Run 16446907

Paper: https://arxiv.org/abs/astro-ph/0604434

Formulas:

1-

\overset{<}{\sim} 1.5 M_{⊙}

2-

(q = M_{2} / M_{1} < 0.02)

3-

\overset{<}{\sim} 10^{- 10} - 10^{- 9}

4-

M_{⊙} {yr}^{- 1}

5-

Z_{conv} > 0.065 R_{⊙}

6-

\frac{d J}{d t} = - {0.510}^{- 28} f^{- 2} k^{2} M_{d} R_{d}^{4} ω^{3} dyn cm,

7-

M_{2}^{m i n} ≃ 0.03

8-

M_{2}^{m i n} ≃ 0.06

9-

M_{2}^{m i n}

10-

M_{2}^{m i n}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1404.3440

Formulas:

1-

I_{1} (x) = \frac{I_{0}}{N^{2}} {[\frac{\sin (π x δ / λ)}{π x δ / λ} \frac{\sin (N π x d / λ)}{\sin (π x d / λ)}]}^{2}

2-

x d / λ = m

3-

Δ x_{1} = \frac{λ}{d} .

4-

ϵ = d - δ ≪ d

5-

{(ϵ / d)}^{2}

6-

\sin^{2} (π x δ / λ) = \sin^{2} (π m - π x ϵ / λ) \approx {(π x ϵ / λ)}^{2}

7-

{(ϵ / d)}^{2}

8-

I (x, y) = \frac{I_{1} (x) I_{1} (y)}{I_{0}} .

9-

{(ϵ / d)}^{4}

10-

Δ x_{2} \approx \frac{1}{N d} = \frac{1}{D}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/physics/0405031

Formulas:

1-

μ (ω; {| H |}^{2}) = 1 + \frac{F ω^{2}}{ω_{0 N L}^{2} ({| H |}^{2}) - ω^{2} + i Γ ω},

2-

ω_{0 N L}

3-

ϵ_{D} ({| E |}^{2}) = ϵ_{D 0} + α {| E |}^{2} / E_{c}^{2},

4-

{| H |}^{2} = α A^{2} \frac{(1 - X^{2}) [{(X^{2} - Ω^{2})}^{2} + Ω^{2} γ^{2}]}{X^{6}},

5-

X = ω_{0 N L} / ω_{0}

6-

A^{2} = 16 ϵ_{D 0}^{3} ω_{0}^{2} h^{2} / c^{2}

7-

Ω = ω / ω_{0}

8-

ω_{0} = (c / a) {[d_{g} / π h ϵ_{D 0}]}^{1 / 2}

9-

Γ^{'} = Γ / ω_{0}

10-

(H, d H / d x)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1412.1382

Formulas:

1-

\bar{Z} = n_{e} / n_{t o t}

2-

n_{t o t} = n_{i} + n_{n}

3-

Δ Φ = C_{e} (\bar{Z} - E r r) \int n_{t o t} 𝑑 ℓ

4-

C_{e} = λ e^{2} / 4 π ϵ_{0} m_{e} c^{2} = 1.58 \times 10^{- 17}

5-

E r r = 0.08

6-

n_{t o t}

7-

\cos (30^{\circ})

8-

v = 2 c_{s} / (γ - 1)

9-

n_{e} = 2 \times 10^{14}

10-

Δ Φ = {7.3}^{\circ}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0906.0719

Formulas:

1-

1.5 \cdot 10^{4} K \leq T_{eff} \leq 3 \cdot 10^{4} K

2-

1.2 \cdot 10^{6} L_{⊙} \leq L \leq 1.9 \cdot 10^{6} L_{⊙}

3-

35.7 M_{⊙} \leq M \leq 49.1 M_{⊙}

4-

70 M_{⊙} \leq M_{ZAMS} \leq 90 M_{⊙}

5-

Δ M_{bol} \leq 0.2

6-

Γ_{1} = 4 / 3

7-

6 day \leq Π \leq 31 day

8-

L \sim 10^{6} L_{⊙}

9-

1.5 \cdot 10^{4} K \leq T_{eff} \leq 3 \cdot 10^{4} K

10-

1.2 \cdot 10^{6} L_{⊙} \leq L \leq 1.9 \cdot 10^{6} L_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0310260

Formulas:

1-

F_{2} (x, Q^{2})

2-

F_{2} (x, Q^{2})

3-

0.45 < Q^{2} \leq 120 G e V^{2}

4-

6.2 \times 10^{- 7} < x \leq 0.01

5-

F_{2} (x, Q^{2})

6-

F_{2} (x, Q^{2}) = \frac{e^{D_{0}} (\frac{Q_{0}^{2}}{Λ^{2}}) x^{- D_{2} + 1}}{1 + D_{3} + D_{1} \log (\frac{1}{x})} [x^{- D_{1} \log (1 + \frac{Q^{2}}{Q_{0}^{2}})} {(1 + \frac{Q^{2}}{Q_{0}^{2}})}^{D_{3} + 1} - 1]

7-

\log (\frac{1}{x})

8-

\log (1 + \frac{Q^{2}}{Q_{0}^{2}}) . \log (\frac{1}{x})

9-

\log (1 + \frac{Q^{2}}{Q_{0}^{2}})

10-

1 + \frac{Q^{2}}{Q_{0}^{2}}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1106.4437

Formulas:

1-

E_{Tamm} = \frac{E_{DBR}}{(1 + η E_{DBR} / E_{plasma})}

2-

η = 2 \frac{| n_{1} - n_{2} |}{π \sqrt{ϵ_{B}}}

3-

E_{Tamm} = 1.359 e V

4-

n_{gold} = 0.03 + 5.72 i

5-

τ_{X_{r e f}} = 1.3 \pm 0.2

6-

τ_{X_{2 D T P}} = 2.5

7-

_{2 D T P}

8-

700 \pm 100 p s

9-

τ_{X_{Q D 1}} = 400

10-

τ_{X_{Q D 2}} = 52

Formulas (html):

