efficient and transition-metal-free method for the synthesis of the structurally diversified pyrroles is described. Various α,β-unsaturated ynones reacted with N-substituted ethyl glycine ethyl ester hydrochlorides in the presence of DBU to form the corresponding products in good yields. This protocol has the advantages of readily available starting materials, mild reaction conditions and a wide scope of substrates, which provides a practical route for the synthesis of polysubstituted pyrroles.     

A Noble‐Metal‐Free Metal–Organic Framework (MOF) Catalyst for the Highly Efficient Conversion of CO2 with Propargylic Alcohols

Cyclization of propargylic alcohols with CO₂ is an important reaction in industry, and noble‐metal catalysts are often employed to ensure the high product yields under environmentally friendly conditions. Herein a porous noble‐metal‐free framework 1 with large 1D channels of 1.66 nm diameter was synthesized for this reaction. Compound 1 exhibits excellent acid/base stability, and is even stable in corrosive triethylamine for one month. Catalytic studies indicate that 1 is an effective catalyst for the cyclization of propargylic alcohols and CO₂ without any solvents under mild conditions, and the turnover number (TON) can reach to a record value of 14 400. Furthermore, this MOF catalyst also has rarely seen catalytic activity when the biological macromolecule ethisterone was used as a substrate. Mechanistic studies reveal that the synergistic catalytic effect between Cu^I and In^III plays a key role in the conversion of CO₂.