Kedar Karhadkar

Kedar Karhadkar

Math PhD student at UCLA

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manifolds

Numerical submersions

1 minute read

Published:

Let $m \geq n$, and let $f: \mathbb{R}^m \to \mathbb{R}^n$ be a submersion. Then $f$ is not necessarily surjective. Consider, for example, $m = n = 1$, and $f(x) = \arctan(x)$. But this function fails to be surjective for a boring reason: it has a vanishing gradient as $x \to \infty$.

optimization

Numerical submersions

1 minute read

Published:

Let $m \geq n$, and let $f: \mathbb{R}^m \to \mathbb{R}^n$ be a submersion. Then $f$ is not necessarily surjective. Consider, for example, $m = n = 1$, and $f(x) = \arctan(x)$. But this function fails to be surjective for a boring reason: it has a vanishing gradient as $x \to \infty$.