In this paper, persistence properties of solutions are investigated for a generalized Camassa-Holm equation (g-kbCH) having (k+1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations. The persistence properties will imply that strong solutions of the g-kbCH equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k. Finally, existence of global solutions is established when b = k+1.

Abstract. In this paper, we prove well-posedness of the Fornberg-Whitham equation in Besov spaces B s 2,r in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-tosolution map provided that the initial data belongs to B s 2,r . We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B s 2,r to C([−T, T ]; B s 2,r ). Furthermore, we prove a Cauchy-Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.

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Objective There is a growing recognition that the use of conventional norms (e.g., age, sex, years of education, race) as proxies to capture a broad range of sociocultural variability on cognitive performance is suboptimal, limiting sample representativeness. The present study evaluated the incremental utility of family income, family conflict, and acculturation beyond the established associations of age, gender,maternal years of education, and race on cognitive performance. Method Hierarchical linear regressions evaluated the incremental utility of sociocultural factors on National Institutes of Health Toolbox in a nationally representative sample of pre-adolescent children (n = 11,878; Mage = 10.0 years; Adolescent Brain Cognitive Development Study). A regression-based norming procedure was implemented for significant models. Paired sample t-tests were used to compare original and newly created demographically corrected T-scores. Results Nearly all regression models predicted performance on the NIH-TB subtests and composite scores (p < .005). Greater family income and lower family conflict predicted better performance, although the effect sizes were small by traditional standards. Acculturation scores did not explain additional variance in cognitive performance. Lastly, there were no significant differences between the original and newly created demographically corrected T-scores (Mdiff < 0.50). Conclusions The present study highlights that, although family income, family conflict, and acculturation have been shown to routinely influence cognitive performance in preadolescent children, the NIH-TB appears to be highly robust to individual differences in sociocultural factors in children between ages 9 and 10. Contextual and temporal implications of the present results are discussed.

Objective Although research shows that abnormal scores are common in adults, fewer studies examine the frequency of abnormal scores within children. The present study assessed how specific tests, cognitive domains, and patient factors affect the likelihood of abnormally low scores among healthy youth athletes. Methods Youth athletes (n = 85, Mage = 11.72, SD = 2.23) completed a comprehensive neuropsychological battery (N = 23 tests) during baseline testing for a sport-related concussion clinical research program. “Abnormal” scores were defined using cutoffs at 1, 1.5, and 2 standard deviations (SDs) below the population mean. Univariate and multivariate Poisson regression models examined abnormal test scores as a function of age, sex, ethnicity, and Full-Scale Intelligence Quotient (FSIQ). Results Using cutoffs of 1, 1.5, and 2 SDs, the percent of healthy participants with at least one abnormal test score was 61.2, 36.5, and 22.4%, respectively, indicating a high proportion of healthy participants with abnormal scores and low specificity for tests using these cutoffs. In univariate Poisson regression analyses, FSIQ and abnormal score were negatively correlated (p < .001), wherein lower FSIQ was associated with more abnormal scores. In multivariate regression analyses, lower performance on executive function, memory, and processing speed tests was significantly associated with an increased frequency of abnormal scores. Conclusions Abnormally low scores in healthy youth occur frequently and do not necessarily represent functional deficits. Cutoffs of 2 SDs should be considered for many tests to increase specificity and reduce the risk of false positive errors that could lead to negative clinical outcomes.