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id="p4.4.m4.1.1.3" xref="p4.4.m4.1.1.3.cmml"><mi id="p4.4.m4.1.1.3.2" xref="p4.4.m4.1.1.3.2.cmml">π</mi><mo id="p4.4.m4.1.1.3.1" xref="p4.4.m4.1.1.3.1.cmml">⁢</mo><msqrt id="p4.4.m4.1.1.3.3" xref="p4.4.m4.1.1.3.3.cmml"><msub id="p4.4.m4.1.1.3.3.2" xref="p4.4.m4.1.1.3.3.2.cmml"><mi id="p4.4.m4.1.1.3.3.2.2" xref="p4.4.m4.1.1.3.3.2.2.cmml">ϵ</mi><mi id="p4.4.m4.1.1.3.3.2.3" xref="p4.4.m4.1.1.3.3.2.3.cmml">B</mi></msub></msqrt></mrow></mfrac></mrow></mrow></math>, <math><mrow id="p4.9.m9.1.1" xref="p4.9.m9.1.1.cmml"><msub id="p4.9.m9.1.1.2" xref="p4.9.m9.1.1.2.cmml"><mi id="p4.9.m9.1.1.2.2" xref="p4.9.m9.1.1.2.2.cmml">E</mi><mtext id="p4.9.m9.1.1.2.3" xref="p4.9.m9.1.1.2.3a.cmml">Tamm</mtext></msub><mo id="p4.9.m9.1.1.1" xref="p4.9.m9.1.1.1.cmml">=</mo><mrow id="p4.9.m9.1.1.3" xref="p4.9.m9.1.1.3.cmml"><mpadded width="+5pt" id="p4.9.m9.1.1.3.2" xref="p4.9.m9.1.1.3.2.cmml"><mn id="p4.9.m9.1.1.3.2a" xref="p4.9.m9.1.1.3.2.cmml">1.359</mn></mpadded><mo id="p4.9.m9.1.1.3.1" 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xref="p11.3.m3.1.1.3.3.cmml">0.2</mn></mrow></mrow></math>, <math><mrow id="p11.4.m4.1.1" xref="p11.4.m4.1.1.cmml"><msub id="p11.4.m4.1.1.2" xref="p11.4.m4.1.1.2.cmml"><mi id="p11.4.m4.1.1.2.2" xref="p11.4.m4.1.1.2.2.cmml">τ</mi><msub id="p11.4.m4.1.1.2.3" xref="p11.4.m4.1.1.2.3.cmml"><mi id="p11.4.m4.1.1.2.3.2" xref="p11.4.m4.1.1.2.3.2.cmml">X</mi><mrow id="p11.4.m4.1.1.2.3.3" xref="p11.4.m4.1.1.2.3.3.cmml"><mn id="p11.4.m4.1.1.2.3.3.2" xref="p11.4.m4.1.1.2.3.3.2.cmml">2</mn><mo id="p11.4.m4.1.1.2.3.3.1" xref="p11.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="p11.4.m4.1.1.2.3.3.3" xref="p11.4.m4.1.1.2.3.3.3.cmml">D</mi><mo id="p11.4.m4.1.1.2.3.3.1a" xref="p11.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="p11.4.m4.1.1.2.3.3.4" xref="p11.4.m4.1.1.2.3.3.4.cmml">T</mi><mo id="p11.4.m4.1.1.2.3.3.1b" xref="p11.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="p11.4.m4.1.1.2.3.3.5" xref="p11.4.m4.1.1.2.3.3.5.cmml">P</mi></mrow></msub></msub><mo id="p11.4.m4.1.1.1" xref="p11.4.m4.1.1.1.cmml">=</mo><mn id="p11.4.m4.1.1.3" xref="p11.4.m4.1.1.3.cmml">2.5</mn></mrow></math>, <math><msub id="S0.F3.9.m4.1.1" xref="S0.F3.9.m4.1.1.cmml"><mi id="S0.F3.9.m4.1.1b" xref="S0.F3.9.m4.1.1.cmml"/><mrow id="S0.F3.9.m4.1.1.1" xref="S0.F3.9.m4.1.1.1.cmml"><mn id="S0.F3.9.m4.1.1.1.2" xref="S0.F3.9.m4.1.1.1.2.cmml">2</mn><mo id="S0.F3.9.m4.1.1.1.1" xref="S0.F3.9.m4.1.1.1.1.cmml">⁢</mo><mi id="S0.F3.9.m4.1.1.1.3" xref="S0.F3.9.m4.1.1.1.3.cmml">D</mi><mo id="S0.F3.9.m4.1.1.1.1b" xref="S0.F3.9.m4.1.1.1.1.cmml">⁢</mo><mi id="S0.F3.9.m4.1.1.1.4" xref="S0.F3.9.m4.1.1.1.4.cmml">T</mi><mo id="S0.F3.9.m4.1.1.1.1c" xref="S0.F3.9.m4.1.1.1.1.cmml">⁢</mo><mi id="S0.F3.9.m4.1.1.1.5" xref="S0.F3.9.m4.1.1.1.5.cmml">P</mi></mrow></msub></math>, <math><mrow id="p12.2.m2.1.1" xref="p12.2.m2.1.1.cmml"><mn id="p12.2.m2.1.1.2" xref="p12.2.m2.1.1.2.cmml">700</mn><mo id="p12.2.m2.1.1.1" xref="p12.2.m2.1.1.1.cmml">±</mo><mrow id="p12.2.m2.1.1.3" xref="p12.2.m2.1.1.3.cmml"><mn id="p12.2.m2.1.1.3.2" xref="p12.2.m2.1.1.3.2.cmml">100</mn><mo id="p12.2.m2.1.1.3.1" xref="p12.2.m2.1.1.3.1.cmml">⁢</mo><mi id="p12.2.m2.1.1.3.3" xref="p12.2.m2.1.1.3.3.cmml">p</mi><mo id="p12.2.m2.1.1.3.1a" xref="p12.2.m2.1.1.3.1.cmml">⁢</mo><mi id="p12.2.m2.1.1.3.4" xref="p12.2.m2.1.1.3.4.cmml">s</mi></mrow></mrow></math>, <math><mrow id="p12.4.m4.1.1" xref="p12.4.m4.1.1.cmml"><msub id="p12.4.m4.1.1.2" xref="p12.4.m4.1.1.2.cmml"><mi id="p12.4.m4.1.1.2.2" xref="p12.4.m4.1.1.2.2.cmml">τ</mi><msub id="p12.4.m4.1.1.2.3" xref="p12.4.m4.1.1.2.3.cmml"><mi id="p12.4.m4.1.1.2.3.2" xref="p12.4.m4.1.1.2.3.2.cmml">X</mi><mrow id="p12.4.m4.1.1.2.3.3" xref="p12.4.m4.1.1.2.3.3.cmml"><mi id="p12.4.m4.1.1.2.3.3.2" xref="p12.4.m4.1.1.2.3.3.2.cmml">Q</mi><mo id="p12.4.m4.1.1.2.3.3.1" xref="p12.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="p12.4.m4.1.1.2.3.3.3" xref="p12.4.m4.1.1.2.3.3.3.cmml">D</mi><mo id="p12.4.m4.1.1.2.3.3.1a" xref="p12.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mn id="p12.4.m4.1.1.2.3.3.4" xref="p12.4.m4.1.1.2.3.3.4.cmml">1</mn></mrow></msub></msub><mo id="p12.4.m4.1.1.1" xref="p12.4.m4.1.1.1.cmml">=</mo><mn id="p12.4.m4.1.1.3" xref="p12.4.m4.1.1.3.cmml">400</mn></mrow></math>, <math><mrow id="p14.2.m2.1.1" xref="p14.2.m2.1.1.cmml"><msub id="p14.2.m2.1.1.2" xref="p14.2.m2.1.1.2.cmml"><mi id="p14.2.m2.1.1.2.2" xref="p14.2.m2.1.1.2.2.cmml">τ</mi><msub id="p14.2.m2.1.1.2.3" xref="p14.2.m2.1.1.2.3.cmml"><mi id="p14.2.m2.1.1.2.3.2" xref="p14.2.m2.1.1.2.3.2.cmml">X</mi><mrow id="p14.2.m2.1.1.2.3.3" xref="p14.2.m2.1.1.2.3.3.cmml"><mi id="p14.2.m2.1.1.2.3.3.2" xref="p14.2.m2.1.1.2.3.3.2.cmml">Q</mi><mo id="p14.2.m2.1.1.2.3.3.1" xref="p14.2.m2.1.1.2.3.3.1.cmml">⁢</mo><mi id="p14.2.m2.1.1.2.3.3.3" xref="p14.2.m2.1.1.2.3.3.3.cmml">D</mi><mo id="p14.2.m2.1.1.2.3.3.1a" xref="p14.2.m2.1.1.2.3.3.1.cmml">⁢</mo><mn id="p14.2.m2.1.1.2.3.3.4" xref="p14.2.m2.1.1.2.3.3.4.cmml">2</mn></mrow></msub></msub><mo id="p14.2.m2.1.1.1" xref="p14.2.m2.1.1.1.cmml">=</mo><mn id="p14.2.m2.1.1.3" xref="p14.2.m2.1.1.3.cmml">52</mn></mrow></math>

