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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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An access theorem for analytic functions
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by Marvin Ortel PDF
Trans. Amer. Math. Soc. 347 (1995), 2213-2223 Request permission

Abstract:

Suppose that $\mathcal {M}$ is an analytic manifold, ${m_0} \in \mathcal {M},f:\mathcal {M} \to \mathbb {R}$, and $f$ is analytic. Then at least one of the following three statements is true: (1) ${m_0}$ is a local maximum of $f$. (2) There is a continuous path $\sigma :[0,1] \to \mathcal {M}$ such that $\sigma (0) = {m_0}$, $f \circ \sigma$ is strictly increasing on $[0,1]$, and $\sigma (1)$ is a local maximum of $f$. (3) There is a continuous path $\sigma :[0,1) \to \mathcal {M}$ with these properties: $\sigma (0) = {m_0};f \circ \sigma$ is strictly increasing on $[0,1)$; whenever $K$ is a compact subset of $\mathcal {M}$, there is a corresponding number $d(K) \in [0,1)$ such that $\sigma (t) \notin K$ for all $t \in [d(K),1)$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 2213-2223
  • MSC: Primary 32C05; Secondary 31B05
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1273513-3
  • MathSciNet review: 1273513