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A two-generator Kleinian group f , g can be naturally associated with a discrete group f , ϕ with the generator ϕ of order two and where f , ϕ f ϕ − 1 = f , g f g − 1 ⊂ f , g , f , ϕ : f , g f g − 1 = 2. This is useful in studying the geometry of the Kleinian groups since f , g will be discrete only if f , ϕ is, and the moduli space of groups f , ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f , g . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen's results on algebraic convergence. [ABSTRACT FROM AUTHOR]

Attention deficit hyperactivity disorder (ADHD) is one of the most common neurodevelopmental brain disorders in childhood. Despite extensive researches, the neurobiological mechanism underlying ADHD is still left unveiled. Since the deficit functions, such as attention, have been demonstrated in ADHD, in our present study, based on the oddball P3 task, the corresponding electroencephalogram (EEG) of both healthy controls (HCs) and ADHD children was first collected. And we then not only focused on the event-related potential (ERP) evoked during tasks but also investigated related brain networks. Although an insignificant difference in behavior was found between the HCs and ADHD children, significant electrophysiological differences were found in both ERPs and brain networks. In detail, the dysfunctional attention occurred during the early stage of the designed task; as compared to HCs, the reduced P2 and N2 amplitudes in ADHD children were found, and the atypical information interaction might further underpin such a deficit. On the one hand, when investigating the cortical activity, HCs recruited much stronger brain activity mainly in the temporal and frontal regions, compared to ADHD children; on the other hand, the brain network showed atypical enhanced long-range connectivity between the frontal and occipital lobes but attenuated connectivity among frontal, parietal, and temporal lobes in ADHD children. We hope that the findings in this study may be instructive for the understanding of cognitive processing in children with ADHD. [ABSTRACT FROM AUTHOR]