We investigate successive matrix squaring (SMS) algorithms for computing the generalized inverseAT,S(2)of a given matrixA∈Cm×n.

Based on the conditions ab 2 = 0 and b π (ab) ∈ A d , we derive that (ab) n , (ba) n , and ab + ba are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A . By using these results, some results on the symmetry representations for the generalized Drazin inverse of ab + ba are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b.

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In this paper, the morphological characteristics of arc plasma and droplet transfer during the alternating magnetic field-assisted narrow gap groove laser-MIG (metal inert gas) hybrid welding process were investigated. The characteristics of arc plasma and droplet transfer, electron temperature, and density were analyzed using a high-speed camera and spectrum diagnosis. Our results revealed that the arc maintained a relatively stable state and rotated at a high speed to enhance the arc stiffness, and further improved the stability of the arc under the alternating magnetic field. The optimum magnetic field parameters in this experiment were B = 16 mT and f = 20 Hz, the electron temperature was 9893.6 K and the electron density was 0.99 × 1017 cm−3 near the bottom of the groove, which improved the temperature distribution inside the narrow gap groove and eliminated the lack of sidewall fusion defect. Compared to those without a magnetic field, the magnetic field could promote droplet transfer, the droplet diameter decreased by 17.6%, and the transition frequency increased by 23.5% (owing to the centrifugal force during droplet spinning and electromagnetic contraction force). The width of the weld bead was increased by 12.4% and the pores were also significantly reduced due to the stirring of the magnetic field on the molten pool.