Perturbation estimates for the nonlinear matrix equation X−A * X q A=Q (0&lt;q&lt;1)

Guan-jun, Jia; Gao, Dongjie

doi:10.1007/s12190-009-0357-z

Cited by 16 publications

(23 citation statements)

References 2 publications

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“…The following definitions and the notations are the same as in [1]. We denote by C n×n the set of n × n complex matrices, and we denote by H n×n the set of n × n Hermitian matrices.…”

Section: Preliminariesmentioning

confidence: 99%

Comment to: Perturbation estimates for the nonlinear matrix equation X−A ∗ X q A=Q (0<q<1) by G. Jia and D. Gao

Berzig

2012

J. Appl. Math. Comput.

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We show that the perturbation estimate for the matrix equation X − A * X q A = Q (0 < q < 1), due to G. Jia and D. Cao, published in this journal, is wrong. Mathematics Subject Classification PreliminariesThe following definitions and the notations are the same as in [1]. We denote by C n×n the set of n × n complex matrices, and we denote by H n×n the set of n × n Hermitian matrices. We write X ≥ Y (X > Y) if X − Y is positive semidefinite (definite). If X, Y ∈ H n×n such that X ≤ Y , then [X, Y ] will be the set of all Z ∈ H n×n satisfying X ≤ Z ≤ Y . We denote by · the spectral norm. For Hermitian matrix N , let λ max (N ) and λ min (N ) be the maximal and minimal eigenvalue of N , respectively.Consider the matrix equationwhere A, Q ∈ C n×n and Q is positive definite matrix. The existence and uniqueness of its positive definite solution X is proved in [2], furthermore X ∈ [βI, αI ], where

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“…The following definitions and the notations are the same as in [1]. We denote by C n×n the set of n × n complex matrices, and we denote by H n×n the set of n × n Hermitian matrices.…”

Section: Preliminariesmentioning

confidence: 99%

Comment to: Perturbation estimates for the nonlinear matrix equation X−A ∗ X q A=Q (0<q<1) by G. Jia and D. Gao

Berzig

2012

J. Appl. Math. Comput.

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“…Analytical calculations show that a magnetic field which is parallel to the electric field can increase the pair-production rate [16,[37][38][39][40]. Recently the enhancement of the pair-production rate due to parallel magnetic fields has been studied in string theory [41][42][43]. The pair production rate is modified by the thermal medium [44][45][46].…”

Section: Introductionmentioning

confidence: 99%

Wigner function and pair production in parallel electric and magnetic fields

et al. 2019

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We derive analytical formulas for the equal-time Wigner function in an electromagnetic field of arbitrary strength. While the magnetic field is assumed to be constant, the electric field is assumed to be space-independent and oriented parallel to the magnetic field. The Wigner function is first decomposed in terms of the so-called Dirac-Heisenberg-Wigner (DHW) functions and then the transverse-momentum dependence is separated using a new set of basis functions which depend on the quantum number n of the Landau levels. Equations for the coefficients are derived and then solved for the case of a constant electric field. The pair-production rate for each Landau level is calculated. In the case of finite temperature and chemical potential, the pair-production rate is suppressed by Pauli's exclusion principle.

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“…Previous reports have shown that central-metal ions can govern the magnetic exchange coupling interaction [24,25]. In particular, isostructural MOFs, which have identical architectures and linkers but different central-metal ions, can exhibit distinctly different properties [26,27]. This means that forming isostructural MOFs by central-metal transformation is also an efficient strategy for integrating functionalities into MOF materials.…”

Section: Introductionmentioning

confidence: 99%