Pervasive Algebras and Maximal Subalgebras.

A uniform algebra A on its Shilov boundary X is maximal if A is not C(X) and no uniform algebra is strictly contained between A and C(X) . It is essentially pervasive if A is dense in C(F) whenever F is a proper closed subset of the essential set of A. If A is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show: (1) If A is pervasive and proper, and has a nonconstant unimodular element, then A contains an infinite descending chain of pervasive subalgebras on X . (2) It is possible to find a compact Hausdorff space X such that there is an isomorphic copy of the lattice of all subsets of N in the family of pervasive subalgebras of C(X). (3) In the other direction, if A is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word “strongly” is removed. We discuss examples involving Dirichlet algebras, A(U) algebras, Douglas algebras, and subalgebras of H∞(D), and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.

Adaptive and Maladaptive Correlates of Repetitive Behavior and Restricted Interests in Persons with Down Syndrome and Developmentally-Matched Typical Children: A Two-Year Longitudinal Sequential Design

We examined the course of repetitive behavior and restricted interests (RBRI) in children with and without Down syndrome (DS) over a two-year time period. Forty-two typically-developing children and 43 persons with DS represented two mental age (MA) levels: `` younger'' 2-4 years; `` older'' 5-11 years. For typically developing younger children some aspects of RBRI increased from Time 1 to Time 2. In older children, these aspects remained stable or decreased over the two-year period. For participants with DS, RBRI remained stable or increased over time. Time 1 RBRI predicted Time 2 adaptive behavior (measured by the Vineland Scales) in typically developing children, whereas for participants with DS, Time 1 RBRI predicted poor adaptive outcome (Child Behavior Checklist) at Time 2. The results add to the body of literature examining the adaptive and maladaptive nature of repetitive behavior.

A Cross-Sectional Survey of Repetitive Behaviors and Restricted Interests in a Typically Developing Turkish Child Population

This study examined compulsive-like behaviors (CLBs) which are higher-order types of Repetitive Behaviors And Restricted Interests (RBRIs) in typically developing children in Turkey. Caregivers of 1,204 children between 8 and 72 months were interviewed with Childhood Routines Inventory (CRI) by trained interviewers in a cross-sectional survey. Factor analysis of the CRI revealed two factor structures comprising "just right behaviors" and "repetitive/sensory sensitivity behaviors". CLB frequency peaked at 2-4 years with declines after age four. In contrast to the previous CRI studies reporting no gender difference, CLBs were more common in males in 12-23 and 48-59 month age groups on both total CLB frequency and repetitive/sensory sensitivity behaviors. Also ages of onsets for CRI items were somewhat later than reported in other samples. Our findings supported the findings of the previous CRI studies while also revealing new perspectives in need of further investigation.

Quotients of Koszul Algebras and 2-D-Determined Algebras

Vatne [13] and Green and Marcos [9] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.