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A condition eqivalent to linear dependence for functions with vanishing Wronskian. (English) Zbl 0671.15005
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. 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.
Reviewer: G.P.Barker

15A03Vector spaces, linear dependence, rank
15A24Matrix equations and identities
Full Text: DOI
[1] Peano, G.: Sur le déterminant wronskien. Mathesis 9, 75-76 (1889) · Zbl 21.0153.01
[2] Bôcher, M.: Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence. Trans. amer. Math. soc. 2, 139-149 (1901) · Zbl 32.0313.02
[3] Curtiss, D. R.: The vanishing of the Wronskian and the problem of linear dependence. Math. ann. 65, 282-298 (1908) · Zbl 39.0354.02
[4] Hurewicz, W.: Lectures on ordinary differential equations. (1958) · Zbl 0082.29702
[5] Meisters, G. H.: Local linear dependence and the vanishing of the Wronskian. Amer. math. Monthly 68, 847-856 (1961) · Zbl 0102.04703
[6] Peano, G.: Sul determinate wronskiano. Atti accad. Naz. lincei rendi. Cl. sci. Fis. mat. Nat., No. 5, 413-415 (1897)
[7] G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Springer-Verlag, New York. · Zbl 1024.00003
[8] Bareket, M.; Dyn, N.: On functions with identically vanishing Wronskian--A supplement to an article by Fréchet. Calcutta math. Soc. bull. 77, 3-6 (1985) · Zbl 0606.34021
[9] K. Wolsson, Linear dependence of a function set of m variables with vanishing generalized Wronskians, this issue. · Zbl 0724.15004