No abstract

We examined three questions. First, do reading difficulties increase children’s risk of behavior difficulties? Second, do behavioral difficulties increase children’s risk of reading difficulties? Third, do mathematics difficulties increase children’s risk of reading or behavioral difficulties? We investigated these questions using a sample of 9,324 children followed from third to fifth grade as they participated in a nationally representative dataset, conducting multilevel logistic regression modeling and including statistical control for many potential confounds. Results indicated that poor readers in third grade were significantly more likely to display poor task management, poor self-control, poor interpersonal skills, internalizing behavior problems, and externalizing behavior problems in fifth grade (odds ratio [OR] range = 1.30 – 1.57). Statistically controlling for a prior history of reading difficulties, children with poor mathematics skills in third grade were also significantly more likely to display poor task management, poor interpersonal skills, internalizing behavior problems, and reading difficulties in fifth grade (OR range = 1.38 – 5.14). In contrast, only those children exhibiting poor task management, but not other types of problem behaviors, in third grade were more likely to be poor readers in fifth grade (OR = 1.49).

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of 12,13] and Lee-Liu-Yu [10] to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.2000 Mathematics Subject Classification. 35Q20; 82C40.

No abstract