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1401.6276

Formulas:

1-

P (X, H, Θ) = P (X | H, Θ) P (H | Θ) P (Θ)

2-

P (X, Θ)

3-

= \int P (X, H, Θ) 𝑑 H

4-

= argmax P (X, Θ),

5-

= \nabla^{2} \log P (X, \hat{Θ})

6-

P (Θ | X) \approx 𝒩 (Θ | \hat{Θ}, - 𝚲^{- 1})

7-

\int P (H | X, Θ^{'}) 𝑑 H = 1

8-

\begin{matrix} \log P (X, Θ) & = \int P (H | X, Θ^{'}) \log P (X, Θ) 𝑑 H \\ = \int P (H | X, Θ^{'}) \log P (X | Θ) 𝑑 H + \log P (Θ) \\ = \int P (H | X, Θ^{'}) \log \frac{P (X, H | Θ)}{P (H | X, Θ)} d H + \log P (Θ) \\ = \int P (H | X, Θ^{'}) \log [\frac{P (X, H | Θ)}{P (H | X, Θ)} \times \frac{P (H | X, Θ^{'})}{P (H | X, Θ^{'})}] 𝑑 H + \log P (Θ) \\ = A (Θ^{'}, Θ) + D (Θ^{'}, Θ) \end{matrix}

9-

A (Θ^{'}, Θ)

10-

A (Θ^{'}, Θ)

Formulas (html):

