Run 16446900

Paper: https://arxiv.org/abs/1109.4011

Formulas:

1-

km s^{- 1} {Mpc}^{- 1}

2-

σ_{c l u s}

3-

0.2 - 5 \times 10^{44} erg / s

4-

R_{200} = 1.73 \frac{σ}{1000 km s^{- 1}} \frac{1}{\sqrt{Ω_{Λ} + Ω_{0} {(1 + z)}^{3}}} h^{- 1} Mpc

5-

BCGprev = (V_{rank2} + V_{rank3}) / 2 - V_{rank1}

6-

P_{int} (c o l, m a g) \equiv \frac{N_{F} (c o l, m a g)}{N_{W} (c o l, m a g)}

7-

(c o l, m a g)

8-

P_{int} \equiv \frac{N_{F} (c o l, m a g) - N_{NO} (c o l, m a g)}{N_{W} (c o l, m a g) - N_{NO} (c o l, m a g) - N_{YES} (c o l, m a g)}

9-

0.5 \leq {(B - V)}_{10.8 k p c} \leq 1.5

10-

- 21.5 \leq M_{V} \leq - 18

Formulas (html):

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xref="S3.E3.m1.6.6.2.2.3.cmml">,</mo><mrow id="S3.E3.m1.6.6.2.2.2.2" xref="S3.E3.m1.6.6.2.2.2.2.cmml"><mi id="S3.E3.m1.6.6.2.2.2.2.2" xref="S3.E3.m1.6.6.2.2.2.2.2.cmml">m</mi><mo id="S3.E3.m1.6.6.2.2.2.2.1" xref="S3.E3.m1.6.6.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.6.6.2.2.2.2.3" xref="S3.E3.m1.6.6.2.2.2.2.3.cmml">a</mi><mo id="S3.E3.m1.6.6.2.2.2.2.1a" xref="S3.E3.m1.6.6.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.6.6.2.2.2.2.4" xref="S3.E3.m1.6.6.2.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E3.m1.6.6.2.2.2.5" xref="S3.E3.m1.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.6.6.3" xref="S3.E3.m1.6.6.3.cmml">≡</mo><mfrac id="S3.E3.m1.4.4" xref="S3.E3.m1.4.4.cmml"><mrow id="S3.E3.m1.2.2.2" xref="S3.E3.m1.2.2.2.cmml"><msub id="S3.E3.m1.2.2.2.4" xref="S3.E3.m1.2.2.2.4.cmml"><mi id="S3.E3.m1.2.2.2.4.2" xref="S3.E3.m1.2.2.2.4.2.cmml">N</mi><mi mathvariant="normal" id="S3.E3.m1.2.2.2.4.3" xref="S3.E3.m1.2.2.2.4.3.cmml">F</mi></msub><mo id="S3.E3.m1.2.2.2.3" xref="S3.E3.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.E3.m1.2.2.2.2.2" xref="S3.E3.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="S3.E3.m1.2.2.2.2.2.3" xref="S3.E3.m1.2.2.2.2.3.cmml">(</mo><mrow id="S3.E3.m1.1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.1.cmml"><mi id="S3.E3.m1.1.1.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.1.1.2.cmml">c</mi><mo id="S3.E3.m1.1.1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E3.m1.1.1.1.1.1.1.3" xref="S3.E3.m1.1.1.1.1.1.1.3.cmml">o</mi><mo id="S3.E3.m1.1.1.1.1.1.1.1a" xref="S3.E3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E3.m1.1.1.1.1.1.1.4" xref="S3.E3.m1.1.1.1.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E3.m1.2.2.2.2.2.4" xref="S3.E3.m1.2.2.2.2.3.cmml">,</mo><mrow id="S3.E3.m1.2.2.2.2.2.2" xref="S3.E3.m1.2.2.2.2.2.2.cmml"><mi id="S3.E3.m1.2.2.2.2.2.2.2" xref="S3.E3.m1.2.2.2.2.2.2.2.cmml">m</mi><mo id="S3.E3.m1.2.2.2.2.2.2.1" xref="S3.E3.m1.2.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.2.2.2.2.2.2.3" xref="S3.E3.m1.2.2.2.2.2.2.3.cmml">a</mi><mo id="S3.E3.m1.2.2.2.2.2.2.1a" xref="S3.E3.m1.2.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.2.2.2.2.2.2.4" xref="S3.E3.m1.2.2.2.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E3.m1.2.2.2.2.2.5" xref="S3.E3.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mrow id="S3.E3.m1.4.4.4" xref="S3.E3.m1.4.4.4.cmml"><msub id="S3.E3.m1.4.4.4.4" xref="S3.E3.m1.4.4.4.4.cmml"><mi id="S3.E3.m1.4.4.4.4.2" xref="S3.E3.m1.4.4.4.4.2.cmml">N</mi><mi mathvariant="normal" id="S3.E3.m1.4.4.4.4.3" xref="S3.E3.m1.4.4.4.4.3.cmml">W</mi></msub><mo id="S3.E3.m1.4.4.4.3" xref="S3.E3.m1.4.4.4.3.cmml">⁢</mo><mrow id="S3.E3.m1.4.4.4.2.2" xref="S3.E3.m1.4.4.4.2.3.cmml"><mo stretchy="false" id="S3.E3.m1.4.4.4.2.2.3" xref="S3.E3.m1.4.4.4.2.3.cmml">(</mo><mrow id="S3.E3.m1.3.3.3.1.1.1" xref="S3.E3.m1.3.3.3.1.1.1.cmml"><mi id="S3.E3.m1.3.3.3.1.1.1.2" xref="S3.E3.m1.3.3.3.1.1.1.2.cmml">c</mi><mo id="S3.E3.m1.3.3.3.1.1.1.1" xref="S3.E3.m1.3.3.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E3.m1.3.3.3.1.1.1.3" xref="S3.E3.m1.3.3.3.1.1.1.3.cmml">o</mi><mo id="S3.E3.m1.3.3.3.1.1.1.1a" xref="S3.E3.m1.3.3.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E3.m1.3.3.3.1.1.1.4" xref="S3.E3.m1.3.3.3.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E3.m1.4.4.4.2.2.4" xref="S3.E3.m1.4.4.4.2.3.cmml">,</mo><mrow id="S3.E3.m1.4.4.4.2.2.2" xref="S3.E3.m1.4.4.4.2.2.2.cmml"><mi id="S3.E3.m1.4.4.4.2.2.2.2" xref="S3.E3.m1.4.4.4.2.2.2.2.cmml">m</mi><mo id="S3.E3.m1.4.4.4.2.2.2.1" xref="S3.E3.m1.4.4.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.4.4.4.2.2.2.3" xref="S3.E3.m1.4.4.4.2.2.2.3.cmml">a</mi><mo id="S3.E3.m1.4.4.4.2.2.2.1a" xref="S3.E3.m1.4.4.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E3.m1.4.4.4.2.2.2.4" xref="S3.E3.m1.4.4.4.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E3.m1.4.4.4.2.2.5" xref="S3.E3.m1.4.4.4.2.3.cmml">)</mo></mrow></mrow></mfrac></mrow></math>, <math><mrow id="S3.SS1.p2.3.m3.2.2.2" xref="S3.SS1.p2.3.m3.2.2.3.cmml"><mo stretchy="false" id="S3.SS1.p2.3.m3.2.2.2.3" xref="S3.SS1.p2.3.m3.2.2.3.cmml">(</mo><mrow id="S3.SS1.p2.3.m3.1.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.1.1.2" xref="S3.SS1.p2.3.m3.1.1.1.1.2.cmml">c</mi><mo id="S3.SS1.p2.3.m3.1.1.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.3.m3.1.1.1.1.3" xref="S3.SS1.p2.3.m3.1.1.1.1.3.cmml">o</mi><mo id="S3.SS1.p2.3.m3.1.1.1.1.1a" xref="S3.SS1.p2.3.m3.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.3.m3.1.1.1.1.4" xref="S3.SS1.p2.3.m3.1.1.1.1.4.cmml">l</mi></mrow><mo id="S3.SS1.p2.3.m3.2.2.2.4" xref="S3.SS1.p2.3.m3.2.2.3.cmml">,</mo><mrow id="S3.SS1.p2.3.m3.2.2.2.2" xref="S3.SS1.p2.3.m3.2.2.2.2.cmml"><mi id="S3.SS1.p2.3.m3.2.2.2.2.2" xref="S3.SS1.p2.3.m3.2.2.2.2.2.cmml">m</mi><mo id="S3.SS1.p2.3.m3.2.2.2.2.1" xref="S3.SS1.p2.3.m3.2.2.2.2.1.cmml">⁢</mo><mi id="S3.SS1.p2.3.m3.2.2.2.2.3" xref="S3.SS1.p2.3.m3.2.2.2.2.3.cmml">a</mi><mo id="S3.SS1.p2.3.m3.2.2.2.2.1a" xref="S3.SS1.p2.3.m3.2.2.2.2.1.cmml">⁢</mo><mi id="S3.SS1.p2.3.m3.2.2.2.2.4" xref="S3.SS1.p2.3.m3.2.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.SS1.p2.3.m3.2.2.2.5" xref="S3.SS1.p2.3.m3.2.2.3.cmml">)</mo></mrow></math>, <math><mrow id="S3.E4.m1.10.11" xref="S3.E4.m1.10.11.cmml"><msub id="S3.E4.m1.10.11.2" xref="S3.E4.m1.10.11.2.cmml"><mi id="S3.E4.m1.10.11.2.2" xref="S3.E4.m1.10.11.2.2.cmml">P</mi><mi id="S3.E4.m1.10.11.2.3" xref="S3.E4.m1.10.11.2.3.cmml">int</mi></msub><mo id="S3.E4.m1.10.11.1" xref="S3.E4.m1.10.11.1.cmml">≡</mo><mfrac id="S3.E4.m1.10.10" xref="S3.E4.m1.10.10.cmml"><mrow id="S3.E4.m1.4.4.4" xref="S3.E4.m1.4.4.4.cmml"><mrow id="S3.E4.m1.2.2.2.2" xref="S3.E4.m1.2.2.2.2.cmml"><msub id="S3.E4.m1.2.2.2.2.4" xref="S3.E4.m1.2.2.2.2.4.cmml"><mi id="S3.E4.m1.2.2.2.2.4.2" xref="S3.E4.m1.2.2.2.2.4.2.cmml">N</mi><mi mathvariant="normal" id="S3.E4.m1.2.2.2.2.4.3" xref="S3.E4.m1.2.2.2.2.4.3.cmml">F</mi></msub><mo id="S3.E4.m1.2.2.2.2.3" xref="S3.E4.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="S3.E4.m1.2.2.2.2.2.2" xref="S3.E4.m1.2.2.2.2.2.3.cmml"><mo stretchy="false" id="S3.E4.m1.2.2.2.2.2.2.3" xref="S3.E4.m1.2.2.2.2.2.3.cmml">(</mo><mrow id="S3.E4.m1.1.1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.E4.m1.1.1.1.1.1.1.1.2" xref="S3.E4.m1.1.1.1.1.1.1.1.2.cmml">c</mi><mo id="S3.E4.m1.1.1.1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.1.1.1.3" xref="S3.E4.m1.1.1.1.1.1.1.1.3.cmml">o</mi><mo id="S3.E4.m1.1.1.1.1.1.1.1.1a" xref="S3.E4.m1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.1.1.1.4" xref="S3.E4.m1.1.1.1.1.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E4.m1.2.2.2.2.2.2.4" xref="S3.E4.m1.2.2.2.2.2.3.cmml">,</mo><mrow id="S3.E4.m1.2.2.2.2.2.2.2" xref="S3.E4.m1.2.2.2.2.2.2.2.cmml"><mi id="S3.E4.m1.2.2.2.2.2.2.2.2" xref="S3.E4.m1.2.2.2.2.2.2.2.2.cmml">m</mi><mo id="S3.E4.m1.2.2.2.2.2.2.2.1" xref="S3.E4.m1.2.2.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.2.2.2.2.2.2.2.3" xref="S3.E4.m1.2.2.2.2.2.2.2.3.cmml">a</mi><mo id="S3.E4.m1.2.2.2.2.2.2.2.1a" xref="S3.E4.m1.2.2.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.2.2.2.2.2.2.2.4" xref="S3.E4.m1.2.2.2.2.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E4.m1.2.2.2.2.2.2.5" xref="S3.E4.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.4.4.4.5" xref="S3.E4.m1.4.4.4.5.cmml">-</mo><mrow id="S3.E4.m1.4.4.4.4" xref="S3.E4.m1.4.4.4.4.cmml"><msub id="S3.E4.m1.4.4.4.4.4" xref="S3.E4.m1.4.4.4.4.4.cmml"><mi id="S3.E4.m1.4.4.4.4.4.2" xref="S3.E4.m1.4.4.4.4.4.2.cmml">N</mi><mi id="S3.E4.m1.4.4.4.4.4.3" xref="S3.E4.m1.4.4.4.4.4.3.cmml">NO</mi></msub><mo id="S3.E4.m1.4.4.4.4.3" xref="S3.E4.m1.4.4.4.4.3.cmml">⁢</mo><mrow id="S3.E4.m1.4.4.4.4.2.2" xref="S3.E4.m1.4.4.4.4.2.3.cmml"><mo stretchy="false" id="S3.E4.m1.4.4.4.4.2.2.3" xref="S3.E4.m1.4.4.4.4.2.3.cmml">(</mo><mrow id="S3.E4.m1.3.3.3.3.1.1.1" xref="S3.E4.m1.3.3.3.3.1.1.1.cmml"><mi id="S3.E4.m1.3.3.3.3.1.1.1.2" xref="S3.E4.m1.3.3.3.3.1.1.1.2.cmml">c</mi><mo id="S3.E4.m1.3.3.3.3.1.1.1.1" xref="S3.E4.m1.3.3.3.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.3.3.3.3.1.1.1.3" xref="S3.E4.m1.3.3.3.3.1.1.1.3.cmml">o</mi><mo id="S3.E4.m1.3.3.3.3.1.1.1.1a" xref="S3.E4.m1.3.3.3.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.3.3.3.3.1.1.1.4" xref="S3.E4.m1.3.3.3.3.