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3
questions
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Deducing properties of the $\ell_3$ norm from the $\ell_1$ and $\ell_2$ norms
Suppose we have a function $f: [0,1] \rightarrow \mathbb{R}$, $f(x) \geq 0$ normalised so that $\|f\|_1 = 1$, where
$$
\| f\|_p = \left( \int_0^1 f(x)^p d x \right)^{1/p}.
$$
Moreover, we know that $\|...
- 1,149
1
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Bounding the dot product of two planar unit vectors.
Does there exist a continuous, monotone increasing function $f\colon[0,2]\to [0,1]$, satisfying $f(0)=0$ and $f(1)=1$, such that for all vectors $(a_1,b_1),(a_2,b_2)\in \mathbb{R}^2$ of unit length, i....
- 2,684
0
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How to bound this function?
I am trying to show that the derivative of the scalar function $V(\mathbf{x})$, $V'(\mathbf{x})$, is such that
$V'(\mathbf{x})=-(x_1^2+x_2^2)+x_2u\leq-||\mathbf{x}||^2+||\mathbf{x}|| \ |u|$
where $\...
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