The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζ...The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζi（G） for all integer i or wi（G） = ζi＋1（G） for every integer i, and w（G/K） = ζ（G/K） for every normal subgroup g in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w（G） = ζ2（G） is given. Finally, the authors prove that the power automorphism group PAut（G） is an elementary abelian p-group if G is a non-abelian p- group with elementary ζ（G） ∩ζ1（G）.展开更多
Suppose that A is an n×n positive definite Hemitain matrix. Let X and Y ben×p and n×q matrices（p+q≤n）, such that X<sup>*</sup>Y=O. The following inequality is provedX<sup>*</sup&...Suppose that A is an n×n positive definite Hemitain matrix. Let X and Y ben×p and n×q matrices（p+q≤n）, such that X<sup>*</sup>Y=O. The following inequality is provedX<sup>*</sup>AY（Y<sup>A</sup>Y）<sup>-</sup>Y<sup>*</sup>AX≤（（λ<sub>1</sub>-λ<sub>n</sub>）/（λ<sub>1</sub>+λ<sub>n</sub>）<sup>2</sup>）X<sup>*</sup>AX,where λ<sub>1</sub> and λ<sub>n</sub> are respectively the largest and smallest eigenvalues of A, and M<sup>-</sup> stands for a generalized inverse of M. This inequality is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation.展开更多
基金supported by the National Natural Science Foundation of China (No. 11071155)the Key Disciplines of Shanghai Municipality (No. S30104)
文摘The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζi（G） for all integer i or wi（G） = ζi＋1（G） for every integer i, and w（G/K） = ζ（G/K） for every normal subgroup g in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w（G） = ζ2（G） is given. Finally, the authors prove that the power automorphism group PAut（G） is an elementary abelian p-group if G is a non-abelian p- group with elementary ζ（G） ∩ζ1（G）.
文摘Suppose that A is an n×n positive definite Hemitain matrix. Let X and Y ben×p and n×q matrices（p+q≤n）, such that X<sup>*</sup>Y=O. The following inequality is provedX<sup>*</sup>AY（Y<sup>A</sup>Y）<sup>-</sup>Y<sup>*</sup>AX≤（（λ<sub>1</sub>-λ<sub>n</sub>）/（λ<sub>1</sub>+λ<sub>n</sub>）<sup>2</sup>）X<sup>*</sup>AX,where λ<sub>1</sub> and λ<sub>n</sub> are respectively the largest and smallest eigenvalues of A, and M<sup>-</sup> stands for a generalized inverse of M. This inequality is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation.