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E4.m1.4.4.4.4.2.2.4" xref="S3.E4.m1.4.4.4.4.2.3.cmml">,</mo><mrow id="S3.E4.m1.4.4.4.4.2.2.2" xref="S3.E4.m1.4.4.4.4.2.2.2.cmml"><mi id="S3.E4.m1.4.4.4.4.2.2.2.2" xref="S3.E4.m1.4.4.4.4.2.2.2.2.cmml">m</mi><mo id="S3.E4.m1.4.4.4.4.2.2.2.1" xref="S3.E4.m1.4.4.4.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.4.4.4.4.2.2.2.3" xref="S3.E4.m1.4.4.4.4.2.2.2.3.cmml">a</mi><mo id="S3.E4.m1.4.4.4.4.2.2.2.1a" xref="S3.E4.m1.4.4.4.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.4.4.4.4.2.2.2.4" xref="S3.E4.m1.4.4.4.4.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E4.m1.4.4.4.4.2.2.5" xref="S3.E4.m1.4.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><mrow id="S3.E4.m1.10.10.10" xref="S3.E4.m1.10.10.10.cmml"><mrow id="S3.E4.m1.6.6.6.2" xref="S3.E4.m1.6.6.6.2.cmml"><msub id="S3.E4.m1.6.6.6.2.4" xref="S3.E4.m1.6.6.6.2.4.cmml"><mi id="S3.E4.m1.6.6.6.2.4.2" xref="S3.E4.m1.6.6.6.2.4.2.cmml">N</mi><mi mathvariant="normal" id="S3.E4.m1.6.6.6.2.4.3" xref="S3.E4.m1.6.6.6.2.4.3.cmml">W</mi></msub><mo id="S3.E4.m1.6.6.6.2.3" xref="S3.E4.m1.6.6.6.2.3.cmml">⁢</mo><mrow id="S3.E4.m1.6.6.6.2.2.2" xref="S3.E4.m1.6.6.6.2.2.3.cmml"><mo stretchy="false" id="S3.E4.m1.6.6.6.2.2.2.3" xref="S3.E4.m1.6.6.6.2.2.3.cmml">(</mo><mrow id="S3.E4.m1.5.5.5.1.1.1.1" xref="S3.E4.m1.5.5.5.1.1.1.1.cmml"><mi id="S3.E4.m1.5.5.5.1.1.1.1.2" xref="S3.E4.m1.5.5.5.1.1.1.1.2.cmml">c</mi><mo id="S3.E4.m1.5.5.5.1.1.1.1.1" xref="S3.E4.m1.5.5.5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.5.5.5.1.1.1.1.3" xref="S3.E4.m1.5.5.5.1.1.1.1.3.cmml">o</mi><mo id="S3.E4.m1.5.5.5.1.1.1.1.1a" xref="S3.E4.m1.5.5.5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.5.5.5.1.1.1.1.4" xref="S3.E4.m1.5.5.5.1.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E4.m1.6.6.6.2.2.2.4" xref="S3.E4.m1.6.6.6.2.2.3.cmml">,</mo><mrow id="S3.E4.m1.6.6.6.2.2.2.2" xref="S3.E4.m1.6.6.6.2.2.2.2.cmml"><mi id="S3.E4.m1.6.6.6.2.2.2.2.2" xref="S3.E4.m1.6.6.6.2.2.2.2.2.cmml">m</mi><mo id="S3.E4.m1.6.6.6.2.2.2.2.1" xref="S3.E4.m1.6.6.6.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.6.6.6.2.2.2.2.3" xref="S3.E4.m1.6.6.6.2.2.2.2.3.cmml">a</mi><mo id="S3.E4.m1.6.6.6.2.2.2.2.1a" xref="S3.E4.m1.6.6.6.2.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.6.6.6.2.2.2.2.4" xref="S3.E4.m1.6.6.6.2.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E4.m1.6.6.6.2.2.2.5" xref="S3.E4.m1.6.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.10.10.10.7" xref="S3.E4.m1.10.10.10.7.cmml">-</mo><mrow id="S3.E4.m1.8.8.8.4" xref="S3.E4.m1.8.8.8.4.cmml"><msub id="S3.E4.m1.8.8.8.4.4" xref="S3.E4.m1.8.8.8.4.4.cmml"><mi id="S3.E4.m1.8.8.8.4.4.2" xref="S3.E4.m1.8.8.8.4.4.2.cmml">N</mi><mi id="S3.E4.m1.8.8.8.4.4.3" xref="S3.E4.m1.8.8.8.4.4.3.cmml">NO</mi></msub><mo id="S3.E4.m1.8.8.8.4.3" xref="S3.E4.m1.8.8.8.4.3.cmml">⁢</mo><mrow id="S3.E4.m1.8.8.8.4.2.2" xref="S3.E4.m1.8.8.8.4.2.3.cmml"><mo stretchy="false" id="S3.E4.m1.8.8.8.4.2.2.3" xref="S3.E4.m1.8.8.8.4.2.3.cmml">(</mo><mrow id="S3.E4.m1.7.7.7.3.1.1.1" xref="S3.E4.m1.7.7.7.3.1.1.1.cmml"><mi id="S3.E4.m1.7.7.7.3.1.1.1.2" xref="S3.E4.m1.7.7.7.3.1.1.1.2.cmml">c</mi><mo id="S3.E4.m1.7.7.7.3.1.1.1.1" xref="S3.E4.m1.7.7.7.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.7.7.7.3.1.1.1.3" xref="S3.E4.m1.7.7.7.3.1.1.1.3.cmml">o</mi><mo id="S3.E4.m1.7.7.7.3.1.1.1.1a" xref="S3.E4.m1.7.7.7.3.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.7.7.7.3.1.1.1.4" xref="S3.E4.m1.7.7.7.3.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E4.m1.8.8.8.4.2.2.4" xref="S3.E4.m1.8.8.8.4.2.3.cmml">,</mo><mrow id="S3.E4.m1.8.8.8.4.2.2.2" xref="S3.E4.m1.8.8.8.4.2.2.2.cmml"><mi id="S3.E4.m1.8.8.8.4.2.2.2.2" xref="S3.E4.m1.8.8.8.4.2.2.2.2.cmml">m</mi><mo id="S3.E4.m1.8.8.8.4.2.2.2.1" xref="S3.E4.m1.8.8.8.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.8.8.8.4.2.2.2.3" xref="S3.E4.m1.8.8.8.4.2.2.2.3.cmml">a</mi><mo id="S3.E4.m1.8.8.8.4.2.2.2.1a" xref="S3.E4.m1.8.8.8.4.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.8.8.8.4.2.2.2.4" xref="S3.E4.m1.8.8.8.4.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E4.m1.8.8.8.4.2.2.5" xref="S3.E4.m1.8.8.8.4.2.3.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.10.10.10.7a" xref="S3.E4.m1.10.10.10.7.cmml">-</mo><mrow id="S3.E4.m1.10.10.10.6" xref="S3.E4.m1.10.10.10.6.cmml"><msub id="S3.E4.m1.10.10.10.6.4" xref="S3.E4.m1.10.10.10.6.4.cmml"><mi id="S3.E4.m1.10.10.10.6.4.2" xref="S3.E4.m1.10.10.10.6.4.2.cmml">N</mi><mi id="S3.E4.m1.10.10.10.6.4.3" xref="S3.E4.m1.10.10.10.6.4.3.cmml">YES</mi></msub><mo id="S3.E4.m1.10.10.10.6.3" xref="S3.E4.m1.10.10.10.6.3.cmml">⁢</mo><mrow id="S3.E4.m1.10.10.10.6.2.2" xref="S3.E4.m1.10.10.10.6.2.3.cmml"><mo stretchy="false" id="S3.E4.m1.10.10.10.6.2.2.3" xref="S3.E4.m1.10.10.10.6.2.3.cmml">(</mo><mrow id="S3.E4.m1.9.9.9.5.1.1.1" xref="S3.E4.m1.9.9.9.5.1.1.1.cmml"><mi id="S3.E4.m1.9.9.9.5.1.1.1.2" xref="S3.E4.m1.9.9.9.5.1.1.1.2.cmml">c</mi><mo id="S3.E4.m1.9.9.9.5.1.1.1.1" xref="S3.E4.m1.9.9.9.5.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.9.9.9.5.1.1.1.3" xref="S3.E4.m1.9.9.9.5.1.1.1.3.cmml">o</mi><mo id="S3.E4.m1.9.9.9.5.1.1.1.1a" xref="S3.E4.m1.9.9.9.5.1.1.1.1.cmml">⁢</mo><mi id="S3.E4.m1.9.9.9.5.1.1.1.4" xref="S3.E4.m1.9.9.9.5.1.1.1.4.cmml">l</mi></mrow><mo id="S3.E4.m1.10.10.10.6.2.2.4" xref="S3.E4.m1.10.10.10.6.2.3.cmml">,</mo><mrow id="S3.E4.m1.10.10.10.6.2.2.2" xref="S3.E4.m1.10.10.10.6.2.2.2.cmml"><mi id="S3.E4.m1.10.10.10.6.2.2.2.2" xref="S3.E4.m1.10.10.10.6.2.2.2.2.cmml">m</mi><mo id="S3.E4.m1.10.10.10.6.2.2.2.1" xref="S3.E4.m1.10.10.10.6.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.10.10.10.6.2.2.2.3" xref="S3.E4.m1.10.10.10.6.2.2.2.3.cmml">a</mi><mo id="S3.E4.m1.10.10.10.6.2.2.2.1a" xref="S3.E4.m1.10.10.10.6.2.2.2.1.cmml">⁢</mo><mi id="S3.E4.m1.10.10.10.6.2.2.2.4" xref="S3.E4.m1.10.10.10.6.2.2.2.4.cmml">g</mi></mrow><mo stretchy="false" id="S3.E4.m1.10.10.10.6.2.2.5" xref="S3.E4.m1.10.10.10.6.2.3.cmml">)</mo></mrow></mrow></mrow></mfrac></mrow></math>, <math><mrow id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mn id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">0.5</mn><mo id="S3.SS2.p1.1.m1.1.1.4" xref="S3.SS2.p1.1.m1.1.1.4.cmml">≤</mo><msub id="S3.SS2.p1.1.m1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.cmml"><mrow id="S3.SS2.p1.1.m1.1.1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS2.p1.1.m1.1.1.1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p1.1.m1.1.1.1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.1.1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.2.cmml">B</mi><mo id="S3.SS2.p1.1.m1.1.1.1.1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.1.cmml">-</mo><mi id="S3.SS2.p1.1.m1.1.1.1.1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.3.cmml">V</mi></mrow><mo stretchy="false" id="S3.SS2.p1.1.m1.1.1.1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.p1.1.m1.1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.1.3.cmml"><mn id="S3.SS2.p1.1.m1.1.1.1.3.2" xref="S3.SS2.p1.1.m1.1.1.1.3.2.cmml">10.8</mn><mo id="S3.SS2.p1.1.m1.1.1.1.3.1" xref="S3.SS2.p1.1.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.1.m1.1.1.1.3.3" xref="S3.SS2.p1.1.m1.1.1.1.3.3.cmml">k</mi><mo id="S3.SS2.p1.1.m1.1.1.1.3.1a" xref="S3.SS2.p1.1.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.1.m1.1.1.1.3.4" xref="S3.SS2.p1.1.m1.1.1.1.3.4.cmml">p</mi><mo id="S3.SS2.p1.1.m1.1.1.1.3.1b" xref="S3.SS2.p1.1.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.1.m1.1.1.1.3.5" xref="S3.SS2.p1.1.m1.1.1.1.3.5.cmml">c</mi></mrow></msub><mo id="S3.SS2.p1.1.m1.1.1.5" xref="S3.SS2.p1.1.m1.1.1.5.cmml">≤</mo><mn id="S3.SS2.p1.1.m1.1.1.6" xref="S3.SS2.p1.1.m1.1.1.6.cmml">1.5</mn></mrow></math>, <math><mrow id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mrow id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml"><mo id="S3.SS2.p2.1.m1.1.1.2.1" xref="S3.SS2.p2.1.m1.1.1.2.1.cmml">-</mo><mn id="S3.SS2.p2.1.m1.1.1.2.2" xref="S3.SS2.p2.1.m1.1.1.2.2.cmml">21.5</mn></mrow><mo id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml">≤</mo><msub id="S3.SS2.p2.1.m1.1.1.4" xref="S3.SS2.p2.1.m1.1.1.4.cmml"><mi id="S3.SS2.p2.1.m1.1.1.4.2" xref="S3.SS2.p2.1.m1.1.1.4.2.cmml">M</mi><mi id="S3.SS2.p2.1.m1.1.1.4.3" xref="S3.SS2.p2.1.m1.1.1.4.3.cmml">V</mi></msub><mo id="S3.SS2.p2.1.m1.1.1.5" xref="S3.SS2.p2.1.m1.1.1.5.cmml">≤</mo><mrow id="S3.SS2.p2.1.m1.1.1.6" xref="S3.SS2.p2.1.m1.1.1.6.cmml"><mo id="S3.SS2.p2.1.m1.1.1.6.1" xref="S3.SS2.p2.1.m1.1.1.6.1.cmml">-</mo><mn id="S3.SS2.p2.1.m1.1.1.6.2" xref="S3.SS2.p2.1.m1.1.1.6.2.cmml">18</mn></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0302029

Formulas:

1-

ρ = 9 \cdot 10^{9} Ω

2-

C_{2} H_{2} F_{4}

3-

V_{g a s}

4-

V_{gas} = V - IR

5-

V_{g a s}

6-

ε (V_{g a s})

7-

Q_{γ} (V_{g a s})

8-

V_{g a s}

9-

R_{irr} = 23.1 M Ω

10-

R_{ref} = 6.6 M Ω

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/cs/0101001

Formulas:

1-

\min {f (x) : x_{l} \leq x \leq x_{u}},

2-

f : ℝ^{n} \mapsto ℝ

3-

\nabla f (x)

4-

\nabla^{2} f (x)

5-

\nabla^{2} f (x)

6-

\min {f (x) : x_{l} \leq x \leq x_{u}, c_{l} \leq c (x) \leq c_{u}},

7-

c : ℝ^{n} \mapsto ℝ^{m}

8-

c^{'} (x)

