A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner’s extension problem in the L–category. The solution of the L–extension problem by Bochner [1] giving the relation between the order of the operator, the dimension, and index p, for which the L–extension pr...

PURPOSE:This study aimed to examine the effects of non-face-to-face learning on health-related physical fitness and balance in adolescents according coronavirus disease 2019 (COVID-19).METHODS:Twenty-nine middle-school students (14 girls 15 boys) were enrolled. We measured students’ (body composition, muscle strength, endurance, flexibility, cardiac endurance) (static dynamic) before after 12 w...

In this paper, we show that if φ(x; y) is a dependent formula, then all φ-types p have an extension to a φ-isolated φ-type, p′. Moreover, we can choose p′ to be a elementary φ-extension of p (see Definition 2.3 below) and so that |dom(p′)− dom(p)| ≤ 2 · ID(φ). We show that this characterizes φ being dependent. Finally, we give some corollaries of this theorem and draw some parallels to the stab...