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Some results of the local theory of smooth functions

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Abstract

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system.

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References

  1. M. W. Hirsch, Differential Topology, Springer, Berlin (1976).

    MATH  Google Scholar 

  2. V. V. Golubev, Lectures on the Analytic Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow-Leningrad (1950).

    MATH  Google Scholar 

  3. J. Milnor, Morse Theory, Princeton University Press, Princeton (1963).

    MATH  Google Scholar 

  4. V. I. Arnol’d, “A remark on the Weierstrass preparation theorem,” Funkts. Analiz, 1, Issue 3, 1–8 (1967).

    Article  MATH  Google Scholar 

  5. V. I. Arnol’d, “Specific features of smooth mappings,” Usp. Mat. Nauk, 23, Issue 1(139), 3–44 (1968).

    MATH  Google Scholar 

  6. C. Houzel, “Géométrie analytique locale. I,” in: Sémin. H. Cartan, No. 18 (1960/1961).

  7. B. Malgrange, “Le théorème de préparation en géométrie diffé rentiable,” in: Sémin. H. Cartan, No. 11, 13–22 (1962/1963).

  8. A. M. Samoilenko, “On the equivalence of a smooth function and a Taylor polynomial in the neighborhood of a critical point of finite type,” Funkts. Anal., 2, Issue 4, 63–69 (1968).

    Google Scholar 

  9. V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Specific Features of Differentiable Mappings [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  10. J. Moser, “On invariant curves of area-preserving mapping of an annulus,” in: Proceedings of Symposium on Nonlinear Differential Equations, Colorado Springs (1961).

  11. I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow (1966).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev

    A. M. Samoilenko

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  1. A. M. Samoilenko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 231–267, February, 2007.

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Samoilenko, A.M. Some results of the local theory of smooth functions. Ukr Math J 59, 243–292 (2007). https://doi.org/10.1007/s11253-007-0019-y

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  • DOI: https://doi.org/10.1007/s11253-007-0019-y

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