9-

L (x, u) = f (x) + ⟨ u, c (x) ⟩

10-

f : ℝ^{n} \mapsto ℝ

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0506031

Formulas:

1-

\dot{P} = d P / d t \sim 10^{- 19 - 21}

2-

B \propto {(P \dot{P})}^{1 / 2} \sim 10^{8 - 9}

3-

τ \equiv P / 2 \dot{P} ≳ 10^{9}

4-

N_{H} = (1.3 \pm 0.3) \times 10^{20}

5-

0 \overset{''}{.} 049

6-

0 \overset{''}{.} 027

7-

5 \overset{''}{.} 5

8-

T_{eff} = (1.56 \pm 0.28) \times 10^{6}

9-

R_{eff} = 0.45 \pm 0.43

10-

F_{X} = (1.4 \pm 0.3) \times 10^{- 14}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1111.1696

Formulas:

1-

K (p, q, p - k q, - 1)

2-

2 \leq q \leq \frac{p}{2}

3-

2 \leq k \leq \frac{p - 2}{q}

4-

p - k q < q

5-

K (p, q, r, - n)

6-

K (p, q, p - k q, - 1)

7-

2 \leq q \leq \frac{p}{2}

8-

2 \leq k \leq \frac{p - 2}{q}

9-

p - k q < q

10-

p - k q < q

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0912.1041

Formulas:

1-

d N_{c h} / d η

2-

d σ^{i n c l} / d η

3-

d σ_{n}^{i n c l} / d η

4-

α_{s} (R)

5-

α_{s} (r_{g})

6-

α_{s} (r_{g})

7-

σ_{n + m} = \frac{1}{n! m!} \int d τ_{n + m} {| A_{2 \to n + m} ({\vec{p}}_{1}, \dots, {\vec{p}}_{n}, {\vec{q}}_{n + 1}, \dots, {\vec{q}}_{n + m}) |}^{2},

8-

A_{2 \to n + m} ({\vec{p}}_{1}, \dots, {\vec{p}}_{n}, {\vec{q}}_{n + 1}, \dots, {\vec{q}}_{n + m})

9-

d τ_{n + m}

10-

σ_{n} = \sum_{m = 0}^{\infty} σ_{n + m} = \frac{1}{n!} \sum_{m = 0}^{\infty} \frac{1}{m!} \int d τ_{n + m} {| A_{2 \to n + m} ({\vec{p}}_{1}, \dots, {\vec{p}}_{n}, {\vec{q}}_{n + 1}, \dots, {\vec{q}}_{n + m}) |}^{2} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0505493

Formulas:

1-

T ≫ m c^{2}

2-

τ_{c} = λ / c

3-

B_{x} = B_{0} \tanh (z)

4-

B_{y} = α B_{0}

5-

f_{\pm} \propto n (z) \exp [- Γ_{β} {ε - β_{\pm} m c u_{y}} / T] + n_{b g} \exp (- ε / 0.1 T)

6-

n (z) = n_{0} \cosh^{- 2} (z)

7-

β_{\pm} = v_{\pm} / c

8-

Γ_{β} = {[1 - β^{2}]}^{- 1 / 2}

9-

𝒖 = {[1 - {(v / c)}^{2}]}^{- 1 / 2} \cdot 𝒗

10-

T = m c^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0409360

Formulas:

1-

α = Δ \log S_{ν} / Δ \log ν

2-

L_{X} \sim 10^{- 6} L_{Edd}

3-

S_{5 G H z}

4-

L_{radio} \propto L_{X}^{b}

5-

0.037 \geq α \geq - 0.030

6-

Γ_{r a d i o} ≲ 1.4

7-

L_{J} = A_{steady} L_{X}^{0.5}

8-

A_{steady} \geq 6 \times 10^{- 3}

9-

L_{J} = 20 Δ t^{2 / 7} L_{radio}^{4 / 7} M^{- 3 / 7} = 2 \times 10^{- 5} Δ t^{2 / 7} d^{8 / 7} S_{5 G H z}^{4 / 7} M^{- 1}

10-

_{5 G H z}

Formulas (html):

