Multiple meanings of vertical bars

Examples from arXiv

arxiv:1008.0249, page 70

• On the region
  $\hat{U}=\{z\mid \Vert P_{c}z{\Vert }_{E_c}\le {\delta }_1,|\eta (t)|\le 1\},$ …
  • set "such that", norm(?), absolute value

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arxiv:0903.4647, page 14

• $\mathrm{Cyl}(L,W):=\{x\in {\mathbb{R}}^d\mid |x_1|\le L,x_2^2+x_3^2+\cdots +x_d^2\le W^2\}$
  • set "such that", absolute value

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arxiv:1912.09447, page 11

• $\mathbb{P}:=\{|\mathbf{R}⟩|⟨\mathbf{R}|\mathbf{R}⟩=1\}$
  • set such that, bra-ket notation, \mid is not used

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

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arxiv:q-alg/971004, 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, page 27, eq 36

• $\mathcal{F}\left({\mathcal{H}}_d\right)=\mathrm{conv}\{U|0⟩⟨0|U^{†}:U\in {\mathcal{C}}_d\}$
  • bra-ket notation only. Set "such that" written with a colon :.

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arxiv:1806.09446, page 12

• ${\mathrm{\Gamma }}_1=\{p|(2|p)=1\}.$ Further
  ${\mathrm{\Gamma }}_s=\{p\mid 2^{s+2}|p-(-1|p)\}$
  • set "such that", divides, Legendre symbol

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arxiv:1912.05696, 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, page 14

• $$\iint \{\omega {\left|v_x^{\epsilon }\right|}^2+\cdots \}$$
  • no set, braces group an expression. Vertical bar is absolute value.

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arxiv:1806.10734, 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, top of page 21

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