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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 \(E^ n\) 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

MSC:

15A03 Vector spaces, linear dependence, rank, lineability
15A24 Matrix equations and identities
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References:

[1] Peano, G., Sur le déterminant Wronskien, Mathesis, 9, 75-76 (1889) · JFM 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) · JFM 32.0313.02
[3] Curtiss, D. R., The vanishing of the Wronskian and the problem of linear dependence, Math. Ann., 65, 282-298 (1908) · JFM 39.0354.02
[4] Hurewicz, W., Lectures on Ordinary Differential Equations (1958), M.I.T. Press: M.I.T. Press Cambridge, Mass · 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., 5, 413-415 (1897) · JFM 28.0145.03
[8] Bareket, M.; Dyn, N., On functions with identically vanishing Wronskian—A supplement to an article by Frechet, Calcutta Math. Soc. Bull., 77, 3-6 (1985) · Zbl 0606.34021
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