Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup&...Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K<sub>o</sub>（R）of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>（R）=0.展开更多
The new concept"derimorphism"generalizing both derivation and homomor-phism is defined.When a derimorphism is invertible,its inverse is a Rota-Baxter operator.The general theory of derimorphism is establishe...The new concept"derimorphism"generalizing both derivation and homomor-phism is defined.When a derimorphism is invertible,its inverse is a Rota-Baxter operator.The general theory of derimorphism is established.The classification of all derimorphisms over an associative unital algebra is obtained.Contrary to the nonexistence of nontriv-ial positive derivations,it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over R[x],the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.展开更多
In this paper , some examples are given to show that the maximal number of admissible preference orderings is greater than 2n-1 when n is greater than 3. And the recurrence formula of local maximal number is presented.
In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our researc...In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our research, we find, by computer enumerating, that some results given by Craven are not correct. By defining a set of constraints, we give the recurrence formula of the local maximal number of admissible preference orderings and some other useful results.展开更多
This short note is devoted to an approach of the quasi-hereditary orderings of An-type algebras with exactly two generators. A necessary and sufficient condition for a quasi-hereditary ordering is obtained. Moreover, ...This short note is devoted to an approach of the quasi-hereditary orderings of An-type algebras with exactly two generators. A necessary and sufficient condition for a quasi-hereditary ordering is obtained. Moreover, the numbers of quasi-hereditary orderings of such algebras are explicitly given.展开更多
We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z...We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.展开更多
By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater tha...By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.展开更多
Layered Double Hydroxides MgFe-LDH and MgAl-LDH have been prepared by th e method involving separate nucleation and ageing steps. The structure analyses for these two materials show that the values of the parameters b...Layered Double Hydroxides MgFe-LDH and MgAl-LDH have been prepared by th e method involving separate nucleation and ageing steps. The structure analyses for these two materials show that the values of the parameters both a and c of M gAl-LDH are smaller than that of MgFe-LDH though their structures are simila r, and MgAl-LDH with higher crystallinity is more easily formed than MgFe-LD H in the same preparing conditions. The IR analyses manifest that the structures of layer sheets and the orderings of the anions in the interlayer regions of Mg Al-LDH are more regular than that of MgFe-LDH. The temperature programmed XR D analyses reveal that the diffraction peak of 003 reflections for MgAl-LDH co uld be seen after calcining at 300℃, while this peak for MgFe-LDH disappears after calcining at 200℃. Together with the TG-DTA analysis it can be conclu ded that the thermal stability of MgAl-LDH is obviously higher than that of Mg Fe-LDH.展开更多
基金Supported by National Nature Science Foundation of China.
文摘Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K<sub>o</sub>（R）of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>（R）=0.
基金Supported by NSFC under Grants 11771280 and 11671258by NSF of Shanghai Municipal under Grant 17ZR1415400.
文摘The new concept"derimorphism"generalizing both derivation and homomor-phism is defined.When a derimorphism is invertible,its inverse is a Rota-Baxter operator.The general theory of derimorphism is established.The classification of all derimorphisms over an associative unital algebra is obtained.Contrary to the nonexistence of nontriv-ial positive derivations,it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over R[x],the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.
文摘In this paper , some examples are given to show that the maximal number of admissible preference orderings is greater than 2n-1 when n is greater than 3. And the recurrence formula of local maximal number is presented.
基金This project is supported by Doctoral Fundation of National Educational Committee.
文摘In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our research, we find, by computer enumerating, that some results given by Craven are not correct. By defining a set of constraints, we give the recurrence formula of the local maximal number of admissible preference orderings and some other useful results.
基金the National Natural Science Foundation of China (No.10601036)
文摘This short note is devoted to an approach of the quasi-hereditary orderings of An-type algebras with exactly two generators. A necessary and sufficient condition for a quasi-hereditary ordering is obtained. Moreover, the numbers of quasi-hereditary orderings of such algebras are explicitly given.
文摘We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.
基金Project supported by the National Natural Science Foundation of China(Grant No.11347026)the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2016AM03 and ZR2017MA011)the Natural Science Foundation of Heze University,China(Grant Nos.XY17KJ09 and XY18PY13).
文摘By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.
文摘Layered Double Hydroxides MgFe-LDH and MgAl-LDH have been prepared by th e method involving separate nucleation and ageing steps. The structure analyses for these two materials show that the values of the parameters both a and c of M gAl-LDH are smaller than that of MgFe-LDH though their structures are simila r, and MgAl-LDH with higher crystallinity is more easily formed than MgFe-LD H in the same preparing conditions. The IR analyses manifest that the structures of layer sheets and the orderings of the anions in the interlayer regions of Mg Al-LDH are more regular than that of MgFe-LDH. The temperature programmed XR D analyses reveal that the diffraction peak of 003 reflections for MgAl-LDH co uld be seen after calcining at 300℃, while this peak for MgFe-LDH disappears after calcining at 200℃. Together with the TG-DTA analysis it can be conclu ded that the thermal stability of MgAl-LDH is obviously higher than that of Mg Fe-LDH.