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id="S1.E2.m2.3.4.3a" xref="S1.E2.m2.3.4.3.cmml"><mo largeop="true" symmetric="true" id="S1.E2.m2.3.4.3.1" xref="S1.E2.m2.3.4.3.1.cmml">∫</mo><mrow id="S1.E2.m2.3.4.3.2" xref="S1.E2.m2.3.4.3.2.cmml"><mi id="S1.E2.m2.3.4.3.2.2" xref="S1.E2.m2.3.4.3.2.2.cmml">P</mi><mo id="S1.E2.m2.3.4.3.2.1" xref="S1.E2.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="S1.E2.m2.3.4.3.2.3.2" xref="S1.E2.m2.3.4.3.2.3.1.cmml"><mo stretchy="false" id="S1.E2.m2.3.4.3.2.3.2.1" xref="S1.E2.m2.3.4.3.2.3.1.cmml">(</mo><mi id="S1.E2.m2.1.1" xref="S1.E2.m2.1.1.cmml">X</mi><mo id="S1.E2.m2.3.4.3.2.3.2.2" xref="S1.E2.m2.3.4.3.2.3.1.cmml">,</mo><mi id="S1.E2.m2.2.2" xref="S1.E2.m2.2.2.cmml">H</mi><mo id="S1.E2.m2.3.4.3.2.3.2.3" xref="S1.E2.m2.3.4.3.2.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E2.m2.3.3" xref="S1.E2.m2.3.3.cmml">Θ</mi><mo rspace="4.2pt" stretchy="false" id="S1.E2.m2.3.4.3.2.3.2.4" xref="S1.E2.m2.3.4.3.2.3.1.cmml">)</mo></mrow><mo id="S1.E2.m2.3.4.3.2.1a" xref="S1.E2.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="S1.E2.m2.3.4.3.2.4" xref="S1.E2.m2.3.4.3.2.4.cmml"><mo rspace="0pt" id="S1.E2.m2.3.4.3.2.4.1" xref="S1.E2.m2.3.4.3.2.4.1.cmml">𝑑</mo><mi id="S1.E2.m2.3.4.3.2.4.2" xref="S1.E2.m2.3.4.3.2.4.2.cmml">H</mi></mrow></mrow></mrow></mstyle></mrow></math>, <math><mrow id="S1.E3.m2.3.3.1" xref="S1.E3.m2.3.3.1.1.cmml"><mrow id="S1.E3.m2.3.3.1.1" xref="S1.E3.m2.3.3.1.1.cmml"><mi id="S1.E3.m2.3.3.1.1.2" xref="S1.E3.m2.3.3.1.1.2.cmml"/><mo id="S1.E3.m2.3.3.1.1.1" xref="S1.E3.m2.3.3.1.1.1.cmml">=</mo><mrow id="S1.E3.m2.3.3.1.1.3" xref="S1.E3.m2.3.3.1.1.3.cmml"><mrow id="S1.E3.m2.3.3.1.1.3.2" xref="S1.E3.m2.3.3.1.1.3.2.cmml"><mo movablelimits="false" id="S1.E3.m2.3.3.1.1.3.2.1" xref="S1.E3.m2.3.3.1.1.3.2.1.cmml">argmax</mo><mo id="S1.E3.m2.3.3.1.1.3.2a" xref="S1.E3.m2.3.3.1.1.3.2.cmml">⁡</mo><mi id="S1.E3.m2.3.3.1.1.3.2.2" xref="S1.E3.m2.3.3.1.1.3.2.2.cmml">P</mi></mrow><mo id="S1.E3.m2.3.3.1.1.3.1" xref="S1.E3.m2.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S1.E3.m2.3.3.1.1.3.3.2" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml"><mo stretchy="false" id="S1.E3.m2.3.3.1.1.3.3.2.1" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">(</mo><mi id="S1.E3.m2.1.1" xref="S1.E3.m2.1.1.cmml">X</mi><mo id="S1.E3.m2.3.3.1.1.3.3.2.2" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E3.m2.2.2" xref="S1.E3.m2.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E3.m2.3.3.1.1.3.3.2.3" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.E3.m2.3.3.1.2" xref="S1.E3.m2.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S1.E3.m5.2.3" xref="S1.E3.m5.2.3.cmml"><mi id="S1.E3.m5.2.3.2" xref="S1.E3.m5.2.3.2.cmml"/><mo id="S1.E3.m5.2.3.1" xref="S1.E3.m5.2.3.1.cmml">=</mo><mrow id="S1.E3.m5.2.3.3" xref="S1.E3.m5.2.3.3.cmml"><mrow id="S1.E3.m5.2.3.3.2" xref="S1.E3.m5.2.3.3.2.cmml"><mrow id="S1.E3.m5.2.3.3.2.1" xref="S1.E3.m5.2.3.3.2.1.cmml"><msup id="S1.E3.m5.2.3.3.2.1.1" xref="S1.E3.m5.2.3.3.2.1.1.cmml"><mo id="S1.E3.m5.2.3.3.2.1.1.2" xref="S1.E3.m5.2.3.3.2.1.1.2.cmml">∇</mo><mn id="S1.E3.m5.2.3.3.2.1.1.3" xref="S1.E3.m5.2.3.3.2.1.1.3.cmml">2</mn></msup><mo id="S1.E3.m5.2.3.3.2.1a" xref="S1.E3.m5.2.3.3.2.1.cmml">⁡</mo><mi id="S1.E3.m5.2.3.3.2.1.2" xref="S1.E3.m5.2.3.3.2.1.2.cmml">log</mi></mrow><mo id="S1.E3.m5.2.3.3.2a" xref="S1.E3.m5.2.3.3.2.cmml">⁡</mo><mi id="S1.E3.m5.2.3.3.2.2" xref="S1.E3.m5.2.3.3.2.2.cmml">P</mi></mrow><mo id="S1.E3.m5.2.3.3.1" xref="S1.E3.m5.2.3.3.1.cmml">⁢</mo><mrow id="S1.E3.m5.2.3.3.3.2" xref="S1.E3.m5.2.3.3.3.1.cmml"><mo stretchy="false" id="S1.E3.m5.2.3.3.3.2.1" xref="S1.E3.m5.2.3.3.3.1.cmml">(</mo><mi id="S1.E3.m5.1.1" xref="S1.E3.m5.1.1.cmml">X</mi><mo id="S1.E3.m5.2.3.3.3.2.2" xref="S1.E3.m5.2.3.3.3.1.cmml">,</mo><mover accent="true" id="S1.E3.m5.2.2" xref="S1.E3.m5.2.2.cmml"><mi mathvariant="normal" id="S1.E3.m5.2.2.2" xref="S1.E3.m5.2.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E3.m5.2.2.1" xref="S1.E3.m5.2.2.1.cmml">^</mo></mover><mo stretchy="false" id="S1.E3.m5.2.3.3.3.2.3" xref="S1.E3.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.E4.m1.3.3" xref="S1.E4.m1.3.3.cmml"><mrow id="S1.E4.m1.2.2.1" xref="S1.E4.m1.2.2.1.cmml"><mi id="S1.E4.m1.2.2.1.3" xref="S1.E4.m1.2.2.1.3.cmml">P</mi><mo id="S1.E4.m1.2.2.1.2" xref="S1.E4.m1.2.2.1.2.cmml">⁢</mo><mrow id="S1.E4.m1.2.2.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.cmml"><mo stretchy="false" id="S1.E4.m1.2.2.1.1.1.2" xref="S1.E4.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.2.2.1.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.2.2.1.1.1.1.2" xref="S1.E4.m1.2.2.1.1.1.1.2.cmml">Θ</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S1.E4.m1.2.2.1.1.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.1.cmml">|</mo><mi id="S1.E4.m1.2.2.1.1.1.1.3" xref="S1.E4.m1.2.2.1.1.1.1.3.cmml">X</mi></mrow><mo stretchy="false" id="S1.E4.m1.2.2.1.1.1.3" xref="S1.E4.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.E4.m1.3.3.3" xref="S1.E4.m1.3.3.3.cmml">≈</mo><mrow id="S1.E4.m1.3.3.2" xref="S1.E4.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E4.m1.3.3.2.3" xref="S1.E4.m1.3.3.2.3.cmml">𝒩</mi><mo id="S1.E4.m1.3.3.2.2" xref="S1.E4.m1.3.3.2.2.cmml">⁢</mo><mrow id="S1.E4.m1.3.3.2.1.1" xref="S1.E4.m1.3.3.2.1.1.1.cmml"><mo stretchy="false" id="S1.E4.m1.3.3.2.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.3.3.2.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.3.3.2.1.1.1.3" xref="S1.E4.m1.3.3.2.1.1.1.3.cmml">Θ</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S1.E4.m1.3.3.2.1.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.2.cmml">|</mo><mrow id="S1.E4.m1.3.3.2.1.1.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.1.2.cmml"><mover accent="true" id="S1.E4.m1.1.1" xref="S1.E4.m1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.1.1.2" xref="S1.E4.m1.1.1.2.cmml">Θ</mi><mo stretchy="false" id="S1.E4.m1.1.1.1" xref="S1.E4.m1.1.1.1.cmml">^</mo></mover><mo id="S1.E4.m1.3.3.2.1.1.1.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.1.2.cmml">,</mo><mrow id="S1.E4.m1.3.3.2.1.1.1.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.cmml"><mo id="S1.E4.m1.3.3.2.1.1.1.1.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.1.cmml">-</mo><msup id="S1.E4.m1.3.3.2.1.1.1.1.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.cmml"><mi id="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.2" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.2.cmml">𝚲</mi><mrow id="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3.cmml"><mo id="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3.1" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3.1.cmml">-</mo><mn id="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3.2" xref="S1.E4.m1.3.3.2.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup></mrow></mrow></mrow><mo stretchy="false" id="S1.E4.m1.3.3.2.1.1.3" xref="S1.E4.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p1.2.m2.2.2" xref="S2.SS1.p1.2.m2.2.2.cmml"><mrow id="S2.SS1.p1.2.m2.2.2.1" xref="S2.SS1.p1.2.m2.2.2.1.cmml"><mo largeop="true" symmetric="true" id="S2.SS1.p1.2.m2.2.2.1.2" xref="S2.SS1.p1.2.m2.2.2.1.2.cmml">∫</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.2.2.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.3.cmml">P</mi><mo id="S2.SS1.p1.2.m2.2.2.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.2.cmml">|</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">X</mi><mo id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml">,</mo><msup id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.2.cmml">Θ</mi><mo id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup></mrow></mrow><mo rspace="4.2pt" stretchy="false" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p1.2.m2.2.2.1.1.2a" xref="S2.SS1.p1.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.4" xref="S2.SS1.p1.2.m2.2.2.1.1.4.cmml"><mo rspace="0pt" id="S2.SS1.p1.2.m2.2.2.1.1.4.1" xref="S2.SS1.p1.2.m2.2.2.1.1.4.1.cmml">𝑑</mo><mi id="S2.SS1.p1.2.m2.2.2.1.1.4.2" xref="S2.SS1.p1.2.m2.2.2.1.1.4.2.cmml">H</mi></mrow></mrow></mrow><mo id="S2.SS1.p1.2.m2.2.2.2" xref="S2.SS1.p1.2.m2.2.2.2.cmml">=</mo><mn id="S2.SS1.p1.2.m2.2.2.3" xref="S2.SS1.p1.2.m2.2.2.3.cmml">1</mn></mrow></math>, <math><mtable columnspacing="0pt" rowspacing="0pt" id="S2.E5.m1.131.131.16" xref="S2.E5.m1.123.123.8.cmml"><mtr id="S2.E5.m1.131.131.16a" xref="S2.E5.m1.123.123.8.cmml"><mtd columnalign="right" id="S2.E5.m1.131.131.16b" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.7.7.7.7.7a" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.7.7.7.7.7a.9" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S2.E5.m1.7.7.7.7.7a.9a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.2.2.2.2.2.2" xref="S2.E5.m1.2.2.2.2.2.2.cmml">P</mi></mrow><mo id="S2.E5.m1.7.7.7.7.7a.8" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.7.7.7.7.7a.10" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.3.3.3.3.3.3" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mi id="S2.E5.m1.4.4.4.4.4.4" xref="S2.E5.m1.4.4.4.4.4.4.cmml">X</mi><mo id="S2.E5.m1.5.5.5.5.5.5" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.6.6.6.6.6.6" xref="S2.E5.m1.6.6.6.6.6.6.