All Questions
4
questions
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Lower bound on the components of a unit vector
Let $u \in \mathbb{R}^n$ be a unit vector: $\|u\| = 1$. Is there a formula for the smallest $C > 0$ such that $|u_i| \geq C$ for some component $u_i$ where $i = 1,...,n$?
For example, if $u \in \...
- 1,093
1
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0
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95
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If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?
Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$?
My Approach:
$$\...
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1
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90
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Randomness in the norm of sum of vectors
Let $x_1,x_2,\ldots,x_n \in \mathbb{R}^d$ be vectors and $a_1, a_2, \ldots, a_n \in \mathbb{R}$ be random iid scalars distributed by $N~(0,\sigma^2).$
Then, is it possible to lower bound the ...
0
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1
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What conditions should vector $x$ satisfy so that $\|[x_2+\alpha x_1, \dots, x_n + \alpha x_1]\|_2$ is bounded by a constant?
Suppose that $x = [x_1,\dots,x_n]$ is a vector with norm less than or equal to one $\|x\|_2^2 \leq 1$. Let $\alpha \in [0,1]$ and define the following vector
$$y = [x_2+\alpha x_1, \dots, x_n + \...
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