Decision implication is a form of decision knowledge represen-tation,which is able to avoid generating attribute implications that occur between condition attributes and between decision attributes.Compared with other...Decision implication is a form of decision knowledge represen-tation,which is able to avoid generating attribute implications that occur between condition attributes and between decision attributes.Compared with other forms of decision knowledge representation,decision implication has a stronger knowledge representation capability.Attribute granularization may facilitate the knowledge extraction of different attribute granularity layers and thus is of application significance.Decision implication canonical basis(DICB)is the most compact set of decision implications,which can efficiently represent all knowledge in the decision context.In order to mine all deci-sion information on decision context under attribute granulating,this paper proposes an updated method of DICB.To this end,the paper reduces the update of DICB to the updates of decision premises after deleting an attribute and after adding granulation attributes of some attributes.Based on this,the paper analyzes the changes of decision premises,examines the properties of decision premises,designs an algorithm for incrementally generating DICB,and verifies its effectiveness through experiments.In real life,by using the updated algorithm of DICB,users may obtain all decision knowledge on decision context after attribute granularization.展开更多
We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
Canonical bases of the tensor powers of the natural $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ -module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This re...Canonical bases of the tensor powers of the natural $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ -module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the ?2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra O q (M m|n ) of a quantum (m,n) × (m,n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ by applying a quantum analogue of the Borel-Weil construction.展开更多
We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these...We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.展开更多
The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. W...The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. We discuss the tight monomials in quantized enveloping algebra of type B3.展开更多
基金supported by the National Natural Science Foundation of China (Nos.61972238,62072294).
文摘Decision implication is a form of decision knowledge represen-tation,which is able to avoid generating attribute implications that occur between condition attributes and between decision attributes.Compared with other forms of decision knowledge representation,decision implication has a stronger knowledge representation capability.Attribute granularization may facilitate the knowledge extraction of different attribute granularity layers and thus is of application significance.Decision implication canonical basis(DICB)is the most compact set of decision implications,which can efficiently represent all knowledge in the decision context.In order to mine all deci-sion information on decision context under attribute granulating,this paper proposes an updated method of DICB.To this end,the paper reduces the update of DICB to the updates of decision premises after deleting an attribute and after adding granulation attributes of some attributes.Based on this,the paper analyzes the changes of decision premises,examines the properties of decision premises,designs an algorithm for incrementally generating DICB,and verifies its effectiveness through experiments.In real life,by using the updated algorithm of DICB,users may obtain all decision knowledge on decision context after attribute granularization.
文摘We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.
基金supported by National Natural Science Foundation of China (Grant No. 10471070)
文摘Canonical bases of the tensor powers of the natural $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ -module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the ?2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra O q (M m|n ) of a quantum (m,n) × (m,n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ by applying a quantum analogue of the Borel-Weil construction.
基金Supported by National Natural Science Foundation of China(Grant Nos.10631060 and 11131008)
文摘We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11226055） and Shanghai Training Foundation for University Young Teacher （No. ZZhy12029）.
文摘The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. We discuss the tight monomials in quantized enveloping algebra of type B3.