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.7.7.7.7.7.7" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow></mtd><mtd columnalign="left" id="S2.E5.m1.131.131.16c" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.124.124.9.116.28.21" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.124.124.9.116.28.21.22" xref="S2.E5.m1.123.123.8a.cmml"/><mo id="S2.E5.m1.8.8.8.8.1.1" xref="S2.E5.m1.8.8.8.8.1.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E5.m1.124.124.9.116.28.21.21" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.124.124.9.116.28.21.21a" xref="S2.E5.m1.123.123.8.cmml"><mo largeop="true" symmetric="true" id="S2.E5.m1.9.9.9.9.2.2" xref="S2.E5.m1.9.9.9.9.2.2.cmml">∫</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.10.10.10.10.3.3" xref="S2.E5.m1.10.10.10.10.3.3.cmml">P</mi><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" 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id="S2.E5.m1.90.90.90.16.16.16.2.2.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml">(</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.2.cmml">|</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.2.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.1.1" xref="S2.E5.m1.90.90.90.16.16.16.1.1.cmml">X</mi><mo id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.2.cmml">,</mo><msup id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.2.cmml">Θ</mi><mo id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.3.cmml">′</mo></msup></mrow></mrow><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.3" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E5.m1.90.90.90.16.16.16.4" xref="S2.E5.m1.90.90.90.16.16.16.4.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.4.4" xref="S2.E5.m1.90.90.90.16.16.16.4.4.cmml">P</mi><mo id="S2.E5.m1.90.90.90.16.16.16.4.3" xref="S2.E5.m1.90.90.90.16.16.16.4.3.cmml">⁢</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.4.2.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.cmml">(</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.2.cmml">|</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.2.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.3.1" xref="S2.E5.m1.90.90.90.16.16.16.3.1.cmml">X</mi><mo id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.2.cmml">,</mo><msup id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.2.cmml">Θ</mi><mo id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.3.cmml">′</mo></msup></mrow></mrow><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.3" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.cmml">)</mo></mrow></mrow></mfrac></mrow><mo rspace="4.2pt" id="S2.E5.m1.91.91.91.17.17.17" xref="S2.E5.m1.123.123.8a.cmml">]</mo></mrow></mrow><mo id="S2.E5.m1.129.129.14.121.27.27.27.2.2.3b" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.129.129.14.121.27.27.27.2.2.4" xref="S2.E5.m1.123.123.8.cmml"><mo rspace="0pt" id="S2.E5.m1.92.92.92.18.18.18" xref="S2.E5.m1.92.92.92.18.18.18.cmml">𝑑</mo><mi id="S2.E5.m1.93.93.93.19.19.19" xref="S2.E5.m1.93.93.93.19.19.19.cmml">H</mi></mrow></mrow></mrow></mstyle><mo id="S2.E5.m1.94.94.94.20.20.20" xref="S2.E5.m1.94.94.94.20.20.20.cmml">+</mo><mrow id="S2.E5.m1.129.129.14.121.27.27.27.3" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.129.129.14.121.27.27.27.3.2" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.95.95.95.21.21.21" xref="S2.E5.m1.95.95.95.21.21.21.cmml">log</mi><mo id="S2.E5.m1.129.129.14.121.27.27.27.3.2a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.96.96.96.22.22.22" xref="S2.E5.m1.96.96.96.22.22.22.cmml">P</mi></mrow><mo id="S2.E5.m1.129.129.14.121.27.27.27.3.1" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.129.129.14.121.27.27.27.3.3" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.97.97.97.23.23.23" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mi mathvariant="normal" id="S2.E5.m1.98.98.98.24.24.24" xref="S2.E5.m1.98.98.98.24.24.24.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.99.99.99.25.25.25" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E5.m1.131.131.16m" xref="S2.E5.m1.123.123.8.cmml"><mtd id="S2.E5.m1.131.131.16n" xref="S2.E5.m1.123.123.8a.cmml"/><mtd columnalign="left" id="S2.E5.m1.131.131.16o" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.131.131.16.123.18.18" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.131.131.16.123.18.18.19" xref="S2.E5.m1.123.123.8a.cmml"/><mo id="S2.E5.m1.100.100.100.1.1.1" xref="S2.E5.m1.100.100.100.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.131.131.16.123.18.18.18" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.130.130.15.122.17.17.17.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.101.101.101.2.2.2" xref="S2.E5.m1.101.101.101.2.2.2.cmml">A</mi><mo id="S2.E5.m1.130.130.15.122.17.17.17.1.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.130.130.15.122.17.17.17.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.102.102.102.3.3.3" xref="S2.E5.m1.123.123.8a.cmml">(</mo><msup id="S2.E5.m1.130.130.15.122.17.17.17.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.103.103.103.4.4.4" xref="S2.E5.m1.103.103.103.4.4.4.cmml">Θ</mi><mo id="S2.E5.m1.104.104.104.5.5.5.1" xref="S2.E5.m1.104.104.104.5.5.5.1.cmml">′</mo></msup><mo id="S2.E5.m1.105.105.105.6.6.6" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.106.106.106.7.7.7" xref="S2.E5.m1.106.106.106.7.7.7.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.107.107.107.8.8.8" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.108.108.108.9.9.9" xref="S2.E5.m1.108.108.108.9.9.9.cmml">+</mo><mrow id="S2.E5.m1.131.131.16.123.18.18.18.2" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.109.109.109.10.10.10" xref="S2.E5.m1.109.109.109.10.10.10.cmml">D</mi><mo id="S2.E5.m1.131.131.16.123.18.18.18.2.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.131.131.16.123.18.18.18.2.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.110.110.110.11.11.11" xref="S2.E5.m1.123.123.8a.cmml">(</mo><msup id="S2.E5.m1.131.131.16.123.18.18.18.2.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.111.111.111.12.12.12" xref="S2.E5.m1.111.111.111.12.12.12.cmml">Θ</mi><mo id="S2.E5.m1.112.112.112.13.13.13.1" xref="S2.E5.m1.112.112.112.13.13.13.1.cmml">′</mo></msup><mo id="S2.E5.m1.113.113.113.14.14.14" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.114.114.114.15.15.15" xref="S2.E5.m1.114.114.114.15.15.15.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.115.115.115.16.16.16" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable></math>, <math><mrow id="S2.SS1.p1.3.m1.2.2" xref="S2.SS1.p1.3.m1.2.2.cmml"><mi id="S2.SS1.p1.3.m1.2.2.3" xref="S2.SS1.p1.3.m1.2.2.3.cmml">A</mi><mo id="S2.SS1.p1.3.m1.2.2.2" xref="S2.SS1.p1.3.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p1.3.m1.2.2.1.1" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">(</mo><msup id="S2.SS1.p1.3.m1.2.2.1.1.1" xref="S2.SS1.p1.3.m1.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S2.SS1.p1.3.m1.2.2.1.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.1.1.2.cmml">Θ</mi><mo id="S2.SS1.p1.3.m1.2.2.1.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.3.m1.2.2.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">,</mo><mi mathvariant="normal" id="S2.SS1.p1.3.m1.1.1" xref="S2.SS1.p1.3.m1.1.1.cmml">Θ</mi><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.4" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml"><mi id="S2.E6.m1.2.2.3" xref="S2.E6.m1.2.2.3.cmml">A</mi><mo id="S2.E6.m1.2.2.2" xref="S2.E6.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.1.1" xref="S2.E6.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.E6.m1.2.2.1.1.2" xref="S2.E6.m1.2.2.1.2.cmml">(</mo><msup id="S2.E6.m1.2.2.1.1.1" xref="S2.E6.m1.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S2.E6.m1.2.2.1.1.1.2" xref="S2.E6.m1.2.2.1.1.1.2.cmml">Θ</mi><mo id="S2.E6.m1.2.2.1.1.1.3" xref="S2.E6.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.E6.m1.2.2.1.1.3" xref="S2.E6.m1.2.2.1.2.cmml">,</mo><mi mathvariant="normal" id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml">Θ</mi><mo stretchy="false" id="S2.E6.m1.2.2.1.1.4" xref="S2.E6.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: stat

