Synthesis,crystal structure and topological analysis of a three‐dimensional polymeric network based on zinc(II), potassium and 5‐sulfobenzene‐1,3‐dicarboxylate (SIP)

Aromatic polycarboxylate linkers provide structural rigidity and strong interactions among the metal centre and the carboxylate O atoms. A new three‐dimensional coordination polymer namely, catena‐poly[potassium [tetraaqua(μ‐5‐sulfobenzene‐1,3‐dicarboxylato)zinc(II)]], {K[Zn(C₈H₃O₇S)(H₂O)₄]}_n or {K[Zn(SIP)(H₂O)₄]}_n, where SIP is 5‐sulfobenzene‐1,3‐dicarboxylate or 5‐sulfoisophthalate, was obtained and characterized by elemental analysis and IR vibrational spectroscopy, and the single‐crystal structure was determined by X‐ray diffraction analysis. The compound crystallizes in the monoclinic space group P2₁/n with Z = 4. Topological analysis revealed that K—O interactions form a two‐dimensional network, which is uninodal 4‐connected and can be described with a point symbol (4⁴.6²), and this plane network is classified as sql/Shubnikov . The layers are connected by Zn²⁺ ions coordinated to the SIP linker, forming a three‐dimensional network. This net is a trinodal (3,5,6)‐connected system with point symbol (3.4⁴.5².6².7³.8³).(3.4⁴.5².6².7).(3.7²).     

有序拓扑向量空间中的共鸣定理

本文研究有序拓扑向量空间中非线性映照的共鸣定理．对于取值于有序拓扑向量空间中的映照，利用序关系，引入了一类广泛的非线性映照．对于这类非线性映照，应用纲理论，并给出了关于点态序有界蕴涵一致序有界的共鸣定理．     

On homeomorphism groups of Menger continua

It is shown that the homeomorphism groups of the (generalized) Sierpinski carpet and the universal Menger continua are not zero-dimensional. These results were corollaries to a 1966 theorem of Brechner. New proofs were needed because we also show that Brechner's proof is inadequate. The method by which we obtain our results, the construction of closed imbeddings of complete Erdos space in the homeomorphism groups, is of independent interest.

     

L-fuzzy拓扑空间中的包含式正则分离性

本文首先在一般L-fuzzy拓扑空间中引入了包含式正则分离性的概念.其次将包含式正则分离性与点式正则分离性及正则分离性作了比较,讨论了它们之间的相互蕴含关系.最后说明了包含式正则分离性与包含式正规分离性及包含式完全正则分离性之间的协调性.     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

Synthesis and structure of the framework coordination polymer based on the cluster anion [Nb<Subscript>4</Subscript>OTe<Subscript>4</Subscript>(CN)<Subscript>12</Subscript>]<Superscript>6−</Superscript> and Mn<Superscript>II</Superscript> aqua complexes

Dark-brown plate crystals of the [Mn₇(H₂O)₂₆{Nb₄OTe₄(CN)₁₂} ₂](OH)₂·11H₂O compound (1) were prepared by the reaction of an aqueous ammonia solution of the K₆[Nb₄OTe₄(CN)₁₂]{K₂CO₃{KOH{8H₂O complex with a glycerol solution of manganese(II) nitrate. The structure of complex 1 was established by X-ray diffraction. Compound 1 has a polymer structure containing four types of manganese atoms. The nitrogen atoms of eight cyano groups of the tetranuclear niobium cluster are coordinated to the manganese atoms to form a {2,3,8}-connected three-dimensional network. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 224–228, February, 2007.     

Epimorphisms and cowellpoweredness for separated metrically generated theories

For metrically generated constructs X we give an internal characterization of the regular closure operator on X, determined by the subconstruct X ₀, consisting of its T ₀ objects. This allows us to describe the epimorphisms in X ₀ and to show that all the constructs of that type are cowellpowered. We capture many known results but our method also gives solutions in cases where the epimorphism problem was still open. The third author is “research assistant” at the FWO-Vlaanderen.     

不变平均和拓扑不变平均

设G为局部紧群，在一致连续函数空间U(G)上，用两种方法证明左不变平均和拓扑左不变的等价性．当G是交换群时．给出一种证明其顺从的方法．     

Existence of Periodic Solutions of an Autonomous Damped Wave Equation in Thin Domains

For a nonlinear autonomous damped wave equation in a thin domain we provide conditions ensuring the existence of periodic solutions in time. Our approach uses both methods developed by Hale and Raugel and methods based on the topological degree theory together with some results on the functionalization of parameter.