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Zbl 0671.15005
Wolsson, Kenneth
A condition eqivalent to linear dependence for functions with vanishing Wronskian. (English)
[J] Linear Algebra Appl. 116, 1-8 (1989). ISSN 0024-3795

Authors summary: It is well known that though the vanishing of the Wronskian W[$\Phi$ ] of a set $\{$ $\Phi$ $\}$ of functions on an interval I is a necessary condition for it to be linearly dependent, it is not a sufficient one. Since Peano in 1889 expressed an interest in finding classes of functions for which W[$\Phi$ ]$\equiv 0$ is sufficient for dependence and offered one such example himself, others (M. Bocher, D. R. Curtiss, W. Hurewicz, and G. H. Meisters) have provided related results. \par Here the author gives a final answer to the question by first generalizing Peano's result using the order of a critical point, thereby obtaining a dense set of intervals of dependence. He then shows that W[$\Phi$ ]$\equiv 0$ together with a condition that the intersection of certain subspaces of $E\sp n$ is nontrivial is equivalent to the linear dependence of $\{$ $\Phi$ $\}$ on I. The above results are used to establish the dynamical theorem that motion of a particle under the action of a central force field is planar so long as the particle is restricted from the origin. The author provides a counterexample for the case in which the particle passes through the origin.
[G.P.Barker]

MSC 2000:: *15A03 Vector spaces
15A24 Matrix equations

Keywords: Wronskian; linearly dependent functions; order of a critical point; intervals of dependence; motion of a particle; counterexample

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Zentralblatt MATH Berlin [Germany]