*Topology representing networks*

A Hebbian adaptation rule with winner-take-all like competition is introduced. It is shown that this competitive Hebbian rule forms so-called Delaunay triangulations, which play an important role in computational geometry for efficiently solving proximity problems. [1]

*Topology optimization approaches*

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research. [2]

*Modeling Internet topology*

The topology of a network, or a group of networks such as the Internet, has a strong bearing on many management and performance issues. Good models of the topological structure of a network are essential for developing and analyzing internetworking technology. This article discusses how graph-based models can be used to represent the topology of large networks, particularly aspects of locality and hierarchy present in the Internet. Two implementations that generate networks whose topology resembles that of typical internetworks are described, together with publicly available source code. [3]

*Countability of Soft Topological Space*

Based on the soft formal analysis, in this article, the countability of soft topological space is studied. The soft neighborhoods basis of soft points is dened in the soft topology, and the properties of soft basis and soft neighborhoods basis are given. The countable axioms are dened in the soft topological space over the soft formal context, the examples are illustrated for the first countable axioms and the second countable axioms. The properties of the countable axioms are studied, and all the results can be implied to the soft rough topological space. [4]

*Compactness of Soft Rough Topological Space*

In this paper, the compactness of soft topological space is discussed in the soft rough formal context T = (G,M,R, F). The countable (finite) cover is de ned over the rough soft formal context. Based on them, the soft compact topological space, compact subset, relative soft topology are de ned, and compact properties of soft rough topological space is discussed over the rough soft formal context. The sufficient and necessary condition is given to check whether a given soft rough topological space is a compact. [5]

Reference

[1] Martinetz, T. and Schulten, K., 1994. Topology representing networks. *Neural Networks*, *7*(3), pp.507-522.

[2] Sigmund, O. and Maute, K., 2013. Topology optimization approaches. *Structural and Multidisciplinary Optimization*, *48*(6), pp.1031-1055.

[3] Calvert, K.L., Doar, M.B. and Zegura, E.W., 1997. Modeling internet topology. *IEEE Communications magazine*, *35*(6), pp.160-163.

[4] Fu, L. and Li, S. (2019) “Countability of Soft Topological Space”, *Journal of Advances in Mathematics and Computer Science*, 33(6), pp. 1-11. doi: 10.9734/jamcs/2019/v33i630198.

[5] Fu, L. and Fu, H. (2016) “Compactness of Soft Rough Topological Space”, *Journal of Advances in Mathematics and Computer Science*, 14(4), pp. 1-10. doi: 10.9734/BJMCS/2016/24096.