The Debreu Gap Lemma and some generalizations

In this paper we study the Debreu Gap Lemma and its generalizations to totally ordered sets more general than (R, less than or equal to). We explain why it is important in economics to study utility functions which may not be real-valued and we build the foundations of a theory of continuity of such generalized utility functions. (C) 2004 Published by Elsevier B.V.

Motor unit synchronization of the vasti muscles in closed and open chain tasks

Objectives: To investigate motor unit synchronization between medial and lateral vasti and whether such synchronization differs in closed and open chain tasks. Design: Electromyographic recordings of single motor unit action potentials were made from the vastus medialis obliquus (VMO) and multiunit recordings from vastus lateralis during isometric contractions at 30 degrees of knee flexion in closed and open chain conditions. Setting: Laboratory. Participants: Five volunteers with no history of knee pain (age, 30 +/- 3.32y). Interventions: Not applicable. Main Outcome Measure: The degree of synchronization between motor unit firing was evaluated by identifying peaks in the electromyographic averages of the vastus lateralis, triggered from motor unit action potentials in the VMO, and the proportion of power in the power spectral density of the triggered average at the firing frequency of the reference motor unit. The proportion of cases in which there was significant power and peaks in the triggered averages was calculated. Results: The proportion of trials with peaks in the triggered averages of the vastus lateralis electromyographic activity was greater than 61.5% in all tasks, and there was a significantly greater proportion of cases where power in the spectrum was greater than 7.5% (P = .01) for the closed chain condition. Conclusions: There was a high proportion of synchronized motor units between the 2 muscles during isometric contractions, with evidence for greater common drive between the VMO and vastus lateralis in closed chain tasks. This has implications for rehabilitation because it suggests that closed chain tasks may generate better coordination between the vasti muscles.