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Granular computing is an emerging computing paradigm of information processing. Rough set theory is a kind of models of granular computing and is used to deal with the vagueness and granularity in information systems. Covering-based rough set theory is one of the most important extensions of the classical Pawlak rough set theory. In the covering-based rough set theory, the covering lower and upper approximation operators are two basic concepts. The axiomatic characterizations of covering-based approximation operators guarantee the existence of coverings reproducing the operators, so the rough set axiomatic system is the foundation of the covering-based rough set theory. In this paper, we present a new covering upper approximation operator and explore the basic properties of the covering upper approximation operator to find an axiomatic set for characterizing the covering upper approximation operator. We also discuss the relationships between the covering upper approximation operator and three other covering upper approximation operators, compare the covering upper approximation operator with the other covering upper approximation operators by defining precision degrees and rough degrees, and present the necessary and sufficient conditions for the covering upper approximation operators to be identical. The results not only are beneficial to enrich the kinds of the covering rough sets, but also have theoretical and actual significance to covering rough sets. [ABSTRACT FROM AUTHOR]

Purpose: To detect the expression of programmed cell death-1 (PD-1) in peripheral blood T lymphocytes of patients with primary liver cancer and its effect on their prognosis., Methods: The medical records of 42 patients with primary liver cancer, 36 patients with chronic hepatitis B (CHB) and 38 healthy volunteers composed the liver cancer group, benign lesion group and control group, respectively. Fluorescence quantitative RT-PCR (qRT-PCR) was used for detecting the expression level of PD-1 mRNA in peripheral blood T lymphocytes of subjects in the three groups, and flow cytometry was used for detecting the positive expression of PD-l protein on the surface of T lymphocytes of subjects., Results: Patients in the control group and the benign lesion group had lower expression level of PD-1 mRNA than those in the liver cancer group (p<0.05). Patients in the control group had lower expression level of PD-1 mRNA than those in the benign lesion group (p<0.05). Patients in the control group and the benign lesion group had lower positive expression rate of PD-1 protein on the surface of T lymphocytes than those in the liver cancer group (p<0.05)., Conclusion: The expression of PD-1 in the peripheral blood was higher in patients with primary liver cancer. Patients in the PD-1 low expression group had significantly better prognosis than those in the PD-1 high expression group. PD-1 may be related to the occurrence and development of primary liver cancer and is worthy of further study.