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American Mathematical Society, Proceedings of the American Mathematical Society, 3(107), p. 697-700, 1989

DOI: 10.1090/s0002-9939-1989-0987612-7

American Mathematical Society, Proceedings of the American Mathematical Society, 3(107), p. 697

DOI: 10.2307/2048167

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Smooth Polynomial Paths with Nonanalytic Tangents

Journal article published in 1989 by Robert M. Mcleod, Gary H. Meisters
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

We prove that there exist C ∞ {C^∞ } functions φ : R t × R x → R φ :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}} such that although φ ( t , x ) φ \left ( {t,x} \right ) is a polynomial in x x for each t t in R , φ ˙ ( 0 , x ) ≡ ( ∂ φ / ∂ t ) ( 0 , x ) {\mathbf {R}},\dot φ \left ( {0,x} \right ) ≡ \left ( {𝟃 φ /𝟃 t} \right )\left ( {0,x} \right ) need not even be analytic in x x let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if φ φ satisfies the group-property (even locally) of flows, namely if φ ( s , φ ( t , x ) ) = φ ( s + t , x ) φ \left ( {s,φ \left ( {t,x} \right )} \right ) = φ \left ( {s + t,x} \right ) .