The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function circle dot with finitely many singularities on the unit circle T; that is, the continuous mappings g : T -> T such that circle dot o g = circle dot on T. These mappings form a group under composition.

Examining The Genetic, Social-Cognitive, And Neural Correlates Of Autism Spectrum Disorder Behaviors In Typically Developing Chi

Autism spectrum disorders (ASD) are pervasive developmental disorders that affect approximately 1 in 50 children (Blumberg et al., 2013). Due to the social nature of the deficits that characterize the disorders, many have classified them as disorders of social cognition, which is the process that individuals use in order to successfully interact with members of their own species (Frith & Frith, 2007). Previous research has typically neglected the spectrum nature of ASD in favor of a more categorical approach of ¿autistic¿ versus ¿non-autistic,¿ but the spectrum requires a more continuous approach. Thus, the present study sought to examine the genetic, social-cognitive, and neural correlates of ASD-like traits as well as the relationship between these dimensions in typically developing children. Parents and children completed several quantitative measures examining several areas of social-cognitive functioning, including theory of mind and social functioning, restricted/repetitive behaviors and interests, and adaptive/maladaptive functioning. Children were also asked to undergo an EEG and both parents and children contributed a saliva sample that was used to sequence four single nucleotide polymorphisms (SNPs) of the OXTR gene, rs1042778, rs53576, rs2254298, and rs237897. We successfully demonstrated a significant relationship between behavioral measures of social-cognition and differences in face perception via the N170. However, the directionality of these relationships varied based on the behavioral measure and particular N170 difference scores. We also found support for the associations between the G_G allelic combination of rs1042778 and the A_A and A_G allelic combinations of rs2254298 and increased ASD-like behavior with decreased social-cognitive functioning. In contrast, our results contradict previous findings with rs237897 and imply that individuals with the A_A and A_G genotypes are less similar to those with ASD and have higher social cognitive functioning than those with the G_G genotype. In conclusion, we have demonstrated the existence of ASD-like traits in typically developing children and have shown a link between behavioral, genetic, and neural correlates of social-cognition. These findings demonstrate the importance of considering autism as a spectrum disorder and provide support for the move to a more continuous approach to neurodevelopmental disorders.