<math><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">α</mi><mo id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml"><mi mathvariant="normal" id="S1.p2.3.m3.1.1.3.2" xref="S1.p2.3.m3.1.1.3.2.cmml">Δ</mi><mo id="S1.p2.3.m3.1.1.3.1" xref="S1.p2.3.m3.1.1.3.1.cmml">⁢</mo><mrow id="S1.p2.3.m3.1.1.3.3" xref="S1.p2.3.m3.1.1.3.3.cmml"><mi id="S1.p2.3.m3.1.1.3.3.1" xref="S1.p2.3.m3.1.1.3.3.1.cmml">log</mi><mo id="S1.p2.3.m3.1.1.3.3a" xref="S1.p2.3.m3.1.1.3.3.cmml">⁡</mo><mrow id="S1.p2.3.m3.1.1.3.3.2" xref="S1.p2.3.m3.1.1.3.3.2.cmml"><mrow id="S1.p2.3.m3.1.1.3.3.2.2" xref="S1.p2.3.m3.1.1.3.3.2.2.cmml"><msub id="S1.p2.3.m3.1.1.3.3.2.2.2" xref="S1.p2.3.m3.1.1.3.3.2.2.2.cmml"><mi id="S1.p2.3.m3.1.1.3.3.2.2.2.2" xref="S1.p2.3.m3.1.1.3.3.2.2.2.2.cmml">S</mi><mi id="S1.p2.3.m3.1.1.3.3.2.2.2.3" xref="S1.p2.3.m3.1.1.3.3.2.2.2.3.cmml">ν</mi></msub><mo id="S1.p2.3.m3.1.1.3.3.2.2.1" xref="S1.p2.3.m3.1.1.3.3.2.2.1.cmml">/</mo><mi mathvariant="normal" id="S1.p2.3.m3.1.1.3.3.2.2.3" xref="S1.p2.3.m3.1.1.3.3.2.2.3.cmml">Δ</mi></mrow><mo id="S1.p2.3.m3.1.1.3.3.2.1" xref="S1.p2.3.m3.1.1.3.3.2.1.cmml">⁢</mo><mrow id="S1.p2.3.m3.1.1.3.3.2.3" xref="S1.p2.3.m3.1.1.3.3.2.3.cmml"><mi id="S1.p2.3.m3.1.1.3.3.2.3.1" xref="S1.p2.3.m3.1.1.3.3.2.3.1.cmml">log</mi><mo id="S1.p2.3.m3.1.1.3.3.2.3a" xref="S1.p2.3.m3.1.1.3.3.2.3.cmml">⁡</mo><mi id="S1.p2.3.m3.1.1.3.3.2.3.2" xref="S1.p2.3.m3.1.1.3.3.2.3.2.cmml">ν</mi></mrow></mrow></mrow></mrow></mrow></math>, <math><mrow id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml"><msub id="S1.p2.5.m5.1.1.2" xref="S1.p2.5.m5.1.1.2.cmml"><mi id="S1.p2.5.m5.1.1.2.2" xref="S1.p2.5.m5.1.1.2.2.cmml">L</mi><mi mathvariant="normal" id="S1.p2.5.m5.1.1.2.3" xref="S1.p2.5.m5.1.1.2.3.cmml">X</mi></msub><mo id="S1.p2.5.m5.1.1.1" xref="S1.p2.5.m5.1.1.1.cmml">∼</mo><mrow id="S1.p2.5.m5.1.1.3" xref="S1.p2.5.m5.1.1.3.cmml"><msup id="S1.p2.5.m5.1.1.3.2" xref="S1.p2.5.m5.1.1.3.2.cmml"><mn id="S1.p2.5.m5.1.1.3.2.2" xref="S1.p2.5.m5.1.1.3.2.2.cmml">10</mn><mrow id="S1.p2.5.m5.1.1.3.2.3" xref="S1.p2.5.m5.1.1.3.2.3.cmml"><mo id="S1.p2.5.m5.1.1.3.2.3.1" xref="S1.p2.5.m5.1.1.3.2.3.1.cmml">-</mo><mn id="S1.p2.5.m5.1.1.3.2.3.2" xref="S1.p2.5.m5.1.1.3.2.3.2.cmml">6</mn></mrow></msup><mo id="S1.p2.5.m5.1.1.3.1" xref="S1.p2.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S1.p2.5.m5.1.1.3.3" xref="S1.p2.5.m5.1.1.3.3.cmml"><mi id="S1.p2.5.m5.1.1.3.3.2" xref="S1.p2.5.m5.1.1.3.3.2.cmml">L</mi><mi id="S1.p2.5.m5.1.1.3.3.3" xref="S1.p2.5.m5.1.1.3.3.3.cmml">Edd</mi></msub></mrow></mrow></math>, <math><msub id="S1.T1.4.4.4.m1.1.1" xref="S1.T1.4.4.4.m1.1.1.cmml"><mi id="S1.T1.4.4.4.m1.1.1.2" xref="S1.T1.4.4.4.m1.1.1.2.cmml">S</mi><mrow id="S1.T1.4.4.4.m1.1.1.3" xref="S1.T1.4.4.4.m1.1.1.3.cmml"><mn id="S1.T1.4.4.4.m1.1.1.3.2" xref="S1.T1.4.4.4.m1.1.1.3.2.cmml">5</mn><mo id="S1.T1.4.4.4.m1.1.1.3.1" xref="S1.T1.4.4.4.m1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S1.T1.4.4.4.m1.1.1.3.3" xref="S1.T1.4.4.4.m1.1.1.3.3.cmml">G</mi><mo id="S1.T1.4.4.4.m1.1.1.3.1a" xref="S1.T1.4.4.4.m1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S1.T1.4.4.4.m1.1.1.3.4" xref="S1.T1.4.4.4.m1.1.1.3.4.cmml">H</mi><mo id="S1.T1.4.4.4.m1.1.1.3.1b" xref="S1.T1.4.4.4.m1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S1.T1.4.4.4.m1.1.1.3.5" xref="S1.T1.4.4.4.m1.1.1.3.5.cmml">z</mi></mrow></msub></math>, <math><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><msub id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2.2" xref="S3.SS1.p1.1.m1.1.1.2.2.cmml">L</mi><mi id="S3.SS1.p1.1.m1.1.1.2.3" xref="S3.SS1.p1.1.m1.1.1.2.3.cmml">radio</mi></msub><mo id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">∝</mo><msubsup id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml"><mi id="S3.SS1.p1.1.m1.1.1.3.2.2" xref="S3.SS1.p1.1.m1.1.1.3.2.2.cmml">L</mi><mi mathvariant="normal" id="S3.SS1.p1.1.m1.1.1.3.2.3" xref="S3.SS1.p1.1.m1.1.1.3.2.3.cmml">X</mi><mi id="S3.SS1.p1.1.m1.1.1.3.3" xref="S3.SS1.p1.1.m1.1.1.3.3.cmml">b</mi></msubsup></mrow></math>, <math><mrow id="S3.SS2.SSS3.p1.1.m1.1.1" xref="S3.SS2.SSS3.p1.1.m1.1.1.cmml"><mn id="S3.SS2.SSS3.p1.1.m1.1.1.2" xref="S3.SS2.SSS3.p1.1.m1.1.1.2.cmml">0.037</mn><mo id="S3.SS2.SSS3.p1.1.m1.1.1.3" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.cmml">≥</mo><mi id="S3.SS2.SSS3.p1.1.m1.1.1.4" xref="S3.SS2.SSS3.p1.1.m1.1.1.4.cmml">α</mi><mo id="S3.SS2.SSS3.p1.1.m1.1.1.5" xref="S3.SS2.SSS3.p1.1.m1.1.1.5.cmml">≥</mo><mrow id="S3.SS2.SSS3.p1.1.m1.1.1.6" xref="S3.SS2.SSS3.p1.1.m1.1.1.6.cmml"><mo id="S3.SS2.SSS3.p1.1.m1.1.1.6.1" xref="S3.SS2.SSS3.p1.1.m1.1.1.6.1.cmml">-</mo><mn id="S3.SS2.SSS3.p1.1.m1.1.1.6.2" xref="S3.SS2.SSS3.p1.1.m1.1.1.6.2.cmml">0.030</mn></mrow></mrow></math>, <math><mrow id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml"><msub id="S4.p2.5.m5.1.1.2" xref="S4.p2.5.m5.1.1.2.cmml"><mi mathvariant="normal" id="S4.p2.5.m5.1.1.2.2" xref="S4.p2.5.m5.1.1.2.2.cmml">Γ</mi><mrow id="S4.p2.5.m5.1.1.2.3" xref="S4.p2.5.m5.1.1.2.3.cmml"><mi id="S4.p2.5.m5.1.1.2.3.2" xref="S4.p2.5.m5.1.1.2.3.2.cmml">r</mi><mo id="S4.p2.5.m5.1.1.2.3.1" xref="S4.p2.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m5.1.1.2.3.3" xref="S4.p2.5.m5.1.1.2.3.3.cmml">a</mi><mo id="S4.p2.5.m5.1.1.2.3.1a" xref="S4.p2.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m5.1.1.2.3.4" xref="S4.p2.5.m5.1.1.2.3.4.cmml">d</mi><mo id="S4.p2.5.m5.1.1.2.3.1b" xref="S4.p2.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m5.1.1.2.3.5" xref="S4.p2.5.m5.1.1.2.3.5.cmml">i</mi><mo id="S4.p2.5.m5.1.1.2.3.1c" xref="S4.p2.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m5.1.1.2.3.6" xref="S4.p2.5.m5.1.1.2.3.6.cmml">o</mi></mrow></msub><mo id="S4.p2.5.m5.1.1.1" xref="S4.p2.5.m5.1.1.1.cmml">≲</mo><mn id="S4.p2.5.m5.1.1.3" xref="S4.p2.5.m5.1.1.3.cmml">1.4</mn></mrow></math>, <math><mrow id="S5.Ex1.m1.1.1" xref="S5.Ex1.m1.1.1.cmml"><msub id="S5.Ex1.m1.1.1.2" xref="S5.Ex1.m1.1.1.2.cmml"><mi id="S5.Ex1.m1.1.1.2.2" xref="S5.Ex1.m1.1.1.2.2.cmml">L</mi><mi mathvariant="normal" id="S5.Ex1.m1.1.1.2.3" xref="S5.Ex1.m1.1.1.2.3.cmml">J</mi></msub><mo id="S5.Ex1.m1.1.1.1" xref="S5.Ex1.m1.1.1.1.cmml">=</mo><mrow id="S5.Ex1.m1.1.1.3" xref="S5.Ex1.m1.1.1.3.cmml"><msub id="S5.Ex1.m1.1.1.3.2" xref="S5.Ex1.m1.1.1.3.2.cmml"><mi id="S5.Ex1.m1.1.1.3.2.2" xref="S5.Ex1.m1.1.1.3.2.2.cmml">A</mi><mi id="S5.Ex1.m1.1.1.3.2.3" xref="S5.Ex1.m1.1.1.3.2.3.cmml">steady</mi></msub><mo id="S5.Ex1.m1.1.1.3.1" xref="S5.Ex1.m1.1.1.3.1.cmml">⁢</mo><msubsup id="S5.Ex1.m1.1.1.3.3" xref="S5.Ex1.m1.1.1.3.3.cmml"><mi id="S5.Ex1.m1.1.1.3.3.2.2" xref="S5.Ex1.m1.1.1.3.3.2.2.cmml">L</mi><mi mathvariant="normal" id="S5.Ex1.m1.1.1.3.3.2.3" xref="S5.Ex1.m1.1.1.3.3.2.3.cmml">X</mi><mn id="S5.Ex1.m1.1.1.3.3.3" xref="S5.Ex1.m1.1.1.3.3.3.cmml">0.5</mn></msubsup></mrow></mrow></math>, <math><mrow id="S5.SS1.p3.1.m1.1.1" xref="S5.SS1.p3.1.m1.1.1.cmml"><msub id="S5.SS1.p3.1.m1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.2.cmml"><mi id="S5.SS1.p3.1.m1.1.1.2.2" xref="S5.SS1.p3.1.m1.1.1.2.2.cmml">A</mi><mi id="S5.SS1.p3.1.m1.1.1.2.3" xref="S5.SS1.p3.1.m1.1.1.2.3.cmml">steady</mi></msub><mo id="S5.SS1.p3.1.m1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.cmml">≥</mo><mrow id="S5.SS1.p3.1.m1.1.1.3" xref="S5.SS1.p3.1.m1.1.1.3.cmml"><mn id="S5.SS1.p3.1.m1.1.1.3.2" xref="S5.SS1.p3.1.m1.1.1.3.2.cmml">6</mn><mo id="S5.SS1.p3.1.m1.1.1.3.1" xref="S5.SS1.p3.1.m1.1.1.3.1.cmml">×</mo><msup id="S5.SS1.p3.1.m1.1.1.3.3" xref="S5.SS1.p3.1.m1.1.1.3.3.cmml"><mn id="S5.SS1.p3.1.m1.1.1.3.3.2" xref="S5.SS1.p3.1.m1.1.1.3.3.2.cmml">10</mn><mrow id="S5.SS1.p3.1.m1.1.1.3.3.3" xref="S5.SS1.p3.1.m1.1.1.3.3.3.cmml"><mo id="S5.SS1.p3.1.m1.1.1.3.3.3.1" xref="S5.SS1.p3.1.m1.1.1.3.3.3.1.cmml">-</mo><mn id="S5.SS1.p3.1.m1.1.1.3.3.3.2" xref="S5.SS1.p3.1.m1.1.1.3.3.3.2.cmml">3</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S5.Ex2.m1.1.1" xref="S5.Ex2.m1.1.1.cmml"><msub id="S5.Ex2.m1.1.1.2" xref="S5.Ex2.m1.1.1.2.cmml"><mi id="S5.Ex2.m1.1.1.2.2" xref="S5.Ex2.m1.1.1.2.2.cmml">L</mi><mi mathvariant="normal" id="S5.Ex2.m1.1.1.2.3" xref="S5.Ex2.m1.1.1.2.3.cmml">J</mi></msub><mo id="S5.Ex2.m1.1.1.3" xref="S5.Ex2.m1.1.1.3.cmml">=</mo><mrow id="S5.Ex2.m1.1.1.4" xref="S5.Ex2.m1.1.1.4.cmml"><mn id="S5.Ex2.m1.1.1.4.2" xref="S5.Ex2.m1.1.1.4.2.cmml">20</mn><mo id="S5.Ex2.m1.1.1.4.1" xref="S5.Ex2.m1.1.1.4.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.Ex2.m1.1.1.4.3" xref="S5.Ex2.m1.1.1.4.3.cmml">Δ</mi><mo id="S5.Ex2.m1.1.1.4.1a" xref="S5.Ex2.m1.1.1.4.1.cmml">⁢</mo><msup id="S5.Ex2.m1.1.1.4.4" xref="S5.Ex2.m1.1.1.4.4.cmml"><mi id="S5.Ex2.m1.1.1.4.4.2" xref="S5.Ex2.m1.1.1.4.4.2.cmml">t</mi><mrow id="S5.Ex2.m1.1.1.4.4.3" xref="S5.Ex2.m1.1.1.4.4.3.cmml"><mn id="S5.Ex2.m1.1.1.4.4.3.2" xref="S5.Ex2.m1.1.1.4.4.3.2.cmml">2</mn><mo id="S5.Ex2.m1.1.1.4.4.3.1" xref="S5.Ex2.m1.1.1.4.4.3.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.4.4.3.3" xref="S5.Ex2.m1.1.1.4.4.3.3.cmml">7</mn></mrow></msup><mo id="S5.Ex2.m1.1.1.4.1b" xref="S5.Ex2.m1.1.1.4.1.cmml">⁢</mo><msubsup id="S5.Ex2.m1.1.1.4.5" xref="S5.Ex2.m1.1.1.4.5.cmml"><mi id="S5.Ex2.m1.1.1.4.5.2.2" xref="S5.Ex2.m1.1.1.4.5.2.2.cmml">L</mi><mi id="S5.Ex2.m1.1.1.4.5.2.3" xref="S5.Ex2.m1.1.1.4.5.2.3.cmml">radio</mi><mrow id="S5.Ex2.m1.1.1.4.5.3" xref="S5.Ex2.m1.1.1.4.5.3.cmml"><mn id="S5.Ex2.m1.1.1.4.5.3.2" xref="S5.Ex2.m1.1.1.4.5.3.2.cmml">4</mn><mo id="S5.Ex2.m1.1.1.4.5.3.1" xref="S5.Ex2.m1.1.1.4.5.3.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.4.5.3.3" xref="S5.Ex2.m1.1.1.4.5.3.3.cmml">7</mn></mrow></msubsup><mo id="S5.Ex2.m1.1.1.4.1c" xref="S5.Ex2.m1.1.1.4.1.cmml">⁢</mo><msup id="S5.Ex2.m1.1.1.4.6" xref="S5.Ex2.m1.1.1.4.6.cmml"><mi id="S5.Ex2.m1.1.1.4.6.2" xref="S5.Ex2.m1.1.1.4.6.2.cmml">M</mi><mrow id="S5.Ex2.m1.1.1.4.6.3" xref="S5.Ex2.m1.1.1.4.6.3.cmml"><mo id="S5.Ex2.m1.1.1.4.6.3.1" xref="S5.Ex2.m1.1.1.4.6.3.1.cmml">-</mo><mrow id="S5.Ex2.m1.1.1.4.6.3.2" xref="S5.Ex2.m1.1.1.4.6.3.2.cmml"><mn id="S5.Ex2.m1.1.1.4.6.3.2.2" xref="S5.Ex2.m1.1.1.4.6.3.2.2.cmml">3</mn><mo id="S5.Ex2.m1.1.1.4.6.3.2.1" xref="S5.Ex2.m1.1.1.4.6.3.2.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.4.6.3.2.3" xref="S5.Ex2.m1.1.1.4.6.3.2.3.cmml">7</mn></mrow></mrow></msup></mrow><mo id="S5.Ex2.m1.1.1.5" xref="S5.Ex2.m1.1.1.5.cmml">=</mo><mrow id="S5.Ex2.m1.1.1.6" xref="S5.Ex2.m1.1.1.6.cmml"><mrow id="S5.Ex2.m1.1.1.6.2" xref="S5.Ex2.m1.1.1.6.2.cmml"><mn id="S5.Ex2.m1.1.1.6.2.2" xref="S5.Ex2.m1.1.1.6.2.2.cmml">2</mn><mo id="S5.Ex2.m1.1.1.6.2.1" xref="S5.Ex2.m1.1.1.6.2.1.cmml">×</mo><msup id="S5.Ex2.m1.1.1.6.2.3" xref="S5.Ex2.m1.1.1.6.2.3.cmml"><mn id="S5.Ex2.m1.1.1.6.2.3.2" xref="S5.Ex2.m1.1.1.6.2.3.2.cmml">10</mn><mrow id="S5.Ex2.m1.1.1.6.2.3.3" xref="S5.Ex2.m1.1.1.6.2.3.3.cmml"><mo id="S5.Ex2.m1.1.1.6.2.3.3.1" xref="S5.Ex2.m1.1.1.6.2.3.3.1.cmml">-</mo><mn id="S5.Ex2.m1.1.1.6.2.3.3.2" xref="S5.Ex2.m1.1.1.6.2.3.3.2.cmml">5</mn></mrow></msup></mrow><mo id="S5.Ex2.m1.1.1.6.1" xref="S5.Ex2.m1.1.1.6.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.Ex2.m1.1.1.6.3" xref="S5.Ex2.m1.1.1.6.3.cmml">Δ</mi><mo id="S5.Ex2.m1.1.1.6.1a" xref="S5.Ex2.m1.1.1.6.1.cmml">⁢</mo><msup id="S5.Ex2.m1.1.1.6.4" xref="S5.Ex2.m1.1.1.6.4.cmml"><mi id="S5.Ex2.m1.1.1.6.4.2" xref="S5.Ex2.m1.1.1.6.4.2.cmml">t</mi><mrow id="S5.Ex2.m1.1.1.6.4.3" xref="S5.Ex2.m1.1.1.6.4.3.cmml"><mn id="S5.Ex2.m1.1.1.6.4.3.2" xref="S5.Ex2.m1.1.1.6.4.3.2.cmml">2</mn><mo id="S5.Ex2.m1.1.1.6.4.3.1" xref="S5.Ex2.m1.1.1.6.4.3.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.6.4.3.3" xref="S5.Ex2.m1.1.1.6.4.3.3.cmml">7</mn></mrow></msup><mo id="S5.Ex2.m1.1.1.6.1b" xref="S5.Ex2.m1.1.1.6.1.cmml">⁢</mo><msup id="S5.Ex2.m1.1.1.6.5" xref="S5.Ex2.m1.1.1.6.5.cmml"><mi id="S5.Ex2.m1.1.1.6.5.2" xref="S5.Ex2.m1.1.1.6.5.2.cmml">d</mi><mrow id="S5.Ex2.m1.1.1.6.5.3" xref="S5.Ex2.m1.1.1.6.5.3.cmml"><mn id="S5.Ex2.m1.1.1.6.5.3.2" xref="S5.Ex2.m1.1.1.6.5.3.2.cmml">8</mn><mo id="S5.Ex2.m1.1.1.6.5.3.1" xref="S5.Ex2.m1.1.1.6.5.3.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.6.5.3.3" xref="S5.Ex2.m1.1.1.6.5.3.3.cmml">7</mn></mrow></msup><mo id="S5.Ex2.m1.1.1.6.1c" xref="S5.Ex2.m1.1.1.6.1.cmml">⁢</mo><msubsup id="S5.Ex2.m1.1.1.6.6" xref="S5.Ex2.m1.1.1.6.6.cmml"><mi id="S5.Ex2.m1.1.1.6.6.2.2" xref="S5.Ex2.m1.1.1.6.6.2.2.cmml">S</mi><mrow id="S5.Ex2.m1.1.1.6.6.2.3" xref="S5.Ex2.m1.1.1.6.6.2.3.cmml"><mn id="S5.Ex2.m1.1.1.6.6.2.3.2" xref="S5.Ex2.m1.1.1.6.6.2.3.2.cmml">5</mn><mo id="S5.Ex2.m1.1.1.6.6.2.3.1" xref="S5.Ex2.m1.1.1.6.6.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.Ex2.m1.1.1.6.6.2.3.3" xref="S5.Ex2.m1.1.1.6.6.2.3.3.cmml">G</mi><mo id="S5.Ex2.m1.1.1.6.6.2.3.1a" xref="S5.Ex2.m1.1.1.6.6.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.Ex2.m1.1.1.6.6.2.3.4" xref="S5.Ex2.m1.1.1.6.6.2.3.4.cmml">H</mi><mo id="S5.Ex2.m1.1.1.6.6.2.3.1b" xref="S5.Ex2.m1.1.1.6.6.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.Ex2.m1.1.1.6.6.2.3.5" xref="S5.Ex2.m1.1.1.6.6.2.3.5.cmml">z</mi></mrow><mrow id="S5.Ex2.m1.1.1.6.6.3" xref="S5.Ex2.m1.1.1.6.6.3.cmml"><mn id="S5.Ex2.m1.1.1.6.6.3.2" xref="S5.Ex2.m1.1.1.6.6.3.2.cmml">4</mn><mo id="S5.Ex2.m1.1.1.6.6.3.1" xref="S5.Ex2.m1.1.1.6.6.3.1.cmml">/</mo><mn id="S5.Ex2.m1.1.1.6.6.3.3" xref="S5.Ex2.m1.1.1.6.6.3.3.cmml">7</mn></mrow></msubsup><mo id="S5.Ex2.m1.1.1.6.1d" xref="S5.Ex2.m1.1.1.6.1.cmml">⁢</mo><msup id="S5.Ex2.m1.1.1.6.7" xref="S5.Ex2.m1.1.1.6.7.cmml"><mi id="S5.Ex2.m1.1.1.6.7.2" xref="S5.Ex2.m1.1.1.6.7.2.cmml">M</mi><mrow id="S5.Ex2.m1.1.1.6.7.3" xref="S5.Ex2.m1.1.1.6.7.3.cmml"><mo id="S5.Ex2.m1.1.1.6.7.3.1" xref="S5.Ex2.m1.1.1.6.7.3.1.cmml">-</mo><mn id="S5.Ex2.m1.1.1.6.7.3.2" xref="S5.Ex2.m1.1.1.6.7.3.2.cmml">1</mn></mrow></msup></mrow></mrow></math>, <math><msub id="S5.SS2.p3.5.m5.1.1" xref="S5.SS2.p3.5.m5.1.1.cmml"><mi id="S5.SS2.p3.5.m5.1.1a" xref="S5.SS2.p3.5.m5.1.1.cmml"/><mrow id="S5.SS2.p3.5.m5.1.1.1" xref="S5.SS2.p3.5.m5.1.1.1.cmml"><mn id="S5.SS2.p3.5.m5.1.1.1.2" xref="S5.SS2.p3.5.m5.1.1.1.2.cmml">5</mn><mo id="S5.SS2.p3.5.m5.1.1.1.1" xref="S5.SS2.p3.5.m5.1.1.1.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.SS2.p3.5.m5.1.1.1.3" xref="S5.SS2.p3.5.m5.1.1.1.3.cmml">G</mi><mo id="S5.SS2.p3.5.m5.1.1.1.1a" xref="S5.SS2.p3.5.m5.1.1.1.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.SS2.p3.5.m5.1.1.1.4" xref="S5.SS2.p3.5.m5.1.1.1.4.cmml">H</mi><mo id="S5.SS2.p3.5.m5.1.1.1.1b" xref="S5.SS2.p3.5.m5.1.1.1.1.cmml">⁢</mo><mi mathvariant="normal" id="S5.SS2.p3.5.m5.1.1.1.5" xref="S5.SS2.p3.5.m5.1.1.1.5.cmml">z</mi></mrow></msub></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1409.3671

