The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree pa...The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns： All sporadic simple groups, the alternating groups Ap （p ≤ 5 is a twin prime） and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups Sp＋3, where p ＋ 2 is a composite number and p ＋ 4 is a prime and 97 〈 p ∈π（1000!）, are 3-fold OD-characterizable. The authors also show that the alternating groups All6 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.展开更多
We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to sev...We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.展开更多
基金the National Natural Science Foundation of China(Nos.11271301,11171364)the National Science Foundation for Distinguished Young Scholars of China(No.11001226)+2 种基金the Fundamental Research Funds for the Central Universities(Nos.XDJK2012D004,XDJK2009C074)the Natural Science Foundation Project of CQ CSTC(Nos.2011jjA00020,2010BB9206)the GraduateInnovation Funds of Science of Southwest University(No.ky2009013)
文摘The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns： All sporadic simple groups, the alternating groups Ap （p ≤ 5 is a twin prime） and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups Sp＋3, where p ＋ 2 is a composite number and p ＋ 4 is a prime and 97 〈 p ∈π（1000!）, are 3-fold OD-characterizable. The authors also show that the alternating groups All6 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.
基金supported by National Natural Science Foundation of China(Grant No.11301022)the State Key Laboratory of Rail Traffic Control and Safety,Beijing Jiaotong University(Grant Nos.RCS2014ZT20 and RCS2014ZZ001)+1 种基金Beijing Natural Science Foundation(Grant No.9144031)the Hong Kong Research Grant Council(Grant Nos.Poly U 501909,502510,502111 and 501212)
文摘We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.
