Netessine and Rudi（2003） consider a consumer-driven substitution problem with an arbitrary number of products under both centralized management and competition. They obtain analytically tractable solutions, establis...Netessine and Rudi（2003） consider a consumer-driven substitution problem with an arbitrary number of products under both centralized management and competition. They obtain analytically tractable solutions, establish concavity of the objective function, i.e., the expected profit function generated by each product and uniqueness of the equilibrium for the decentralize case. For the centralized case, they indicate that the objective function, i.e., the expected profit function, might not be concave and not even quasiconcave. In this note we show, for the centralize case, that the objective function is submodular, and partially verify Netessine and Rudi＇s observation that the objective function be unimodal in each of the decision variables for some case.展开更多
This paper is concerned with the enumeration of a special kind of labeled connected graphs of which the cutpoint-graphs are trees.A new method—treelization is introduced, by which the enumeration of this special kind...This paper is concerned with the enumeration of a special kind of labeled connected graphs of which the cutpoint-graphs are trees.A new method—treelization is introduced, by which the enumeration of this special kind of graphs can be solved. The enumerative formula with generating function is derived. The method of treelization is powerful in solving enumeration problems of graphs and deserves further research. For example, using the similar way, another special kind of labeled connected graphs of which the block-graphs are trees can be enumerated.展开更多
基金This study was supported by the Oklahoma Applied Research Support （OARS）, Oklahoma Center for the Advancement of Science and Technology （OCAST）, the State of Oklahoma through the Project AR062-034, and the United States Department of Energy under the Genomics： GTL program through the Virtual Institute of Microbial Stress and Survival （VIMSShttp：//vimss.lbl.gov）, Environmental Remediation Science Program （ERSP）, Office of Biological and Environmental Research, Office of Science.
文摘Netessine and Rudi（2003） consider a consumer-driven substitution problem with an arbitrary number of products under both centralized management and competition. They obtain analytically tractable solutions, establish concavity of the objective function, i.e., the expected profit function generated by each product and uniqueness of the equilibrium for the decentralize case. For the centralized case, they indicate that the objective function, i.e., the expected profit function, might not be concave and not even quasiconcave. In this note we show, for the centralize case, that the objective function is submodular, and partially verify Netessine and Rudi＇s observation that the objective function be unimodal in each of the decision variables for some case.
文摘This paper is concerned with the enumeration of a special kind of labeled connected graphs of which the cutpoint-graphs are trees.A new method—treelization is introduced, by which the enumeration of this special kind of graphs can be solved. The enumerative formula with generating function is derived. The method of treelization is powerful in solving enumeration problems of graphs and deserves further research. For example, using the similar way, another special kind of labeled connected graphs of which the block-graphs are trees can be enumerated.
