Smooth polynomial paths with nonanalytic tangents
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- by Robert M. McLeod and Gary H. Meisters PDF
- Proc. Amer. Math. Soc. 107 (1989), 697-700 Request permission
Abstract:
We prove that there exist ${C^\infty }$ functions $\varphi :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}}$ such that although $\varphi \left ( {t,x} \right )$ is a polynomial in $x$ for each $t$ in ${\mathbf {R}},\dot \varphi \left ( {0,x} \right ) \equiv \left ( {\partial \varphi /\partial t} \right )\left ( {0,x} \right )$ need not even be analytic in $x$ let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if $\varphi$ satisfies the group-property (even locally) of flows, namely if $\varphi \left ( {s,\varphi \left ( {t,x} \right )} \right ) = \varphi \left ( {s + t,x} \right )$ .References
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Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 697-700
- MSC: Primary 26E10; Secondary 14E07, 58C27
- DOI: https://doi.org/10.1090/S0002-9939-1989-0987612-7
- MathSciNet review: 987612