Systematic thermodynamic analysis reveals that an essential condition for the thermodynamically valid chemographic projec-tions proposed by Greenwood is completely excessive.In other words,the phases or components fro...Systematic thermodynamic analysis reveals that an essential condition for the thermodynamically valid chemographic projec-tions proposed by Greenwood is completely excessive.In other words,the phases or components from which the projection is made need not be pure,nor have their chemical potentials fixed over the whole chemographic diagram.To facilitate the analy-sis of phase assemblages in multicomponent systems,all phases and components in the system are divided into internal and external ones in terms of their thermodynamic features and roles,where the external phases are those common to all assem-blages in the system,and the external components include excess components and the components whose chemical potentials(or relevant intensive properties of components) are used to define the thermodynamic conditions of the system.This general classification overcomes the difficulties and defects in the previous classifications,and is easier to use than the previous ones.According to the above classification,the phase rule is transformed into a new form.This leads to two findings:(1) the degree of freedom of the system under the given conditions is only determined by the internal components and phases;(2) different external phases can be identified conveniently according to the conditions of the system before knowing the real phase rela-tions.Based on the above results,a simple but general approach is proposed for the treatment of phases and components:all external phases and components can be eliminated from the system without affecting the phase relations,where the external components can be eliminated by appropriate chemographic projections.The projections have no restriction on the states of the phases or the chemical potentials of components from which the projections are made.The present work can give a unified ex-planation of the previous treatments of phases and components in the analysis of phase assemblages under various specific conditions.It helps to avoid potential misunderstandings or errors in the topological analysis of phase relations.展开更多
Univariate Gonarov polynomials arose from the Goncarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose ord...Univariate Gonarov polynomials arose from the Goncarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Goncarov polynomials,which form a basis of solutions for multivariate Goncarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Goncarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Goncarov polynomials.展开更多
基金supported by National Natural Science Founda-tion of China (Grant No.40873018)Open Foundation of the State Key La-boratory of Ore Deposit Geochemistry,Guiyang Institute of Geochemistry,Chinese Academy of Sciences (Grant No.200807)+1 种基金the Open Fund (Grant No.PLC201001) of the State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology)the Natural Science Foundation of Hebei Province (Grant No.D2008000535)
文摘Systematic thermodynamic analysis reveals that an essential condition for the thermodynamically valid chemographic projec-tions proposed by Greenwood is completely excessive.In other words,the phases or components from which the projection is made need not be pure,nor have their chemical potentials fixed over the whole chemographic diagram.To facilitate the analy-sis of phase assemblages in multicomponent systems,all phases and components in the system are divided into internal and external ones in terms of their thermodynamic features and roles,where the external phases are those common to all assem-blages in the system,and the external components include excess components and the components whose chemical potentials(or relevant intensive properties of components) are used to define the thermodynamic conditions of the system.This general classification overcomes the difficulties and defects in the previous classifications,and is easier to use than the previous ones.According to the above classification,the phase rule is transformed into a new form.This leads to two findings:(1) the degree of freedom of the system under the given conditions is only determined by the internal components and phases;(2) different external phases can be identified conveniently according to the conditions of the system before knowing the real phase rela-tions.Based on the above results,a simple but general approach is proposed for the treatment of phases and components:all external phases and components can be eliminated from the system without affecting the phase relations,where the external components can be eliminated by appropriate chemographic projections.The projections have no restriction on the states of the phases or the chemical potentials of components from which the projections are made.The present work can give a unified ex-planation of the previous treatments of phases and components in the analysis of phase assemblages under various specific conditions.It helps to avoid potential misunderstandings or errors in the topological analysis of phase relations.
基金supported by the National Priority Research Program (Grant No. #[5101-1-025]) from the Qatar National Research Fund (a member of Qatar Foundation)
文摘Univariate Gonarov polynomials arose from the Goncarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Goncarov polynomials,which form a basis of solutions for multivariate Goncarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Goncarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Goncarov polynomials.