Formulas:

1-

{| D_{i} ⟩}

2-

N = N_{α} + N_{β}

3-

{ϕ_{1}, \dots, ϕ_{M}}

4-

| Ψ_{0} ⟩ = \sum_{i} c_{i} | D_{i} ⟩

5-

H_{i j} = ⟨ D_{i} | H | D_{j} ⟩

6-

⟨ D_{i} | D_{j} ⟩ = δ_{i j}

7-

| Ψ_{0}^{(n)} ⟩ = \sum_{i \in S_{n}} c_{i}^{(n)} | D_{i} ⟩

8-

| Ψ_{0}^{(n)} ⟩

9-

⟨ Ψ_{0}^{(n)} | H | D_{i_{c}} ⟩ \neq 0

10-

δ e (| D_{i_{c}} ⟩) = - \frac{{⟨ Ψ_{0}^{(n)} | H | D_{i_{c}} ⟩}^{2}}{⟨ D_{i_{c}} | H | D_{i_{c}} ⟩ - E_{0}^{(n)}}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0301154

Formulas:

1-

M_{p} \sim > 0.1

2-

Ω_{p} = \sqrt{G M_{*} / r_{p}^{3}}

3-

(r, θ, ϕ)

4-

𝐮 = (u_{r}, u_{θ}, u_{ϕ})

5-

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ 𝐮) = 0,

6-

\frac{\partial s_{r}}{\partial t} + \nabla \cdot (s_{r} 𝐮)

7-

ρ r \sin^{2} θ {(\frac{u_{ϕ}}{r \sin θ} + Ω_{p})}^{2} + \frac{ρ u_{θ}^{2}}{r}

8-

- \frac{\partial p}{\partial r} - ρ \frac{\partial Φ}{\partial r} + f_{r},

9-

\frac{\partial s_{θ}}{\partial t} + \nabla \cdot (s_{θ} 𝐮)

10-

ρ r^{2} \sin θ \cos θ {(\frac{u_{ϕ}}{r \sin θ} + Ω_{p})}^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0006253

Formulas:

1-

M_{b} / M_{D M}

2-

ϕ [μ_{B} (0)]

3-

ρ_{d i s k + I r r}

4-

0.0084 h_{70}^{- 1}

5-

ϕ [μ_{B} (0)]

6-

ρ_{L S B} = 17 \times ρ_{H S B}

7-

ρ_{d i s k + I r r}

8-

0.016 h_{70}^{- 1}

9-

ϕ [μ_{B} (0)]

10-

ρ_{L S B} = 22 \times ρ_{H S B}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/nucl-ex/0207018

Formulas:

1-

E_{Z D C}

2-

100 ≲ N_{p} ≲ 200

3-

E_{Z D C}

4-

E_{Z D C}

5-

⟨ D Y (E_{T}) ⟩ = \frac{\int_{0}^{\infty} d^{2} b P (E_{T} | b) ⟨ D Y (b) ⟩ P_{i n t} (b)}{\int_{0}^{\infty} d^{2} b P (E_{T} | b) P_{i n t} (b)}

6-

⟨ D Y (b) ⟩

7-

P_{i n t} (b)

8-

P (E_{T} | b)

9-

⟨ E_{T} (b) ⟩ = q N_{p} (b) σ_{E_{T}} (b) = q \sqrt{a N_{p} (b)}

10-

N_{p} (b)

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1511.01265

Formulas:

1-

1 f_{5 / 2}

2-

1 f_{5 / 2}

3-

2 p_{1 / 2}

4-

2 p_{1 / 2}

5-

2 p_{3 / 2}

6-

2 p_{3 / 2}

7-

2 p_{1 / 2}

8-

1 f_{5 / 2}

9-

1 f_{7 / 2}

10-

1 d_{3 / 2}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/1802.02726

Formulas:

1-

⟨ A x - A y, x - y ⟩ \geq α {∥ A x - A y ∥}^{2},

2-

⟨ A x - A y, x - y ⟩ \geq α {∥ x - y ∥}^{2} .

3-

⟨ B x - B y, x - y ⟩ \geq (- u) {∥ B x - B y ∥}^{2} + v {∥ x - y ∥}^{2},

4-

V I (C, A)

5-

lim_{n} ∥ A z_{n} - A u ∥ = 0,

6-

lim_{n} ∥ z_{n} - u ∥ = 0,

7-

Γ = \cap_{n = 1}^{\infty} F (T_{n}) \cap Ω \cap V I (E, B)

8-

Γ = \cap_{n = 1}^{\infty} F (T_{n}) \cap Ω \cap V I (E, B)

9-

lim_{n} ∥ B z_{n} - B p ∥ = 0 .

10-

∥ B x - B y ∥ \geq α ∥ x - y ∥ .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0702531

Formulas:

1-

Δ N_{e} / N_{e} \sim 0.22 %

2-

| Δ C_{l}^{T T} / C_{l}^{T T} | \sim 0.15 %

3-

| Δ C_{l}^{E E} / C_{l}^{E E} | \sim 0.36 %

4-

| Δ N_{e} / N_{e} | \sim few \times 10^{- 6}

5-

Δ N_{e} / N_{e} \sim 10^{- 6}

6-

1 s \to 2 s

7-

Ly n \to Ly (n - 1)

8-

Δ I_{ν} (z_{em})

9-

z_{f} \approx z_{\max} ν_{Ly (n - 1)} / ν_{Ly n}

10-

Ly n \to Ly (n - 1)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0610308

Formulas:

1-

h_{1}^{a} (x)

2-

H_{1}^{a} (z)

3-

B_{G} (z) = {(1 + z^{2} ⟨ 𝐩_{h_{1}}^{2} ⟩ / ⟨ 𝐊_{H_{1}}^{2} ⟩)}^{- 1 / 2}

4-

H_{1}^{a} (z) \equiv H_{1}^{⟂ (1 / 2) a} (z) =

5-

\int d^{2} 𝐊_{T} \frac{| 𝐊_{T} |}{2 z m_{π}} H_{1}^{⟂ a} (z, 𝐊_{T})

6-

A_{U T}^{\sin (ϕ + ϕ_{S})} = 2 \frac{\sum_{a} e_{a}^{2} x h_{1}^{a} (x) B_{G} H_{1}^{a} (z)}{\sum_{a} e_{a}^{2} x f_{1}^{a} (x) D_{1}^{a} (z)} .

7-

H_{1}^{a} (z)

8-

e^{+} e^{-} \to \bar{q} q \to 2 j e t s

9-

d σ = d σ_{unp} \underset{\equiv A_{1}}{\underset{⏟}{[1 + \cos (2 ϕ_{1}) \frac{\sin^{2} θ}{1 + \cos^{2} θ} C_{G} \times \frac{\sum_{a} e_{a}^{2} H_{1}^{a} H_{1}^{\bar{a}}}{\sum_{a} e_{a}^{2} D_{1}^{a} D_{1}^{\bar{a}}}]}}

10-

C_{G} (z_{1}, z_{2}) = \frac{16}{π} z_{1} z_{2} / (z_{1}^{2} + z_{2}^{2})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0507129

Formulas:

1-

\frac{d \vec{p}}{d t}

2-

q (\vec{v} \times \vec{B})

3-

v \equiv ∥ \vec{v} ∥

4-

p \equiv ∥ \vec{p} ∥

5-

p \sin θ_{B} = q ∥ \vec{B} ∥ R

6-

\frac{d \vec{u}}{d l}

7-

\frac{q}{p} (\vec{u} \times \vec{B})

8-

\frac{d \vec{x}}{d l} = \frac{\vec{v}}{v}

9-

d l \equiv v d t

10-

{\vec{u}}_{0} + \frac{q}{p} \int_{l_{0}}^{l_{1}} 𝑑 l^{'} (\vec{u} \times \vec{B}) (l^{'})

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1002.2927

Formulas:

1-

f (a) = n \frac{1}{\sqrt{2 π} σ a} \exp (- \frac{{(\ln a - \ln a_{m})}^{2}}{2 σ^{2}}) .

2-

n ({cm}^{- 3})

3-

a_{m} (K)

4-

f (a_{c}) = A a_{c}^{B},

5-

\frac{2.55 \cdot 10^{- 5}}{100^{B}} {cm}^{- 3} μ m^{- 1}

6-

\approx - 20 {}^{\circ}C

7-

L \leq 200 μ m

8-

Γ (L) = 2,

9-

Γ (L) = 5 \cdot L^{0.59}

10-

L > 200 μ m

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9807199

Formulas:

1-

c z = H_{0} r

2-

^{[8, 9, 10, 11, 12]}

3-

^{[8, 11, 13]}

4-

100 km s^{- 1} {Mpc}^{- 1}

5-

δ (𝐫) \equiv (ρ (𝐫) - ⟨ ρ ⟩) / ⟨ ρ ⟩

6-

^{[3, 22, 23, 24]}

7-

c z = H_{0} r + \hat{𝐫} \cdot [𝐯 (𝐫) - 𝐯 (𝟎)],

8-

δ_{g} = b δ_{D M}

9-

^{[41, 42, 43]}

10-

\nabla \cdot 𝐯 (𝐫) = - H_{0} Ω^{0.6} δ_{D M} (𝐫)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1712.07984

Formulas:

1-

G > 2 e^{2} / h

2-

2 e^{2} / h

3-

G \leq 2 e^{2} / h

4-

G ≪ 2 e^{2} / h

5-

n_{s} = 7 \div 8 \times 10^{11}

6-

μ = 1.5 \times 10^{6}

7-

n_{s} = 5 \div 6 \times 10^{11}

8-

μ = 4 \times 10^{5}

9-

J = (10^{- 10} \div 10^{- 8})

10-

G_{d a r k} (V_{g}^{e f f}) / G_{0}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1206.3597

Formulas:

1-

H \bar{H} 0

2-

H \bar{H} 0

3-

ℏ ω_{o} = 50

4-

S (Q, ω) = \sum_{n} S_{n} (Q, ω)

5-

S_{n} (Q, ω) = \frac{1}{n!} {(\frac{ℏ Q^{2}}{2 m ω_{o}})}^{n} e x p (\frac{- ℏ Q^{2}}{2 m ω_{o}}) δ (ℏ ω - n ℏ ω_{o})

6-

S_{n} (Q) = A_{n} Q^{2 n} e x p (- C Q^{2}) + B_{n}

7-

Γ (n + 1)

8-

ℏ ω_{o} = 50

9-

e x p (\frac{- {(ℏ ω - n ℏ ω_{o} - \frac{ℏ^{2} Q^{2}}{2 M})}^{2}}{2 σ^{2}})

10-

n ω_{o} + \frac{ℏ^{2} Q^{2}}{2 M}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0104359

Formulas:

1-

t_{1 / 2} > \sim 10^{9}

2-

9 < \sim T_{nuc} (Gy) < \sim 27

3-

10^{5} < \sim t_{1 / 2} < \sim 10^{8}

4-

0.8 < \sim P_{235} / P_{238} < \sim 1.2

5-

1.4 < \sim P_{232} / P_{238} < \sim 2.7

6-

1 < P_{235} / P_{238} < 1.5

7-

1.7 < P_{232} / P_{238} < 2

8-

t_{1 / 2} \approx 42

9-

λ = {(1.666 \pm 0.010)}^{- 11}

10-

T > \sim 10^{8}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1103.1865

Formulas:

1-

≃ 40 k_{B} T

2-

≃ 60 k_{B} T

3-

a ≃ 60 n m

4-

\sim 100 k_{B} T

5-

⟨ S (q) ⟩

6-

\dot{γ} = 1700 s^{- 1}

7-

\dot{γ} = 1700 s^{- 1}

8-

ϕ_{e f f}

9-

R_{g} ≃ 35 \pm 3 μ m

10-

\dot{γ} = 1700 s^{- 1}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1802.09855

Formulas:

1-

\hat{H} (𝐫) ψ_{n} (𝐫) = E_{n} ψ_{n} (𝐫),

2-

\hat{H} (𝐫)

3-

ψ_{n} (𝐫)

4-

E_{n} = k_{n}^{2} .

