Some results of the local theory of smooth functions
- 22 Downloads
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.
KeywordsAnalytic Function Smooth Function Form Versus Negative Eigenvalue Local Theory
Unable to display preview. Download preview PDF.
- 6.C. Houzel, “Géométrie analytique locale. I,” in: Sémin. H. Cartan, No. 18 (1960/1961).Google Scholar
- 7.B. Malgrange, “Le théorème de préparation en géométrie diffé rentiable,” in: Sémin. H. Cartan, No. 11, 13–22 (1962/1963).Google Scholar
- 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).Google Scholar