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.

INTERPOLATING BLASCHKE PRODUCTS AND ANGULAR DERIVATIVES

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H-infinity[(b) over bar : b has finite angular derivative everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Hearing Delzant Polytopes From the Equivariant Spectrum

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu asked whether the spectrum of the Laplace operator $\Delta_g$ on $\mathcal{C}^\infty(M)$ determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of M^4 is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of M and the spectrum of its associated real manifold M_R determine its polygon, up to translation and a small number of choices. For M of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of M.

Interpolating Blaschke Products and Angular Derivatives

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra generated by the algebra of bounded analytic functions and the conjugates of Blaschke products with angular derivative finite everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

INCREASING PARTICIPATION IN THE CLASSROOM FOR CHILDREN WITH AUTISM SPECTRUM DISORDERS

Active participation is as essential a skill to children with autism as it is for children without autism, as children are expected to engage in these skills both in and outside the classroom. Without participation skills, children are at a disadvantage when it comes to school and other settings, such as extracurricular activities and the workforce. Recent research has shown that there are interventions available that aim to improve the social skills of children in the home and in the school. These interventions can be delivered in varying forms with the primary caregiver as the interventionist, the specialist as the interventionist, and naturalistic interventions. The purpose of this study was to investigate one of the naturalistic interventions, the Competent Learner Model, and determine its effects on the participation and social skills of students with autism. Three middle school male students diagnosed with autism from a rural northeast middle school participated in the study. They were assessed using the Competent Learner Repertoire Assessments of the Competent Learner Model and the adaptive measures of the Vineland-II and ABAS-II. The results showed improvement for one of the three students and little to no improvement for the other two students.

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.

Design Simulation for Solar Energy Research and Education

Solar research is primarily conducted in regions with consistent sunlight, severely limiting research opportunities in many areas. Unfortunately, the unreliable weather in Lewisburg, PA, can prove difficult for such testing to be conducted. As such, a solar simulator was developed for educational purposes for the Mechanical Engineering department at Bucknell University. The objective of this work was to first develop a geometric model to evaluate a one sun solar simulator. This was intended to provide a simplified model that could be used without the necessity of expensive software. This model was originally intended to be validated experimentally, but instead was done using a proven ray tracing program, TracePro. Analyses with the geometrical model and TracePro demonstrated the influence the geometrical properties had results, specifically the reflector (aperture) diameter and the rim angle. Subsequently, the two were approaches were consistent with one another for aperture diameters 0.5 m and larger, and for rim angles larger than 45°. The constructed prototype, that is currently untested, was designed from information provided by the geometric model, includes a metal halide lamp with a 9.5 mm arc diameter and parabolic reflector with an aperture diameter of 0.631 meters. The maximum angular divergence from the geometrical model was predicted to be 30 mRadians. The average angular divergence in TraceProof the system was 19.5 mRadians, compared to the sun’s divergence of 9.2 mRadians. Flux mapping in TracePro showed an intensity of 1000 W/m2 over the target plane located 40 meters from the lamp. The error between spectrum of the metal halide lamp and the solar spectrum was 10.9%, which was found by comparing their respective Plank radiation distributions. The project did not satisfy the original goal of matching the angular divergence of sunlight, although the system could still to be used for optical testing. The geometric model indicated performance in this area could be improved by increasing the diameter of the reflector, as well as decreasing the source diameter. Although ray tracing software provides more information to analyze the simulator system, the geometrical model is adequate to provide enough information to design a system.