Through polarization and spectra analysis of the magnetic field observed by the Van Allen Probe A, we present two typical cases of O+ band electromagnetic ion cyclotron (EMIC) waves in the outer plasmasphere or plasma trough. Although such O+ band EMIC waves are rarely observed, 18 different events of O+ band EMIC waves (16 events in the outer plasmasphere and two events in the plasma trough) are found from September 2012 to August 2014 with observations of the Van Allen Probe A. We find that the preferred region for the occurrence of O+ band EMIC waves is in L = 2–5 and magnetic local time  = 03–13, 19–20, which is in accordance with the occurrence region of O+ ion torus. Therefore, our result suggests that the O+ ion torus in the outer plasmasphere during geomagnetic activities should play an important role in the generation of EMIC waves in O+ band.

Electromagnetic ion cyclotron (EMIC) waves can precipitate the ring current ions and relativistic electrons and heat the cold electrons in the magnetosphere. This requires comprehensive knowledge of the occurrence and wave properties of EMIC waves. In the present study, we used the data from one new mission, the Magnetospheric Multiscale (MMS) mission launched in March 2015, to investigate the occurrence and wave properties of H+‐band and He+‐band EMIC waves in the magnetosphere. Our statistical results show the following: (1) H+‐band EMIC waves mostly occur in the higher L‐shells (L > 5) while He+‐band EMIC waves are mostly observed in the lower L‐shells (L < 6). (2) The occurrence rate of H+‐band EMIC waves in the dayside is higher than that in the nightside. The highest peak of occurrence rate of H+‐band EMIC waves is in the postnoon sector (5–8 L‐shells), and the secondary peak lies in the small area of the dawn sector. (3) The wave power spectral density peaks in the postnoon and predusk sectors, while the wave normal angles are largest in the dawn sector. (4) Linear and right‐hand polarized H+‐band EMIC waves are mainly in the regions of peak occurrence, while linear polarized waves are seen to also dominate outside of the regions of peak occurrence. The highest occurrence rate of linear polarized He+‐band EMIC waves is observed in the dawn sector. We discussed the results and compared with previous findings.

Using the method of the integral factor, this work proves the Hyers-Ulam stability of linear differential equations of first order and extends the existing results.

The correlations between channel-bottom light intensity and channel-base current of all discharge processes of a rocket-and-wire-triggered lightning flash, including initial continuous current (ICC) pulses, ICC pulse background continuing current (IBCC), return strokes, M components, and M component background continuing currents (MBCC), have been investigated. A rough linear correlation has been found between the current squared and the light intensity for ICC pulses (including peaks of different ICC pulses), IBCC, the initial rising stage (IRS) of return strokes (including current peaks of different strokes), and MBCC. The slopes of the correlation regression lines for the current squared versus light intensity of ICC pulses and IBCC are similar, but they are about 2-3 times smaller than the slopes of MBCC and 5-7 times smaller than the slopes of the IRS of return strokes. In contrast, a rough linear correlation has been found between the current and the light intensity for the later slow decay stage of return strokes and for the M components. The slopes of the correlation regression lines of the current versus the light intensity for these latter two processes are found to be similar. No simple correlation has been found between the current and the light intensity for the initial fast decay stage (IFDS) of return strokes. The duration of the IFDS of return strokes generally lasts from several microseconds to several tens of microseconds and is more or less directly proportional to the corresponding peak return stroke current squared. A time delay ranging from 12 μs to 300 μs has been found between the current and the light intensity of all ICC pulses and M components. The time delay decreases as the corresponding peak current increases.