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1711.02986

Formulas:

1-

R (x + y) = R (x) + R (y)

2-

\dot{I_{e}} = \frac{3 I_{0}}{4 a τ_{L} ρ_{0}} [1 - \frac{1}{2 α_{L}^{2} a^{2}} (1 - e^{- 2 α_{L} a} (1 + 2 α_{L} a))] = \frac{3 I_{0}}{4 a τ_{L} ρ_{0}} C (α_{L} a),

3-

J o u l e / c m^{2}

4-

\dot{f} (t)

5-

ρ \ddot{𝒖} = - \nabla_{𝒖} P,

6-

ρ = ρ (t)

7-

ρ_{0} r^{2} = u^{2} ρ \frac{\partial u}{\partial r}

8-

Δ \dot{u} = \ddot{u} Δ t

9-

\dot{v} = v \nabla_{𝒖} \cdot \dot{𝒖},

10-

\dot{v} = \frac{v}{ρ_{0} r^{2}} [2 \dot{u} u \frac{\partial u}{\partial r} + u^{2} \frac{\partial \dot{u}}{\partial r}] .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0910.2375

Formulas:

1-

(x_{0}, y_{0})

2-

(x_{0}, y_{0})

3-

I (x, y) = a + b x + c y + d x^{2} + e x y + f y^{2}

4-

(x_{s}, y_{s})

5-

(x_{s}, y_{s}) = (\frac{c e - b f}{2 d f - 2 e^{2}}, \frac{b e - c d}{2 d f - 2 e^{2}})

6-

\sqrt{{(x_{s} - x_{0})}^{2} + {(y_{s} - y_{0})}^{2}}

7-

S = A \times 2 π σ^{2}

8-

A e^{\frac{- {(x - x_{s})}^{2} - {(y - y_{s})}^{2}}{2 σ^{2}}} + μ_{s k y}

9-

μ_{s k y}

10-

(x_{0}, y_{0})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0411213

Formulas:

1-

H = J \sum_{⟨ i j ⟩} 𝐒_{i} \cdot 𝐒_{j},

2-

Θ_{C W} = 390

3-

⟨ 𝐒_{i} \cdot 𝐒_{j} - 𝐒_{k} \cdot 𝐒_{l} ⟩

4-

2 (R_{L T} - R_{H T})

5-

F d \bar{3} m

6-

2 (R_{LT} - R_{HT})

7-

R = {| \frac{1 - \sqrt{ϵ}}{1 + \sqrt{ϵ}} |}^{2}

8-

ϵ (ω, T) = ϵ_{\infty} + \sum_{j} \frac{S_{j} ω_{j}^{2}}{ω_{j}^{2} - ω^{2} - i ω γ_{j}}

9-

ϵ_{\infty} (T) \equiv ϵ (\infty, T)

10-

S_{j} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1112.2513

Formulas:

1-

\sim 0 \overset{''}{.} 1

2-

μ (= \cos θ)

3-

0 \overset{''}{.} 0405

4-

Θ^{'} < θ < Θ

5-

\tan Θ = \tan φ + \frac{a}{h \cos φ}; \tan Θ^{'} = \tan φ - \frac{a}{h \cos φ},

6-

0 < Θ < 90^{\circ}

7-

0 \leq Θ^{'} < Θ

8-

l = \frac{a}{\cos φ} - h ∣ \tan θ - \tan φ ∣ .

9-

FCS = f ⟨ l ⟩ .

10-

h / a, φ_{0}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0601010

Formulas:

1-

ρ \propto \exp (a E)

2-

T_{C} = 1 / a

3-

Z (T) = T_{C} T / (T_{C} - T)

4-

Δ S = \frac{Q}{T_{A}} .

5-

ρ (Q) = S_{0} e^{Q / T_{A}} \approx K e^{E / T_{A}} .

6-

Z (T) = \int e^{E / T_{A}} e^{- E / T} 𝑑 E = \int e^{- (\frac{1}{T} - \frac{1}{T_{A}}) E} 𝑑 E = \frac{T_{A} T}{T_{A} - T}

7-

0 \leq T < T_{A}

8-

ρ_{T} (x) = ρ_{A} (E - x) ρ_{B} (x)

9-

ρ_{A} = e^{ϵ / T_{A}}

10-

ρ_{T} (x) = \exp (\frac{E - x}{T_{A}}) ρ_{B} (x) = e^{E / T_{A}} e^{- x / T_{A}} ρ_{B} (x) .

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0803.1671

Formulas:

1-

\frac{\partial 𝒖}{\partial t} + 𝒖 \cdot \nabla 𝒖 = - \nabla h - \nabla Φ,

2-

\frac{\partial h}{\partial t} + 𝒖 \cdot \nabla h = - c_{s}^{2} \nabla \cdot 𝒖,

3-

h = c_{s}^{2} \log ρ

4-

Φ (r, z)

5-

(r, ϕ, z)

6-

q = q_{0} + q^{'}

7-

𝒖_{0} = 𝛀 \times 𝒓 = r Ω (r) 𝒆_{ϕ}

8-

h_{0} (r, z)

9-

\nabla h_{0} = r Ω^{2} 𝒆_{r} - \nabla Φ

10-

q^{'} (r, ϕ, z, t) = Re [\tilde{q^{'}} (r, z) \exp (i m ϕ - i ω t)],

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0705.4627

Formulas:

1-

H = H_{0} + V = h_{0} + H_{A} + V

2-

H_{A} = \sum_{i = 1}^{A} h_{i} + \sum_{i < j}^{A} v_{i j}

3-

V = \sum_{i = 1}^{A} v_{0 i}

4-

\sum_{i = 1}^{A} T_{0 i}

5-

V \equiv \sum_{i = 1}^{A} v_{0 i}

6-

T \equiv \sum_{i = 1}^{A} T_{0 i}

7-

H_{A} | ϕ_{n}^{A} ⟩ = E_{n} | ϕ_{n}^{A} ⟩

8-

h_{0} | 𝐤 ⟩ = E_{k} | 𝐤 ⟩

9-

H_{0} = h_{0} + H_{A}

10-

H_{0} | Φ ⟩ = E | Φ ⟩

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/1712.04626

Formulas:

1-

M_{Z}^{2} ≃ - 2 μ^{2} (M_{Z}) - 2 m_{H_{u}}^{2} (M_{Z})

2-

M_{Z}^{2} ≃ - 1.9 μ^{2} + 5.9 M_{3}^{2} - 1.2 m_{H_{u}}^{2} + 1.5 m_{\tilde{t}}^{2} - 0.8 A_{t} M_{3} + 0.2 A_{t}^{2} + \dots,

