Let A be a normal class of algebras. In the present paper, we characterize the following four problems for A: for which radical class R, there holds that(1) R(i1∧i2) = R(i1)∧R(i2);(2) R(i1∨i2) = R(i1)∨R(i2);(3) (i...Let A be a normal class of algebras. In the present paper, we characterize the following four problems for A: for which radical class R, there holds that(1) R(i1∧i2) = R(i1)∧R(i2);(2) R(i1∨i2) = R(i1)∨R(i2);(3) (i1∧i2) =i1∧i2;(4) (i1∨i2) = i1∨i2, for arbitrary algebra a∈A and any i1,i2∈La,where j denotes the ideal of a uniquely determined by R(a/j) = j/j?展开更多
Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general st...Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general strongly semisimple radicals and investigate some properties of them in normal classes of algebras. This extends some known studies on the theory of radicals of various algebraic strutures.展开更多
In object oriented paradigm, cohesion of a class refers to the degree to which members of the class are interrelated. Metrics have been defined to measure cohesiveness of a class both at design and source code levels....In object oriented paradigm, cohesion of a class refers to the degree to which members of the class are interrelated. Metrics have been defined to measure cohesiveness of a class both at design and source code levels. In comparison to source code level class cohesion metrics, only a few design level class cohesion metrics have been proposed. Design level class cohesion metrics are based on the assumption that if all the methods of a class have access to similar para-meter types then they all process closely related information. A class with a large number of parameter types common in its methods is more cohesive than a class with less number of parameter types common in its methods. In this paper, we review the design level class cohesion metrics with a special focus on metrics which use similarity of parameter types of methods of a class as the basis of its cohesiveness. Basically three metrics fall in this category: Cohesion among Methods of a Class (CAMC), Normalized Hamming Distance (NHD), and Scaled NHD (SNHD). Keeping in mind the anomalies in the definitions of the existing metrics, a variant of the existing metrics is introduced. It is named NHD Modified (NHDM). An automated metric collection tool is used to collect the metric data from an open source software program. The metric data is then subjected to statistical analysis.展开更多
Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and ...Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.展开更多
基金The NSF (2024201051) of Liaoning Education Department.
文摘Let A be a normal class of algebras. In the present paper, we characterize the following four problems for A: for which radical class R, there holds that(1) R(i1∧i2) = R(i1)∧R(i2);(2) R(i1∨i2) = R(i1)∨R(i2);(3) (i1∧i2) =i1∧i2;(4) (i1∨i2) = i1∨i2, for arbitrary algebra a∈A and any i1,i2∈La,where j denotes the ideal of a uniquely determined by R(a/j) = j/j?
文摘Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general strongly semisimple radicals and investigate some properties of them in normal classes of algebras. This extends some known studies on the theory of radicals of various algebraic strutures.
文摘In object oriented paradigm, cohesion of a class refers to the degree to which members of the class are interrelated. Metrics have been defined to measure cohesiveness of a class both at design and source code levels. In comparison to source code level class cohesion metrics, only a few design level class cohesion metrics have been proposed. Design level class cohesion metrics are based on the assumption that if all the methods of a class have access to similar para-meter types then they all process closely related information. A class with a large number of parameter types common in its methods is more cohesive than a class with less number of parameter types common in its methods. In this paper, we review the design level class cohesion metrics with a special focus on metrics which use similarity of parameter types of methods of a class as the basis of its cohesiveness. Basically three metrics fall in this category: Cohesion among Methods of a Class (CAMC), Normalized Hamming Distance (NHD), and Scaled NHD (SNHD). Keeping in mind the anomalies in the definitions of the existing metrics, a variant of the existing metrics is introduced. It is named NHD Modified (NHDM). An automated metric collection tool is used to collect the metric data from an open source software program. The metric data is then subjected to statistical analysis.
文摘Let（T<sub>1</sub>,T<sub>2</sub>）and （S<sub>1</sub><sup>*</sup>,S<sub>2</sub><sup>*</sup>） be double commuting subnormal operator pairs,X and K be bounded linear operators with T<sub>1</sub>KS<sub>1</sub>+T<sub>2</sub>KS<sub>2</sub>=0. Then （1）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K||≥||K||and （2）||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub>+K<sub>1</sub>|<sub>2</sub><sup>2</sup>=||T<sub>1</sub>XS<sub>1</sub>+T<sub>2</sub>XS<sub>2</sub> ||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if K is Hilbert-Schmidt operator,Let T and S* be dominant operator and subnormal operator, respectively, and K is Hilbert-Schmidt operator, then （3）||TX-XS+K||<sub>2</sub><sup>2</sup>=||TX-XS||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup>if TK+KS;and （4）||TXS-X+K||<sub>2</sub><sup>2</sup>=||TXS-X||<sub>2</sub><sup>2</sup>+||K||<sub>2</sub><sup>2</sup> if TKS=K. These generalize the results of （1）and （3）.