5-

[- \frac{d^{2}}{d x^{2}} + V (x)] ψ_{n} (x) = k_{n}^{2} ψ_{n} (x),

6-

ψ_{n} (x) \propto e^{i k_{n} | x |} for | x | \to \infty,

7-

E_{n} = {(p_{n} + i ϰ_{n})}^{2} = (p_{n}^{2} - ϰ_{n}^{2}) + 2 i p_{n} ϰ_{n},

8-

k_{n} = p_{n} + i ϰ_{n}

9-

E_{n} = - ϰ_{n}^{2} < 0

10-

ψ_{n} (x) \propto e^{- ϰ_{n} | x |} \to 0

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1703.09292

Formulas:

1-

u, u^{e}, u^{e^{2}}, \dots

2-

N (e, q)

3-

N (e, q)

4-

N (e, q) = \sum_{d ∣ ρ} \frac{φ (d)}{ℓ_{e} (d)},

5-

ℓ_{a} (b)

6-

N (e, q)

7-

ℓ_{a} (b)

8-

N (e, p) \leq p^{o (1)}

9-

N (e, p)

10-

π (x; k, a)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/hep-lat/0409046

Formulas:

1-

I m (V_{t s}^{*} V_{t d}) [P^{(1 / 2)} - P^{(3 / 2)}]

2-

P^{(1 / 2)}

3-

r \sum_{i = 3}^{10} y_{i} (μ) {⟨ O_{i} ⟩}_{0} (μ) (1 - Ω_{η + η^{'}})

4-

P^{(3 / 2)}

5-

\frac{r}{ω} \sum_{i = 7}^{10} y_{i} (μ) {⟨ O_{i} ⟩}_{2} (μ)

6-

\frac{G_{F} ω}{2 | ϵ | R e A_{0}}

7-

Δ I = 1 / 2

8-

Ω_{η + η^{'}} = 0.060 (77)

9-

P^{(1 / 2)}

10-

P^{(3 / 2)}

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0910.1551

Formulas:

1-

𝒪 (α_{s})

2-

γ γ, Z Z, W^{+} W^{-}

3-

w a r p e d

4-

d s^{2} = e^{- 2 𝒦 R_{c} | ϕ |} η_{μ ν} d x^{μ} d x^{ν} + R_{c}^{2} d ϕ^{2}

5-

\frac{M_{P l}}{M_{E W}} \sim 10^{15}

6-

h_{μ ν}^{(n)}

7-

ℒ_{i n t}

8-

- \frac{1}{\bar{M_{P l}}} T^{μ ν} (x) h_{μ ν}^{(0)} (x) - \frac{e^{π 𝒦 R_{c}}}{\bar{M_{P l}}} \sum_{n = 1}^{\infty} T^{μ ν} (x) h_{μ ν}^{(n)} (x) .

9-

h_{μ ν}^{(n)}

10-

x_{n} 𝒦 e^{- π 𝒦 R_{c}},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0510340

Formulas:

1-

1.78 \pm 0.23 M_{⊙}

2-

0.60 \pm 0.13 M_{⊙}

3-

\sim 4 \times 10^{36} ergs s^{- 1}

4-

α_{J2000} = 21^{h} 44^{m} 40^{s}

5-

δ_{J2000} = 38 ° 19' 00 ″

6-

{5.2}_{- 1.1}^{+ 7.9} \times 10^{- 11}

7-

(1.1 \pm 0.3) \times 10^{- 8}

8-

(1.0 \pm 0.3) \times 10^{- 8}

9-

< 13 \times 10^{- 3}

10-

(1.0 \pm 0.2) 10^{36} erg s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1002.4626

Formulas:

1-

L \sim 0.1 L_{H α}^{*}

2-

P A = 0^{o}

3-

P A = - 60^{o}

4-

P A = - 60^{o}

5-

P A = 0^{o}

6-

P A = - 60^{o}

7-

P A = 0^{o}

8-

P A = - 60^{o}

9-

\log N = 20.65 \pm 0.05

10-

P A = 0^{o}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1902.06870

Formulas:

1-

\partial_{t} w + 𝐯 \cdot \nabla w

2-

- β \partial_{y} φ = 0,

3-

w ≐ \nabla^{2} φ

4-

𝐱 \equiv (x, y)

5-

β ≐ a / L_{n}

6-

L_{n} ≐ {(- d \ln n_{0} / d x)}^{- 1}

7-

ρ_{s} ≐ c_{s} / Ω_{ci}

8-

φ (t, 𝐱)

9-

T_{e} ρ_{s} / (e a)

10-

𝐯 ≐ (- \partial_{y} φ, \partial_{x} φ)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/hep-th/0006041

Formulas:

1-

M^{4} \times R^{1} \times ℋ_{21}

2-

| Δ \vec{y} |

3-

| Δ \vec{y} | \to 0

4-

| Δ \vec{y} |

5-

| Δ \vec{y} |

6-

A d S_{5} \times S^{5}

7-

g_{s} = e^{Φ} = g_{Y M}^{2}

8-

d s^{2} = e^{2 Φ} (\pm d t^{2} + d x_{i} d x_{i}) + 4 l_{c}^{2} e^{4 Φ} {(d Φ)}^{2} + d {\vec{y}}_{21}^{2},

9-

i = 1, 2, 3

10-

Λ_{Q C D}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/2004.04144

Formulas:

1-

I (t), S (t), R (t), M (t)

2-

I (t), S (t), R (t),

3-

𝜹 = (δ_{1}, \dots, δ_{n})

4-

\frac{d I (t)}{d t}

5-

= I (t) 𝒞 ℳ_{S (t)} \cdot \frac{1}{m} - α I (t)

6-

\frac{d S (t)}{d t}

7-

= - I (t) 𝒞 ℳ_{S (t)} \cdot \frac{1}{m}

8-

\frac{d R (t)}{d t}

9-

= (α 𝟏 - 𝜹) ⊙ I (t)

10-

\frac{d M (t)}{d t}

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: q-bio

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0108172

Formulas:

1-

S O (10)

2-

S O (10)

3-

S O (10)

4-

Y_{B} = \frac{n_{B} - n_{\bar{B}}}{s} = \frac{n_{B} - n_{\bar{B}}}{7 n_{γ}} = \frac{η}{7},

5-

n_{B, \bar{B}, γ}

6-

Y_{B} = 10^{- 11} - 10^{- 10}

7-

Y_{B} = \frac{a}{a - 1} Y_{L}, a = \frac{8 N_{f} + 4 N_{H}}{22 N_{f} + 13 N_{H}},

8-

Y_{B} ≃ - Y_{L} / 2

9-

Y_{L} = d \frac{ϵ_{1}}{g^{*}}

10-

ϵ_{1} = \frac{1}{8 π v^{2} {(M_{D}^{†} M_{D})}_{11}} \sum_{j = 2, 3} Im {[{(M_{D}^{†} M_{D})}_{j 1}]}^{2} f (\frac{M_{j}^{2}}{M_{1}^{2}}),

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1410.3796

Formulas:

1-

D S (L, R)

2-

D S (L, R)

3-

K_{1, n - 1}

4-

K_{1, n - 1}

5-

G \neq K_{1, n - 1}

6-

D S (L, R)

7-

G \neq K_{1, n - 1}

8-

G \neq K_{1, 3}

9-

V (G) ∖ V (H)

10-

| V (G) | - 5

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1712.08631

Formulas:

1-

Q_{int} \sim 1.7 \cdot 10^{6}

2-

Δ E_{dc} / E_{dc}

3-

Δ B_{dc} / B_{dc}

4-

R R R = 300

5-

L_{x} \times L_{y} \times L_{z} ≃ 25.6 mm \times 7 mm \times 14 mm

6-

δ x_{S} = 2 mm

7-

E_{mw} = | \sin (3 π x / L_{x}) \sin (π z / L_{z}) |

8-

V_{1} = - 1 V

9-

V_{2} = 1 V

10-

E_{dc} = 0.95 V / cm

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1902.03148

Formulas:

1-

\dot{x} (t) = v (t) p (t) + γ^{- 1} F + η (t),

2-

F = - \partial ϕ / \partial x

3-

⟨ η (t) η (t^{'}) ⟩ = 2 D_{t} δ (t - t^{'})

4-

β ϕ (x) = \frac{1}{2} ω_{0} x^{2} - α {| x |}^{3},

5-

β = 1 / k_{B} T

6-

ϕ^{e f f}

7-

v (t) = v_{0} e^{- {(t / t_{0})}^{n}},

8-

v (t) = v_{0}

9-

D = D_{t} + D_{a} = D_{t} + \frac{v_{0}^{2} τ}{2},

10-

D_{a} = v_{0}^{2} τ / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1310.8584

Formulas:

1-

S_{L M (k)} = \sum_{i = 1}^{𝐤} \frac{1}{λ_{i}} | λ_{i} ⟩ ⟨ λ_{i} |,

2-

γ_{5} D_{C I}

3-

S_{R D (k)} = S_{F u l l} - S_{L M (k)},

4-

M_{π} = 322 (5)

5-

a = 0.144 (1)

6-

C_{i j} (t) = ⟨ 0 | 𝒪_{i} (t) 𝒪_{j}^{†} (0) | 0 ⟩,

7-

C (t) {\vec{υ}}_{n} = {\tilde{λ}}^{(n)} (t) C (t_{0}) {\vec{υ}}_{n},

8-

{\tilde{λ}}^{(n)} (t, t_{0}) = e^{- E_{n} (t - t_{0})} (1 + 𝒪 (e^{- Δ E_{n} (t - t_{0})}))

9-

{\bar{q}}_{w} γ_{k} γ_{t} q_{w}

10-

{\bar{q}}_{n} γ_{k} γ_{5} q_{n}

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/2008.12046

Formulas:

1-

𝐦 \in ℝ^{3 \times 6704}

2-

𝐂 \in ℝ^{300 \times 6704}

3-

𝐥_{o p} \in ℝ^{2 \times 5}

4-

𝐋_{o p} \in ℝ^{3 \times 5}

5-

𝐥_{o p} = 𝐀 \cdot 𝐋_{o p} + 𝐭,

6-

𝐀 \in ℝ^{2 \times 3}

7-

𝐭 \in ℝ^{2 \times 5}

8-

\underset{𝐀}{\arg \min} {∥ 𝐥_{o p} - 𝐀 \cdot 𝐋_{o p} ∥}_{2}^{2},

9-

𝐀 = 𝐥_{o p} \cdot 𝐋_{o p}^{+}

10-

𝐭 = 𝐥_{o p} - 𝐀 \cdot 𝐋_{o p}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1607.00659

Formulas:

1-

𝐬 = {[x_{1}, y_{1}, \dots, x_{n}, y_{n}]}^{T}

2-

{𝐬, 𝐡^{1}, 𝐡^{2}}

3-

\begin{matrix} E_{D B M_{s}} (𝐬, 𝐡^{1}, 𝐡^{2}; θ_{s}) = & \frac{1}{2} \sum_{i} \frac{{(s_{i} - b_{s_{i}})}^{2}}{σ_{s i}^{2}} - \sum_{i, j} W_{i j}^{1} s_{i} h_{j}^{1} \\ - \sum_{j, l} W_{j l}^{2} h_{j}^{1} h_{l}^{2} \end{matrix}

4-

θ_{s} = {𝐖^{1}, 𝐖^{2}, σ_{s}, 𝐛_{s}}

5-

{𝐬, 𝐡^{1}, 𝐡^{2}}

6-

P (𝐬; θ_{s}) = \sum_{𝐡^{1}, 𝐡^{2}} \frac{\exp (- E_{D B M_{s}} (𝐬, 𝐡^{1}, 𝐡^{2}; θ_{s}))}{Z (θ_{s})}

7-

Z (θ_{s})

8-

{𝐚, \tilde{𝐚}, 𝐦, 𝐠_{m}, 𝐠_{a}^{1}, 𝐠_{a}^{2}}

9-

\begin{matrix} E_{R D B M_{g}} (𝐚, \tilde{𝐚}, 𝐦, 𝐠_{m}, 𝐠_{a}^{1}, 𝐠_{a}^{2}; θ_{a}) = \sum_{i} \frac{γ_{i}^{2} m_{i} {(a_{i} - {\tilde{a}}_{i})}^{2}}{2 σ_{g_{i}}^{2}} \\ + \sum_{i} \frac{{(a_{i} - b_{g_{i}})}^{2}}{2 σ_{g_{i}}^{2}} - \sum_{i, j} V_{i j}^{1} a_{i} g_{a j}^{1} - \sum_{j, l} V_{j l}^{2} g_{a j}^{1} g_{a l}^{2} \\ - \sum_{i, k} U_{i k} m_{i} g_{m k} + \sum_{i} \frac{{({\tilde{a}}_{i} - {\tilde{b}}_{g_{i}})}^{2}}{2 {\tilde{σ}}_{g_{i}}^{2}} \end{matrix}

10-

θ_{a} = {𝐕^{1}, 𝐕^{2}, 𝐔, σ_{g}, 𝐛_{g}, {\tilde{σ}}_{g}, {\tilde{𝐛}}_{g}}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1401.0926

Formulas:

1-

\begin{matrix} x (k + 1) = A x (k) \\ y (k) = H x (k) \end{matrix}

2-

\begin{matrix} where & y (k) = {(y_{1}^{T} (k), \dots, y_{m}^{T} (k))}^{T} with y_{i} (k) = H_{i} x (k), \\ x (k) \in ℝ^{n}, y_{i} (k) \in ℝ^{r_{i}} \end{matrix}

3-

𝒢 = (𝕍, 𝔼)

4-

{\hat{x}}_{i} (k)

5-

{lim}_{k \to \infty} || {\hat{x}}_{i} (k) - x (k) || = 0

6-

𝕍 \overset{d e f}{=} {1, \dots, m}

7-

𝒢 = (𝕍, 𝔼)

8-

(i, i) \notin \bar{𝔼}

9-

s p (M)

10-

s p (M) = {λ \in ℂ ∣ d e t (M - λ I) = 0}

Formulas (html):

