Local Bifurcation of Critical Periods for a Class of Liénard Equations

In this paper,we study the local bifurcation of critical periods near the nondegenerate center(the origin) of a class of Liénard equations with degree 2n,and prove that at most 2n-2 critical periods(taken into account multiplicity) can be produced from a weak center of finite order.We also prove that it can have exactly2n-2 critical periods near the origin.     

Optimizing inventory decisions in a two-level supply chain with order quantity constraints

A fundamental assumption in traditional inventory models is that all of the ordered items are of perfect quality. A two-level supply chain is considered consists of one retailer and a collection of suppliers that operate within a finite planning horizon, including multiple periods, and a model is formulated that simultaneously determines both supplier selection and inventory allocation problems in the supply chain. It is supposed that the ordered products dependent on the suppliers include a certain percentage of imperfect quality products and have different prices. In this paper, we study the impact of the retailer’s financial constraint. On the other hand, suppliers have restricted capacities and set minimum order quantity (MOQ) policy for the retailer’s order amount happened in each period. So, the problem is modeled as a mixed integer nonlinear programming. The purpose of this model is to maximize the total profit. The nutrients, fishery and fruitage industries give good examples for the proposed model. A numerical example is presented to indicate the efficiency of the proposed model. Considering the complexity of the model, a genetic algorithm (GA) is presented to solve the model. We demonstrate analytically that the proposed genetic algorithm is suitable in the feasible situations.     

Atomic emission and cavity field spectra of the Jaynes－Cummings model

Atomic emission and cavity field spectra of the Jaynes－Cummings model are analytically compared. We show that the two spectra are in general different, except for the special case where the interaction is resonant, the cavity is initially empty and the retarded time of one-photon emission is considered in calculations of the spectra. In this case, the two spectra are the same. We also show that this result comes from the fact that the two-time autocorrelation function of the cavity field is not directly proportional to that of the atom.     

Seasonal characteristics and provenance of organic aerosols in the urban atmosphere of Liaocheng in the North China Plain: Significant effect of biomass burning

To better understand the seasonal characteristics of urban organic aerosol (OA) in the North China Plain (NCP), PM_2.5 samples in the urban atmosphere of Liaocheng were collected and analyzed. The molecular distribution of the organic markers in the urban atmosphere of Liaocheng reveals that n-alkanes (39.3%) was the most abundant species all year round, followed by saccharides (28.2%), phthalic acids (Ph, 20.8%), biogenic secondary organic aerosol (BSOA) tracers (9.4%), and polycyclic aromatic hydrocarbon (PAHs, 2.3%). PM_2.5, organic carbon (OC), elemental carbon (EC), and primary organic markers exhibit the highest concentrations in winter, due largely to the increased biomass burning and coal combustion for house heating in local and surrounding regions. However, the concentration and relative abundance of BSOA are significantly higher in summer than other seasons, induced by the more favorable meteorological conditions that would promote the emissions of biogenic volatile organic compounds (BVOCs) and the secondary production of BSOA. The ratios of OC/EC and 3-methyl-1,2,3-butanetricarboxylic acid to cis-pinic acid plus cis-pinonic acid (MBTCA/(PA + PNA) are higher in the warm seasons than those in the cold seasons, indicating that the oxidation of OA is sensitive to air temperature. Compared to 2017, the concentration level of PAHs during wintertime decreased by 40.8%, confirming that the stringent regulation of coal burning is effective. The highest concentration of high molecular weight (HMW) n-alkanes and three anhydrosugars in winter, and the close correlation of levoglucosan with HMW n-alkanes suggests that the impact of biomass burning was more significant in winter. The same seasonal characteristic of the ratios of high-/low-NO_x products with NO_x and the strong correlation of high-/low-NO_x products with levoglucosan indicate that the formation of isoprene SOA (SOA_I) tracers was significantly influenced by anthropogenic emissions. The molecular compositions, the distributions of fire spots, backward trajectories of air masses, and correlation analysis suggest that air pollution events in spring were primarily resulted from biomass burning and secondary oxidation, while pollution events in winter were largely driven by the increased combustion sources, and promoted aqueous secondary formation. Our results suggest that the reduction of biomass and coal combustion should be taken into account to improve the urban air quality in the NCP.