Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid

Stability and change in screen-based sedentary behaviours and associated factors among Norwegian children in the transition between childhood and adolescence

<p>Abstract</p> <p>Background</p> <p>In order to inform interventions to prevent sedentariness, more longitudinal studies are needed focusing on stability and change over time in multiple sedentary behaviours. This paper investigates patterns of stability and change in TV/DVD use, computer/electronic game use and total screen time (TST) and factors associated with these patterns among Norwegian children in the transition between childhood and adolescence.</p> <p>Methods</p> <p>The baseline of this longitudinal study took place in September 2007 and included 975 students from 25 control schools of an intervention study, the HEalth In Adolescents (HEIA) study. The first follow-up took place in May 2008 and the second follow-up in May 2009, with 885 students participating at all time points (average age at baseline = 11.2, standard deviation ± 0.3). Time used for/spent on TV/DVD and computer/electronic games was self-reported, and a TST variable (hours/week) was computed. Tracking analyses based on absolute and rank measures, as well as regression analyses to assess factors associated with change in TST and with tracking high TST were conducted.</p> <p>Results</p> <p>Time spent on all sedentary behaviours investigated increased in both genders. Findings based on absolute and rank measures revealed a fair to moderate level of tracking over the 2 year period. High parental education was inversely related to an increase in TST among females. In males, self-efficacy related to barriers to physical activity and living with married or cohabitating parents were inversely related to an increase in TST. Factors associated with tracking high vs. low TST in the multinomial regression analyses were low self-efficacy and being of an ethnic minority background among females, and low self-efficacy, being overweight/obese and not living with married or cohabitating parents among males.</p> <p>Conclusions</p> <p>Use of TV/DVD and computer/electronic games increased with age and tracked over time in this group of 11-13 year old Norwegian children. Interventions targeting these sedentary behaviours should thus be introduced early. The identified modifiable and non-modifiable factors associated with change in TST and tracking of high TST should be taken into consideration when planning such interventions.</p