# Multiple meanings of vertical bars

### Examples from arXiv

[arxiv:1008.0249](https://arxiv.org/pdf/1008.0249.pdf), page 70
* On the region $\hat{U}=\left\{z \mid \|P_{c} z\|_{E_{c}} \leq \delta_{1},| \eta(t) | \leq 1\right\},$ ...
* set "such that", norm(?), absolute value

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[arxiv:0903.4647](https://arxiv.org/pdf/0903.4647.pdf), page 14
* $\operatorname{Cyl}(L, W):=\left\{x \in \mathbb{R}^{d} \mid | x_{1}| \leq L, x_{2}^{2}+x_{3}^{2}+\cdots+x_{d}^{2} \leq W^{2}\right\}$
* set "such that", absolute value

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[arxiv:1912.09447](https://arxiv.org/pdf/1912.09447.pdf), page 11
* $\mathbb{P}:=\{|\mathbf{R}\rangle|\langle\mathbf{R}|\mathbf{R}\rangle=1\}$
* set such that, bra-ket notation, `\mid` is not used

* $\{g|\mathbf{R}\rangle| g \in \mathrm{U}(1)\}$
* set such that, bra-ket notation, `\mid` is not used, **2nd** vertbar is set fence

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[arxiv:q-alg/971004](https://arxiv.org/pdf/q-alg/9710004.pdf), page 12, example 13
* $\{x(1 | 2)\}\{y(2 | 3)\}\{a|b, c| d, e, f, g\}$
* multibraces, partitions

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[arxiv:1806.06107](https://arxiv.org/pdf/1806.06107.pdf), page 27, eq 36
* $\mathcal{F}\left(\mathcal{H}_{d}\right)=\operatorname{conv}\left\{U|0\rangle\langle 0| U^{\dagger}: U \in \mathcal{C}_{d}\right\}$
* bra-ket notation **only**. Set "such that" written with a colon `:`.

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[arxiv:1806.09446](https://arxiv.org/pdf/1806.09446.pdf), page 12
* $\Gamma_{1}=\{p\,\,|(2|p)=1\} .$ Further $\Gamma_{s}=\left\{p \mid 2^{s+2}| p-(-1|p)\right\}$
* set "such that", divides, Legendre symbol

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[arxiv:1912.05696](https://arxiv.org/pdf/1912.05696.pdf), page 15, theorem 5
* Let a coherent assessment $(x, \mu)$ on $\{C|A, D|(C|A)\}$ be given.
* iterated conditional, no "such that"

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[arxiv:0505018](https://arxiv.org/pdf/math/0505018.pdf), page 14

* $$\iint\left\{\omega\left|v_{x}^{\varepsilon}\right|^{2}+\cdots \right\}$$

* no set, braces group an expression. Vertical bar is absolute value.

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[arxiv:1806.10734](https://arxiv.org/pdf/1806.10734.pdf), page 12
* We will use symbols of the form $\{N|A|d\}$ to indicate the order of the stencil in (37)
* no set, special symbol notation
* example use: $\{N|A: 0| d: 0\}$

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[arxiv:1804.03896](https://arxiv.org/pdf/1804.03896.pdf), top of page 21

* The convergence of the sequence$\left\{n\left|X^{n, *}(T)\right|^{2}\right\}$ to zero
* "sequence" brace notation. vertical bar is a matrix norm or absolute value.