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id="S1.Ex1.m1.59.59.5.57.18.18.18.2.1.2"><mi id="S1.Ex1.m1.47.47.47.9.9.9" xref="S1.Ex1.m1.47.47.47.9.9.9.cmml">y</mi><mi id="S1.Ex1.m1.48.48.48.10.10.10.1" xref="S1.Ex1.m1.48.48.48.10.10.10.1.cmml">i</mi></msub><mo id="S1.Ex1.m1.59.59.5.57.18.18.18.2.1.1" xref="S1.Ex1.m1.56.56.2.3.cmml">⁢</mo><mrow id="S1.Ex1.m1.59.59.5.57.18.18.18.2.1.3"><mo stretchy="false" id="S1.Ex1.m1.49.49.49.11.11.11" xref="S1.Ex1.m1.56.56.2.3.cmml">(</mo><mi id="S1.Ex1.m1.50.50.50.12.12.12" xref="S1.Ex1.m1.50.50.50.12.12.12.cmml">k</mi><mo stretchy="false" id="S1.Ex1.m1.51.51.51.13.13.13" xref="S1.Ex1.m1.56.56.2.3.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.52.52.52.14.14.14" xref="S1.Ex1.m1.52.52.52.14.14.14.cmml">∈</mo><msup id="S1.Ex1.m1.59.59.5.57.18.18.18.2.2"><mi id="S1.Ex1.m1.53.53.53.15.15.15" xref="S1.Ex1.m1.53.53.53.15.15.15.cmml">ℝ</mi><msub id="S1.Ex1.m1.54.54.54.16.16.16.1" xref="S1.Ex1.m1.54.54.54.16.16.16.1.cmml"><mi id="S1.Ex1.m1.54.54.54.16.16.16.1.2" xref="S1.Ex1.m1.54.54.54.16.16.16.1.2.cmml">r</mi><mi id="S1.Ex1.m1.54.54.54.16.16.16.1.3" xref="S1.Ex1.m1.54.54.54.16.16.16.1.3.cmml">i</mi></msub></msup></mrow></mrow></mtd></mtr></mtable></math>, <math><mrow id="S1.p2.1.m1.2.3" xref="S1.p2.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.1.m1.2.3.2" xref="S1.p2.1.m1.2.3.2.cmml">𝒢</mi><mo id="S1.p2.1.m1.2.3.1" xref="S1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S1.p2.1.m1.2.3.3.2" xref="S1.p2.1.m1.2.3.3.1.cmml"><mo id="S1.p2.1.m1.2.3.3.2.1" xref="S1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">𝕍</mi><mo id="S1.p2.1.m1.2.3.3.2.2" xref="S1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S1.p2.1.m1.2.2" xref="S1.p2.1.m1.2.2.cmml">𝔼</mi><mo id="S1.p2.1.m1.2.3.3.2.3" xref="S1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p3.1.m1.1.2" xref="S1.p3.1.m1.1.2.cmml"><msub id="S1.p3.1.m1.1.2.2" xref="S1.p3.1.m1.1.2.2.cmml"><mover accent="true" id="S1.p3.1.m1.1.2.2.2" xref="S1.p3.1.m1.1.2.2.2.cmml"><mi id="S1.p3.1.m1.1.2.2.2.2" xref="S1.p3.1.m1.1.2.2.2.2.cmml">x</mi><mo stretchy="false" id="S1.p3.1.m1.1.2.2.2.1" xref="S1.p3.1.m1.1.2.2.2.1.cmml">^</mo></mover><mi id="S1.p3.1.m1.1.2.2.3" xref="S1.p3.1.m1.1.2.2.3.cmml">i</mi></msub><mo id="S1.p3.1.m1.1.2.1" xref="S1.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.1.2.3.2" xref="S1.p3.1.m1.1.2.cmml"><mo stretchy="false" id="S1.p3.1.m1.1.2.3.2.1" xref="S1.p3.1.m1.1.2.cmml">(</mo><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">k</mi><mo stretchy="false" id="S1.p3.1.m1.1.2.3.2.2" xref="S1.p3.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p3.4.m4.3.3" xref="S1.p3.4.m4.3.3.cmml"><mrow id="S1.p3.4.m4.3.3.1" xref="S1.p3.4.m4.3.3.1.cmml"><msub id="S1.p3.4.m4.3.3.1.2" xref="S1.p3.4.m4.3.3.1.2.cmml"><mo id="S1.p3.4.m4.3.3.1.2.2" xref="S1.p3.4.m4.3.3.1.2.2.cmml">lim</mo><mrow id="S1.p3.4.m4.3.3.1.2.3" xref="S1.p3.4.m4.3.3.1.2.3.cmml"><mi id="S1.p3.4.m4.3.3.1.2.3.2" xref="S1.p3.4.m4.3.3.1.2.3.2.cmml">k</mi><mo id="S1.p3.4.m4.3.3.1.2.3.1" xref="S1.p3.4.m4.3.3.1.2.3.1.cmml">→</mo><mi mathvariant="normal" id="S1.p3.4.m4.3.3.1.2.3.3" xref="S1.p3.4.m4.3.3.1.2.3.3.cmml">∞</mi></mrow></msub><mo id="S1.p3.4.m4.3.3.1a" xref="S1.p3.4.m4.3.3.1.cmml">⁡</mo><mrow id="S1.p3.4.m4.3.3.1.1.1" xref="S1.p3.4.m4.3.3.1.1.2.cmml"><mo fence="true" id="S1.p3.4.m4.3.3.1.1.1.2" xref="S1.p3.4.m4.3.3.1.1.2.1.cmml">||</mo><mrow id="S1.p3.4.m4.3.3.1.1.1.1" xref="S1.p3.4.m4.3.3.1.1.1.1.cmml"><mrow id="S1.p3.4.m4.3.3.1.1.1.1.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.cmml"><msub id="S1.p3.4.m4.3.3.1.1.1.1.2.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.2.cmml"><mover accent="true" id="S1.p3.4.m4.3.3.1.1.1.1.2.2.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.2.2.cmml"><mi id="S1.p3.4.m4.3.3.1.1.1.1.2.2.2.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.2.2.2.cmml">x</mi><mo stretchy="false" id="S1.p3.4.m4.3.3.1.1.1.1.2.2.2.1" xref="S1.p3.4.m4.3.3.1.1.1.1.2.2.2.1.cmml">^</mo></mover><mi id="S1.p3.4.m4.3.3.1.1.1.1.2.2.3" xref="S1.p3.4.m4.3.3.1.1.1.1.2.2.3.cmml">i</mi></msub><mo id="S1.p3.4.m4.3.3.1.1.1.1.2.1" xref="S1.p3.4.m4.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S1.p3.4.m4.3.3.1.1.1.1.2.3.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.cmml"><mo stretchy="false" id="S1.p3.4.m4.3.3.1.1.1.1.2.3.2.1" xref="S1.p3.4.m4.3.3.1.1.1.1.2.cmml">(</mo><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">k</mi><mo stretchy="false" id="S1.p3.4.m4.3.3.1.1.1.1.2.3.2.2" xref="S1.p3.4.m4.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.4.m4.3.3.1.1.1.1.1" xref="S1.p3.4.m4.3.3.1.1.1.1.1.cmml">-</mo><mrow id="S1.p3.4.m4.3.3.1.1.1.1.3" xref="S1.p3.4.m4.3.3.1.1.1.1.3.cmml"><mi id="S1.p3.4.m4.3.3.1.1.1.1.3.2" xref="S1.p3.4.m4.3.3.1.1.1.1.3.2.cmml">x</mi><mo id="S1.p3.4.m4.3.3.1.1.1.1.3.1" xref="S1.p3.4.m4.3.3.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S1.p3.4.m4.3.3.1.1.1.1.3.3.2" xref="S1.p3.4.m4.3.3.1.1.1.1.3.cmml"><mo stretchy="false" id="S1.p3.4.m4.3.3.1.1.1.1.3.3.2.1" xref="S1.p3.4.m4.3.3.1.1.1.1.3.cmml">(</mo><mi id="S1.p3.4.m4.2.2" xref="S1.p3.4.m4.2.2.cmml">k</mi><mo stretchy="false" id="S1.p3.4.m4.3.3.1.1.1.1.3.3.2.2" xref="S1.p3.4.m4.3.3.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S1.p3.4.m4.3.3.1.1.1.3" xref="S1.p3.4.m4.3.3.1.1.2.1.cmml">||</mo></mrow></mrow><mo id="S1.p3.4.m4.3.3.2" xref="S1.p3.4.m4.3.3.2.cmml">=</mo><mn id="S1.p3.4.m4.3.3.3" xref="S1.p3.4.m4.3.3.3.cmml">0</mn></mrow></math>, <math><mrow id="S2.SS1.p2.3.3.2.m1.3.4" xref="S2.SS1.p2.3.3.2.m1.3.4.cmml"><mi id="S2.SS1.p2.3.3.2.m1.3.4.2" xref="S2.SS1.p2.3.3.2.m1.3.4.2.cmml">𝕍</mi><mo id="S2.SS1.p2.3.3.2.m1.3.4.1" xref="S2.SS1.p2.3.3.2.m1.3.4.1.cmml">⁢</mo><mover accent="true" id="S2.SS1.p2.3.3.2.m1.3.4.3" xref="S2.SS1.p2.3.3.2.m1.3.4.3.cmml"><mo id="S2.SS1.p2.3.3.2.m1.3.4.3.2" xref="S2.SS1.p2.3.3.2.m1.3.4.3.2.cmml">=</mo><mrow id="S2.SS1.p2.3.3.2.m1.3.4.3.1" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.cmml"><mi id="S2.SS1.p2.3.3.2.m1.3.4.3.1.2" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.2.cmml">d</mi><mo id="S2.SS1.p2.3.3.2.m1.3.4.3.1.1" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.1.cmml">⁢</mo><mi id="S2.SS1.p2.3.3.2.m1.3.4.3.1.3" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.3.cmml">e</mi><mo id="S2.SS1.p2.3.3.2.m1.3.4.3.1.1a" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.1.cmml">⁢</mo><mi id="S2.SS1.p2.3.3.2.m1.3.4.3.1.4" xref="S2.SS1.p2.3.3.2.m1.3.4.3.1.4.cmml">f</mi></mrow></mover><mo id="S2.SS1.p2.3.3.2.m1.3.4.1a" xref="S2.SS1.p2.3.3.2.m1.3.4.1.cmml">⁢</mo><mrow id="S2.SS1.p2.3.3.2.m1.3.4.4.2" xref="S2.SS1.p2.3.3.2.m1.3.4.4.1.cmml"><mo stretchy="false" id="S2.SS1.p2.3.3.2.m1.3.4.4.2.1" xref="S2.SS1.p2.3.3.2.m1.3.4.4.1.cmml">{</mo><mn id="S2.SS1.p2.3.3.2.m1.1.1" xref="S2.SS1.p2.3.3.2.m1.1.1.cmml">1</mn><mo id="S2.SS1.p2.3.3.2.m1.3.4.4.2.2" xref="S2.SS1.p2.3.3.2.m1.3.4.4.1.cmml">,</mo><mi mathvariant="normal" id="S2.SS1.p2.3.3.2.m1.2.2" xref="S2.SS1.p2.3.3.2.m1.2.2.cmml">⋯</mi><mo id="S2.SS1.p2.3.3.2.m1.3.4.4.2.3" xref="S2.SS1.p2.3.3.2.m1.3.4.4.1.cmml">,</mo><mi id="S2.SS1.p2.3.3.2.m1.3.3" xref="S2.SS1.p2.3.3.2.m1.3.3.cmml">m</mi><mo stretchy="false" id="S2.SS1.p2.3.3.2.m1.3.4.4.2.4" xref="S2.SS1.p2.3.3.2.m1.3.4.4.1.cmml">}</mo></mrow></mrow></math>, <math><mrow id="S2.SS1.p2.5.5.1.m1.2.3" xref="S2.SS1.p2.5.5.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.5.5.1.m1.2.3.2" xref="S2.SS1.p2.5.5.1.m1.2.3.2.cmml">𝒢</mi><mo id="S2.SS1.p2.5.5.1.m1.2.3.1" xref="S2.SS1.p2.5.5.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS1.p2.5.5.1.m1.2.3.3.2" xref="S2.SS1.p2.5.5.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.p2.5.5.1.m1.2.3.3.2.1" xref="S2.SS1.p2.5.5.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS1.p2.5.5.1.m1.1.1" xref="S2.SS1.p2.5.5.1.m1.1.1.cmml">𝕍</mi><mo id="S2.SS1.p2.5.5.1.m1.2.3.3.2.2" xref="S2.SS1.p2.5.5.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.5.5.1.m1.2.2" xref="S2.SS1.p2.5.5.1.m1.2.2.cmml">𝔼</mi><mo id="S2.SS1.p2.5.5.1.m1.2.3.3.2.3" xref="S2.SS1.p2.5.5.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="footnote5.m2.2.3" xref="footnote5.m2.2.3.cmml"><mrow id="footnote5.m2.2.3.2.2" xref="footnote5.m2.2.3.2.1.cmml"><mo stretchy="false" id="footnote5.m2.2.3.2.2.1" xref="footnote5.m2.2.3.2.1.cmml">(</mo><mi id="footnote5.m2.1.1" xref="footnote5.m2.1.1.cmml">i</mi><mo id="footnote5.m2.2.3.2.2.2" xref="footnote5.m2.2.3.2.1.cmml">,</mo><mi id="footnote5.m2.2.2" xref="footnote5.m2.2.2.cmml">i</mi><mo stretchy="false" id="footnote5.m2.2.3.2.2.3" xref="footnote5.m2.2.3.2.1.cmml">)</mo></mrow><mo id="footnote5.m2.2.3.1" xref="footnote5.m2.2.3.1.cmml">∉</mo><mover accent="true" id="footnote5.m2.2.3.3" xref="footnote5.m2.2.3.3.cmml"><mi id="footnote5.m2.2.3.3.2" xref="footnote5.m2.2.3.3.2.cmml">𝔼</mi><mo id="footnote5.m2.2.3.3.1" xref="footnote5.m2.2.3.3.1.cmml">¯</mo></mover></mrow></math>, <math><mrow id="S2.SS1.p2.11.11.1.m1.1.2" xref="S2.SS1.p2.11.11.1.m1.1.2.cmml"><mi id="S2.SS1.p2.11.11.1.m1.1.2.2" xref="S2.SS1.p2.11.11.1.m1.1.2.2.cmml">s</mi><mo id="S2.SS1.p2.11.11.1.m1.1.2.1" xref="S2.SS1.p2.11.11.1.m1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.p2.11.11.1.m1.1.2.3" xref="S2.SS1.p2.11.11.1.m1.1.2.3.cmml">p</mi><mo id="S2.SS1.p2.11.11.1.m1.1.2.1a" xref="S2.SS1.p2.11.11.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.11.11.1.m1.1.2.4.2" xref="S2.SS1.p2.11.11.1.m1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p2.11.11.1.m1.1.2.4.2.1" xref="S2.SS1.p2.11.11.1.m1.1.2.cmml">(</mo><mi id="S2.SS1.p2.11.11.1.m1.1.1" xref="S2.SS1.p2.11.11.1.m1.1.1.cmml">M</mi><mo stretchy="false" id="S2.SS1.p2.11.11.1.m1.1.2.4.2.2" xref="S2.SS1.p2.11.11.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.SS1.p2.13.13.3.m2.3.3" xref="S2.SS1.p2.13.13.3.m2.3.3.cmml"><mrow id="S2.SS1.p2.13.13.3.m2.3.3.4" xref="S2.SS1.p2.13.13.3.m2.3.3.4.cmml"><mi id="S2.SS1.p2.13.13.3.m2.3.3.4.2" xref="S2.SS1.p2.13.13.3.m2.3.3.4.2.cmml">s</mi><mo id="S2.SS1.p2.13.13.3.m2.3.3.4.1" xref="S2.SS1.p2.13.13.3.m2.3.3.4.1.cmml">⁢</mo><mi id="S2.SS1.p2.13.13.3.m2.3.3.4.3" xref="S2.SS1.p2.13.13.3.m2.3.3.4.3.cmml">p</mi><mo id="S2.SS1.p2.13.13.3.m2.3.3.4.1a" xref="S2.SS1.p2.13.13.3.m2.3.3.4.1.cmml">⁢</mo><mrow id="S2.SS1.p2.13.13.3.m2.3.3.4.4.2" xref="S2.SS1.p2.13.13.3.m2.3.3.4.cmml"><mo stretchy="false" id="S2.SS1.p2.13.13.3.m2.3.3.4.4.2.1" xref="S2.SS1.p2.13.13.3.m2.3.3.4.cmml">(</mo><mi id="S2.SS1.p2.13.13.3.m2.1.1" xref="S2.SS1.p2.13.13.3.m2.1.1.cmml">M</mi><mo stretchy="false" id="S2.SS1.p2.13.13.3.m2.3.3.4.4.2.2" xref="S2.SS1.p2.13.13.3.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.13.13.3.m2.3.3.3" xref="S2.SS1.p2.13.13.3.m2.3.3.3.cmml">=</mo><mrow id="S2.SS1.p2.13.13.3.m2.3.3.2.2" xref="S2.SS1.p2.13.13.3.m2.3.3.2.3.cmml"><mo stretchy="false" id="S2.SS1.p2.13.13.3.m2.3.3.2.2.3" xref="S2.SS1.p2.13.13.3.m2.3.3.2.3.1.cmml">{</mo><mrow id="S2.SS1.p2.13.13.3.m2.2.2.1.1.1" xref="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.cmml"><mi id="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.2" xref="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.2.cmml">λ</mi><mo id="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.1" xref="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.1.cmml">∈</mo><mi id="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.3" xref="S2.SS1.p2.13.13.3.m2.2.2.1.1.1.3.cmml">ℂ</mi></mrow><mo 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id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.2" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.cmml"><mi id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.2" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.2.cmml">M</mi><mo id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.1" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.1.cmml">-</mo><mrow id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.cmml"><mi id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.2" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.2.cmml">λ</mi><mo id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.1" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.3" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.3.3.cmml">I</mi></mrow></mrow><mo stretchy="false" id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.3" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.2" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.2.cmml">=</mo><mn id="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.3" xref="S2.SS1.p2.13.13.3.m2.3.3.2.2.2.3.cmml">0</mn></mrow><mo stretchy="false" id="S2.SS1.p2.13.13.3.m2.3.3.2.2.5" xref="S2.SS1.p2.13.13.3.m2.3.3.2.3.1.cmml">}</mo></mrow></mrow></math>