3-

m_{\tilde{t}}^{2} \equiv (m_{{\tilde{t}}_{L}}^{2} + m_{{\tilde{t}}_{R}}^{2}) / 2

4-

m_{{\tilde{t}}_{L}} ≃ m_{{\tilde{t}}_{R}}

5-

\ln (M_{G} / M_{Z})

6-

X_{t} / m_{\tilde{t}} = 0

7-

X_{t} / m_{\tilde{t}} ≃ - 1

8-

X_{t} / m_{\tilde{t}} ≃ - 2

9-

X_{t} / m_{\tilde{t}}

10-

f (λ_{t}, α_{3}, Λ)

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/0710.5781

Formulas:

1-

110 \times 270 \times 90 μ

2-

B_{local} (H)

3-

d B_{local} / d H

4-

H_{fp} / H_{c1} = \tanh (\sqrt{0.36 b / a}),

5-

H_{fp} / H_{c1} = 0.50

6-

H_{c1} (T)

7-

H_{c1} (T)

8-

H_{c1} (T)

9-

λ^{- 2} (T)

10-

d H_{c1} / d T = - 38

Formulas (html):

<math><mrow id="p4.9.m9.1.1" xref="p4.9.m9.1.1.cmml"><mrow id="p4.9.m9.1.1.2" xref="p4.9.m9.1.1.2.cmml"><mn id="p4.9.m9.1.1.2.2" xref="p4.9.m9.1.1.2.2.cmml">110</mn><mo id="p4.9.m9.1.1.2.1" xref="p4.9.m9.1.1.2.1.cmml">×</mo><mn id="p4.9.m9.1.1.2.3" xref="p4.9.m9.1.1.2.3.cmml">270</mn><mo id="p4.9.m9.1.1.2.1a" xref="p4.9.m9.1.1.2.1.cmml">×</mo><mpadded width="+3.3pt" id="p4.9.m9.1.1.2.4" xref="p4.9.m9.1.1.2.4.cmml"><mn id="p4.9.m9.1.1.2.4a" xref="p4.9.m9.1.1.2.4.cmml">90</mn></mpadded></mrow><mo id="p4.9.m9.1.1.1" xref="p4.9.m9.1.1.1.cmml">⁢</mo><mi id="p4.9.m9.1.1.3" xref="p4.9.m9.1.1.3.cmml">μ</mi></mrow></math>, <math><mrow id="p5.1.m1.1.2" xref="p5.1.m1.1.2.cmml"><msub id="p5.1.m1.1.2.2" xref="p5.1.m1.1.2.2.cmml"><mi id="p5.1.m1.1.2.2.2" xref="p5.1.m1.1.2.2.2.cmml">B</mi><mi id="p5.1.m1.1.2.2.3" xref="p5.1.m1.1.2.2.3.cmml">local</mi></msub><mo id="p5.1.m1.1.2.1" xref="p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="p5.1.m1.1.2.3.2" xref="p5.1.m1.1.2.cmml"><mo stretchy="false" 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<math><mrow id="S0.E1.m1.3.3.1" xref="S0.E1.m1.3.3.1.1.cmml"><mrow id="S0.E1.m1.3.3.1.1" xref="S0.E1.m1.3.3.1.1.cmml"><mrow id="S0.E1.m1.3.3.1.1.2" xref="S0.E1.m1.3.3.1.1.2.cmml"><msub id="S0.E1.m1.3.3.1.1.2.2" xref="S0.E1.m1.3.3.1.1.2.2.cmml"><mi id="S0.E1.m1.3.3.1.1.2.2.2" xref="S0.E1.m1.3.3.1.1.2.2.2.cmml">H</mi><mi id="S0.E1.m1.3.3.1.1.2.2.3" xref="S0.E1.m1.3.3.1.1.2.2.3.cmml">fp</mi></msub><mo id="S0.E1.m1.3.3.1.1.2.1" xref="S0.E1.m1.3.3.1.1.2.1.cmml">/</mo><msub id="S0.E1.m1.3.3.1.1.2.3" xref="S0.E1.m1.3.3.1.1.2.3.cmml"><mi id="S0.E1.m1.3.3.1.1.2.3.2" xref="S0.E1.m1.3.3.1.1.2.3.2.cmml">H</mi><mi id="S0.E1.m1.3.3.1.1.2.3.3" xref="S0.E1.m1.3.3.1.1.2.3.3.cmml">c1</mi></msub></mrow><mo id="S0.E1.m1.3.3.1.1.1" xref="S0.E1.m1.3.3.1.1.1.cmml">=</mo><mrow id="S0.E1.m1.3.3.1.1.3.2" xref="S0.E1.m1.3.3.1.1.3.1.cmml"><mi id="S0.E1.m1.1.1" xref="S0.E1.m1.1.1.cmml">tanh</mi><mo id="S0.E1.m1.3.3.1.1.3.2a" xref="S0.E1.m1.3.3.1.1.3.1.cmml">⁡</mo><mrow id="S0.E1.m1.3.3.1.1.3.2.1" xref="S0.E1.m1.3.3.1.1.3.1.cmml"><mo maxsize="120%" minsize="120%" id="S0.E1.m1.3.3.1.1.3.2.1.1" xref="S0.E1.m1.3.3.1.1.3.1.cmml">(</mo><msqrt id="S0.E1.m1.2.2" xref="S0.E1.m1.2.2.cmml"><mrow id="S0.E1.m1.2.2.2" xref="S0.E1.m1.2.2.2.cmml"><mrow id="S0.E1.m1.2.2.2.2" xref="S0.E1.m1.2.2.2.2.cmml"><mpadded width="+5pt" id="S0.E1.m1.2.2.2.2.2" xref="S0.E1.m1.2.2.2.2.2.cmml"><mn id="S0.E1.m1.2.2.2.2.2a" xref="S0.E1.m1.2.2.2.2.2.cmml">0.36</mn></mpadded><mo id="S0.E1.m1.2.2.2.2.1" xref="S0.E1.m1.2.2.2.2.1.cmml">⁢</mo><mi id="S0.E1.m1.2.2.2.2.3" xref="S0.E1.m1.2.2.2.2.3.cmml">b</mi></mrow><mo id="S0.E1.m1.2.2.2.1" xref="S0.E1.m1.2.2.2.1.cmml">/</mo><mi id="S0.E1.m1.2.2.2.3" xref="S0.E1.m1.2.2.2.3.cmml">a</mi></mrow></msqrt><mo maxsize="120%" minsize="120%" id="S0.E1.m1.3.3.1.1.3.2.1.2" xref="S0.E1.m1.3.3.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S0.E1.m1.3.3.1.2" xref="S0.E1.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="p6.1.m1.1.1" xref="p6.1.m1.1.1.cmml"><mrow id="p6.1.m1.1.1.2" xref="p6.1.m1.1.1.2.cmml"><msub id="p6.1.m1.1.1.2.2" xref="p6.1.m1.1.1.2.2.cmml"><mi id="p6.1.m1.1.1.2.2.2" xref="p6.1.m1.1.1.2.2.2.cmml">H</mi><mi id="p6.1.m1.1.1.2.2.3" xref="p6.1.m1.1.1.2.2.3.cmml">fp</mi></msub><mo id="p6.1.m1.1.1.2.1" xref="p6.1.m1.1.1.2.1.cmml">/</mo><msub id="p6.1.m1.1.1.2.3" xref="p6.1.m1.1.1.2.3.cmml"><mi id="p6.1.m1.1.1.2.3.2" xref="p6.1.m1.1.1.2.3.2.cmml">H</mi><mi id="p6.1.m1.1.1.2.3.3" xref="p6.1.m1.1.1.2.3.3.cmml">c1</mi></msub></mrow><mo id="p6.1.m1.1.1.1" xref="p6.1.m1.1.1.1.cmml">=</mo><mn id="p6.1.m1.1.1.3" xref="p6.1.m1.1.1.3.cmml">0.50</mn></mrow></math>, <math><mrow id="p6.2.m2.1.2" xref="p6.2.m2.1.2.cmml"><msub id="p6.2.m2.1.2.2" xref="p6.2.m2.1.2.2.cmml"><mi id="p6.2.m2.1.2.2.2" xref="p6.2.m2.1.2.2.2.cmml">H</mi><mi id="p6.2.m2.1.2.2.3" xref="p6.2.m2.1.2.2.3.cmml">c1</mi></msub><mo id="p6.2.m2.1.2.1" xref="p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="p6.2.m2.1.2.3.2" xref="p6.2.m2.1.2.cmml"><mo stretchy="false" id="p6.2.m2.1.2.3.2.1" 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xref="p6.7.m7.1.2.1.cmml">⁢</mo><mrow id="p6.7.m7.1.2.3.2" xref="p6.7.m7.1.2.cmml"><mo stretchy="false" id="p6.7.m7.1.2.3.2.1" xref="p6.7.m7.1.2.cmml">(</mo><mi id="p6.7.m7.1.1" xref="p6.7.m7.1.1.cmml">T</mi><mo stretchy="false" id="p6.7.m7.1.2.3.2.2" xref="p6.7.m7.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.15.m15.1.2" xref="p6.15.m15.1.2.cmml"><msup id="p6.15.m15.1.2.2" xref="p6.15.m15.1.2.2.cmml"><mi id="p6.15.m15.1.2.2.2" xref="p6.15.m15.1.2.2.2.cmml">λ</mi><mrow id="p6.15.m15.1.2.2.3" xref="p6.15.m15.1.2.2.3.cmml"><mo id="p6.15.m15.1.2.2.3.1" xref="p6.15.m15.1.2.2.3.1.cmml">-</mo><mn id="p6.15.m15.1.2.2.3.2" xref="p6.15.m15.1.2.2.3.2.cmml">2</mn></mrow></msup><mo id="p6.15.m15.1.2.1" xref="p6.15.m15.1.2.1.cmml">⁢</mo><mrow id="p6.15.m15.1.2.3.2" xref="p6.15.m15.1.2.cmml"><mo stretchy="false" id="p6.15.m15.1.2.3.2.1" xref="p6.15.m15.1.2.cmml">(</mo><mi id="p6.15.m15.1.1" xref="p6.15.m15.1.1.cmml">T</mi><mo stretchy="false" id="p6.15.m15.1.2.3.2.2" xref="p6.15.m15.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.18.m18.1.1" xref="p6.18.m18.1.1.cmml"><mrow id="p6.18.m18.1.1.2" xref="p6.18.m18.1.1.2.cmml"><mrow id="p6.18.m18.1.1.2.2" xref="p6.18.m18.1.1.2.2.cmml"><mrow id="p6.18.m18.1.1.2.2.2" xref="p6.18.m18.1.1.2.2.2.cmml"><mi mathvariant="normal" id="p6.18.m18.1.1.2.2.2.2" xref="p6.18.m18.1.1.2.2.2.2.cmml">d</mi><mo id="p6.18.m18.1.1.2.2.2.1" xref="p6.18.m18.1.1.2.2.2.1.cmml">⁢</mo><msub id="p6.18.m18.1.1.2.2.2.3" xref="p6.18.m18.1.1.2.2.2.3.cmml"><mi id="p6.18.m18.1.1.2.2.2.3.2" xref="p6.18.m18.1.1.2.2.2.3.2.cmml">H</mi><mi id="p6.18.m18.1.1.2.2.2.3.3" xref="p6.18.m18.1.1.2.2.2.3.3.cmml">c1</mi></msub></mrow><mo id="p6.18.m18.1.1.2.2.1" xref="p6.18.m18.1.1.2.2.1.cmml">/</mo><mi mathvariant="normal" id="p6.18.m18.1.1.2.2.3" xref="p6.18.m18.1.1.2.2.3.cmml">d</mi></mrow><mo id="p6.18.m18.1.1.2.1" xref="p6.18.m18.1.1.2.1.cmml">⁢</mo><mi id="p6.18.m18.1.1.2.3" xref="p6.18.m18.1.1.2.3.cmml">T</mi></mrow><mo id="p6.18.m18.1.1.1" xref="p6.18.m18.1.1.1.cmml">=</mo><mrow id="p6.18.m18.1.1.3" xref="p6.18.m18.1.1.3.cmml"><mo id="p6.18.m18.1.1.3.1" xref="p6.18.m18.1.1.3.1.cmml">-</mo><mn id="p6.18.m18.1.1.3.2" xref="p6.18.m18.1.1.3.2.cmml">38</mn></mrow></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/9804070

