Balance Properties and Stabilization of Min-Max Systems

A variety of problems in operations research, performance analysis, manufacturing, and communication networks, etc., can be modelled as discrete event systems with minimum and maximum constraints. When such systems require only maximum constraints （or dually, only minimum constraints）, they can be studied using linear methods based on max-plus algebra. Systems with mixed constraints are called min-max systems in which rain, max and addition operations appear simultaneously. A significant amount of work on such systems can be seen in literature. In this paper we provide some new results with regard to the balance problem of min-max functions; these are the structure properties of min-max systems. We use these results in the structural stabilization. Our main results are two sufficient conditions for the balance and one sufficient condition for the structural stabilization. The block technique is used to analyse the structure of the systems. The proposed methods, based on directed graph and max-plus algebra are constructive in nature. We provide several examples to demonstrate how the methods work in practice.

Integrated genetic and epigenetic analysis reveals DNA repair alterations in multifocal hepatocellular carcinoma

Dear Editor,Hepatocellular carcinoma(HCC)ranks the fourth most lethal cancer worldwide and over 50%of cases are diagnosed as multifocal HCC(mHCC)with dismal prognosis.1 mHCC displays more complicated intratumor heterogeneity(ITH)and clonal evolution course which decreases the efficacy of clinical treatments.

Stabilization technology and corrosion mechanism of rust layer on Q370 weathering steel surface

A newly developed rust layer on Q370qENH weathering steel(sample C)was studied under dry–wet cycle test of 3.5 wt.%NaCl solution and salt spray test,compared with the bare weathering steel(sample A)and the weathering steel with additive contents(CuSO_(4)0.2,FeSO_(4)0.1,NaHSO_(3)0.02,NaCl 0.02,Na_(2)HPO_(4)0.01,and water balance,in mass%,termed as sample B).Corrosion mechanism of Q370qENH weathering steel after different surface treatments was investigated by means of surface potential scanning,mass change,polarization curve,and X-ray diffraction.The results of X-ray diffraction indicate that rust of bare weathering steel has rare Fe_(3)O_(4),and a dense oxide layer mainly composed of Fe_(3)O_(4) is formed on rust of both sample B and sample C.The surface potential of sample A gradually increases to−0.2 V in the dry–wet cycle test,while the surface potential of sample A maintains at about−0.6 V in the salt spray test.The surface potential of sample B and sample C is higher than that of sample A in the early stage and remains stable during the short-term accelerated corrosion test.Moreover,the mass change of sample C gradually stabilizes with time in the dry–wet cycle and salt spray test.The corrosion current of the sample A is lower than that of sample C in the initial stage of short-term accelerated corrosion test.However,the corrosion current of sample C is smaller than that of the sample A in the experiment.The main components of the deep rust of sample A areγ-FeOOH andα-FeOOH,while those of the deep rust of sample B and sample C are Fe_(3)O_(4).