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Solutions of minus partial ordering equations over von Neumann regular rings

Yu Guan;  Zhaojia Tong — 2015

Open Mathematics

In this paper, we mainly derive the general solutions of two systems of minus partial ordering equations over von Neumann regular rings. Meanwhile, some special cases are correspondingly presented. As applications, we give some necessary and sufficient conditions for the existence of solutions. It can be seen that some known results can be regarded as the special cases of this paper.

The logarithmic Sobolev constant of some finite Markov chains

Guan-Yu Chen;  Wai-Wai Liu;  Laurent Saloff-Coste — 2008

Annales de la faculté des sciences de Toulouse Mathématiques

The logarithmic Sobolev constant is always bounded above by half the spectral gap. It is natural to ask when this inequality is an equality. We consider this question in the context of reversible Markov chains on small finite state spaces. In particular, we prove that equality holds for simple random walk on the five cycle and we discuss assorted families of chains on three and four points.

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