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1102.0762

Formulas:

1-

H = H_{C} + H_{Q C},

2-

= \sum_{i = 1}^{N} J_{i} 𝐬_{i} \cdot 𝐬_{i + 1} .

3-

J_{i} = J_{0} > 0

4-

i = 1, \dots, N - 1

5-

𝐬_{N + 1} \equiv 𝐬_{1}

6-

H_{Q C} = J_{A, i} 𝐒_{A} \cdot 𝐬_{i} + J_{B, j} 𝐒_{B} \cdot 𝐬_{j} .

7-

0 < λ_{α} \equiv J_{α, i} / J_{0} ≪ 1

8-

J_{0} = ℏ = 1

9-

| Ψ (0) ⟩ = (a | 0_{A} ⟩ + b | 1_{A} ⟩) \otimes | 0_{C} ⟩ \otimes | 0_{B} ⟩

10-

a = \cos \frac{θ}{2}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1811.01797

Formulas:

1-

ℒ_{S Y M} = \frac{1}{4} F_{μ ν} F^{μ ν} + \frac{1}{2} \bar{λ} (D̸ + m_{g}) λ .

2-

N_{c} \oplus {\bar{N}}_{c}

3-

ℒ_{S Q C D} =

4-

ℒ_{S Y M} + {| D_{μ} Φ_{1} |}^{2} + {| D_{μ} Φ_{2}^{†} |}^{2} + \bar{ψ} (γ_{μ} D_{f}^{μ} + m) ψ + m^{2} {| Φ_{1} |}^{2} + m^{2} {| Φ_{2} |}^{2}

5-

+ i \sqrt{2} g {\bar{λ}}^{a} (Φ_{1}^{†} P_{+} + Φ_{2} P_{-}) T^{a} ψ - i \sqrt{2} g \bar{ψ} T^{a} (P_{-} Φ_{1} + P_{+} Φ_{2}^{†}) λ^{a}

6-

+ \frac{g^{2}}{2} {(Φ_{1}^{†} T^{a} Φ_{1} - Φ_{2} T^{a} Φ_{2}^{†})}^{2} .

7-

SU (N_{c})

8-

\frac{3}{2} N_{c} < N_{f} < 3 N_{c}

9-

SU (N_{c})

10-

SU (N_{c} - N_{f})

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/0006075

Formulas:

1-

m^{I} = \int d^{3} x \frac{ρ}{\sqrt{1 - v^{2} / c^{2}}} ≃ m + \frac{m}{5} {(\frac{Ω R}{c})}^{2},

2-

m^{I} = m [1 + η^{s p} \frac{1}{5} {(\frac{Ω R}{c})}^{2}] .

3-

M^{p} = m [1 + η^{e p} η^{s p} \frac{1}{5} {(\frac{Ω R}{c})}^{2}],

4-

a = g \frac{M^{p}}{m^{I}} ≃ g [1 + (η^{e p} - 1) η^{s p} \frac{1}{5} {(\frac{Ω R}{c})}^{2}] .

5-

\frac{Δ a}{a} ≃ \frac{(η^{e p} - 1) η^{s p}}{5} {(\frac{2 π R n}{c})}^{2} .

6-

n = 4 \times 10^{4} rpm, R = 10 c m

7-

\frac{Δ a}{a} = (η^{e p} - 1) η^{s p} \times 4 \times 10^{- 13} .

8-

(η^{e p} - 1) η^{s p} \neq 0

9-

m^{s} = E^{s} / c^{2} .

10-

M^{s} = η^{s} E^{s} / c^{2},

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/math/0003136

Formulas:

1-

G L_{2} (𝐅_{p})

2-

𝐙_{p} [[T_{1}, T_{2}, T_{3}]]

3-

27 + 4 a^{3}

4-

x^{3} + a x + 1

5-

27 + 4 a^{3}

6-

27 + 4 a^{3} < 10^{15} / 2

7-

x^{3} + 7 x - 12

8-

{(𝔭_{1} 𝔭_{2} 𝔭_{3})}^{2}

9-

𝔭_{1}, 𝔭_{2}, 𝔭_{3}

10-

x^{3} + a x + 1 = 0

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1002.0190

Formulas:

1-

{n_{1}, n_{2}, n_{3}, n_{4}}

2-

{n_{1}, \dots, n_{k}}

3-

{n_{1}, \dots, n_{k}}

4-

[k] := {1, 2, \dots, k}

5-

{n_{1}, \dots, n_{k}}

6-

K (n_{1}, \dots, n_{k})

7-

{n_{1}, \dots, n_{k}}

8-

{n_{1}, \dots, n_{k}}

9-

{3, 3, 3, 3}

10-

{1, 1, 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1406.3991

Formulas:

1-

f : ℝ^{n} \to ℝ

2-

{\underline{κ}}_{i} {< \frac{\partial f}{\partial x_{i}} |}_{x} < {\bar{κ}}_{i}, i = 1, \dots, n

3-

{\underline{M}}_{i j} {< \frac{\partial^{2} f}{\partial x_{j} \partial x_{i}} |}_{x} < {\bar{M}}_{i j}, i, j = 1, \dots, n

4-

x_{a}, x_{b} \in 𝒳

5-

f (x_{b}) - f (x_{a}) \geq \sum_{i = 1}^{n} min [\begin{matrix} {\underline{κ}}_{i} (x_{b, i} - x_{a, i}), \\ {\bar{κ}}_{i} (x_{b, i} - x_{a, i}) \end{matrix}],

6-

f (x_{b}) - f (x_{a}) \leq \sum_{i = 1}^{n} max [\begin{matrix} {\underline{κ}}_{i} (x_{b, i} - x_{a, i}), \\ {\bar{κ}}_{i} (x_{b, i} - x_{a, i}) \end{matrix}],

7-

\begin{matrix} f (x_{b}) - f (x_{a}) \geq \nabla f {(x_{a})}^{T} (x_{b} - x_{a}) + \\ \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} min [\begin{matrix} {\underline{M}}_{i j} (x_{b, i} - x_{a, i}) (x_{b, j} - x_{a, j}), \\ {\bar{M}}_{i j} (x_{b, i} - x_{a, i}) (x_{b, j} - x_{a, j}) \end{matrix}], \end{matrix}

8-

\begin{matrix} f (x_{b}) - f (x_{a}) \leq \nabla f {(x_{a})}^{T} (x_{b} - x_{a}) + \\ \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} max [\begin{matrix} {\underline{M}}_{i j} (x_{b, i} - x_{a, i}) (x_{b, j} - x_{a, j}), \\ {\bar{M}}_{i j} (x_{b, i} - x_{a, i}) (x_{b, j} - x_{a, j}) \end{matrix}], \end{matrix}

9-

\hat{f} (γ) = f (x (γ)),

10-

x (γ) = x_{a} + γ (x_{b} - x_{a}), γ \in [0, 1]

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1602.02657

Formulas:

1-

| 1, 1 ⟩ - | 0, 2 ⟩

2-

| 1, 2 ⟩ - | 0, 3 ⟩

3-

| 1, 0 ⟩ - | 0, 1 ⟩

4-

\hat{H} = - J \sum_{⟨ i, j ⟩} {\hat{a}}_{i}^{†} {\hat{a}}_{j} + \sum_{i} \frac{U (t)}{2} {\hat{n}}_{i} ({\hat{n}}_{i} - 1) .

5-

{\hat{n}}_{i} = {\hat{a}}_{i}^{†} {\hat{a}}_{i}

6-

U (t) = U + δ U \sin (2 π f_{\mod} t)

7-

h f_{\mod} ≫ U, J

8-

{\hat{H}}_{eff} = - J \sum_{⟨ i, j ⟩} {\hat{a}}_{i}^{†} J_{0} (K ({\hat{n}}_{i} - {\hat{n}}_{j})) {\hat{a}}_{j} + \sum_{i} \frac{U}{2} {\hat{n}}_{i} ({\hat{n}}_{i} - 1),

9-

J_{eff} = J \times J_{0} (K Δ n)

10-

K = δ U / (h f_{\mod})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0809.0255

Formulas:

1-

_{5 G H z}

2-

_{5 G H z}

3-

_{1.4 G H z}

4-

_{5 G H z}

5-

_{5 G H z}

6-

_{5 G H z}

7-

{}_{H}^{g a l}

8-

_{20 - 100 k e V}

9-

_{20 - 100 k e V}

10-

_{c o r e}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0901.1638

Formulas:

1-

\frac{\partial Σ_{g}}{\partial t}

2-

- (𝒖 \cdot \nabla) Σ_{g} - Σ_{g} \nabla \cdot 𝒖 + f_{D} (Σ_{g})

3-

\frac{\partial 𝒖}{\partial t}

4-

- (𝒖 \cdot \nabla) 𝒖 - \frac{1}{Σ_{g}} \nabla P - \nabla Φ - \frac{Σ_{p}}{Σ_{g}} 𝒇_{d}

5-

+ 2 Σ_{g}^{- 1} \nabla \cdot (ν Σ_{g} 𝐒) + f_{ν} (𝒖, Σ_{g})

6-

\frac{d 𝒙_{p}}{d t}

7-

\frac{d 𝒗_{p}}{d t}

8-

- \nabla Φ + 𝒇_{d}

9-

Φ_{sg} - \frac{G M_{⊙}}{r}

10-

4 π G (Σ_{g} + Σ_{p}) δ (z)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1905.04089

Formulas:

1-

{< K >}_{O} = {(- A^{3})}^{- w r (K)} Σ_{s \in S (K)} A^{σ (s)} {(- A^{2} - A^{- 2})}^{p (s)} O^{q (s)},

2-

w r (K)

3-

A O + A^{- 1}

4-

+ O 1 + U 2 U 1 O 2

5-

J (K) = \sum_{c \in O d d (K)} s i g n (c)

6-

O d d (K)

7-

- A^{2} - A^{- 2}

8-

{< K >}_{P} = \sum_{S} A^{n (S)} {(- A^{2} - A^{- 2})}^{l (S)} G (S)

9-

P_{K} = {(- A^{3})}^{- w r (K)} {< K >}_{P}

10-

< (Σ, K) > = \sum_{(Σ, s)} A^{σ (s)} d^{| s (c) |} [s (c)],

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Run 16446900 (Agent672)