Sleep Patterns in Elementary School Children (Grades 2-5) and Adolescents (Grade 10)

Background: To date, there is limited research examining sleep patterns in elementary school children. Previous researchers focused on parental responses rather than student responses to determine factors that affect sleep. The presented study surveyed sleep patterns and examined external factors affecting total sleep time among elementary school children and adolescents. Methods: Students in grades 2-5 (n=885) and grade 10 (n=190) enrolled in a public school system in the Northeast, completed a district administered survey that included questions on sleep duration and hygiene. Results. Average reported sleep duration decreased with increasing grade level. Children in grades 2-5 woke up earlier (31.7-72.4%) and on their own in comparison to adolescents in grade 10 (6.8%). Significantly shorter sleep durations were associated with having a television (grades 2, 4, 5, p< 0.01) or a cell phone in the room (grades 3, 4; p < 0.05), playing on the computer or video games (grades 3, 4, p<.001) before going to bed. In contrast, students in grade 2, 3, & 4 who reported reading a book before going to bed slept on average 21 minutes more per night (p=.029, .007, .009, respectively). For tenth graders, only consumption of energy drinks led to significant reduction in sleep duration (p<.0001). Conclusion. Sleep is a fundamental aspect in maintaining a healthy and adequate life style. Understanding sleep patterns will assist parents, health care providers, and educators in promoting quality sleep hygiene in school-aged children.

Typability and Type Checking in the Second-Order Λ-Calculus Are Equivalent and Undecidable (Preliminary Draft)

We consider the problems of typability[1] and type checking[2] in the Girard/Reynolds second-order polymorphic typed λ-calculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pure λ -terms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lower-bound complexity results for System F have been achieved, but they have remained "embarrassing open problems"[3] for System F itself. We first prove that type checking in System F is undecidable by a reduction from semi-unification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructing λ-terms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require that specific subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment may be simulated. We develop this method, which we call "constants for free", for both the λK and λI calculi.

New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi

Two new notions of reduction for terms of the λ-calculus are introduced and the question of whether a λ-term is beta-strongly normalizing is reduced to the question of whether a λ-term is merely normalizing under one of the new notions of reduction. This leads to a new way to prove beta-strong normalization for typed λ-calculi. Instead of the usual semantic proof style based on Girard's "candidats de réductibilité'', termination can be proved using a decreasing metric over a well-founded ordering in a style more common in the field of term rewriting. This new proof method is applied to the simply-typed λ-calculus and the system of intersection types.