Formulas:

1-

L, L - q_{l}, L - 2 q_{l}, \dots \dots, q_{l}

2-

(L - 1) q_{l}

3-

F \propto \frac{1}{L - 1} - \frac{1}{L} (1)

4-

F = G (q_{l}) m_{1} m_{2} \frac{1}{L (L - q_{l})} (2)

5-

G (q_{l})

6-

\frac{d^{2} u}{d θ^{2}} + u = \frac{k^{2}}{h^{2}} (3)

7-

k^{2} = G M

8-

h = \frac{2 π a b}{τ}

9-

\frac{d^{2} u}{d θ^{2}} + (1 - \frac{q_{l} k^{2}}{h^{2}}) u = \frac{k^{2}}{h^{2}} (4)

10-

r = \frac{p}{1 + A p \cos x θ} (5)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/math/0211183

Formulas:

1-

v_{1}, \dots, v_{n}

2-

{G |}_{v_{1}, \dots, v_{k}}

3-

x_{1}, x_{2}, \dots

4-

a, a^{'} \in A

5-

b, b^{'} \in B

6-

\bar{a^{'} b^{'}}

7-

(a, b, b^{'})

8-

(b^{'}, a, a^{'})

9-

ϕ : [0, 1] \to B

10-

\bar{a ϕ (t)}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1401.5271

Formulas:

1-

θ_{L} = {(2 U_{0} / E)}^{1 / 2}

2-

U_{cr} (x)

3-

z = 2.5 Δ z

4-

Δ z = a / \tan θ_{m}

5-

(N - 1) a

6-

γ m \frac{d^{2} x}{d t^{2}} = - \frac{d U_{cr}}{d x},

7-

z (t) = v_{0} t

8-

\tan θ_{i} = v_{x i} (τ) / v_{0}

9-

v_{x i} (τ) = {(d x / d t)}_{τ}

10-

θ_{m} = 10 μ

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0009312

Formulas:

1-

3 - 9 M ☉

2-

3 - 9 M ☉

3-

{\dot{ρ}}_{*} (z)

4-

ℜ_{I a} (z)

5-

{\dot{ρ}}_{*} (z)

6-

ℜ_{I a} (z)

7-

{\dot{ρ}}_{*} (z)

8-

{\dot{ρ}}_{*} (z)

9-

ℜ_{I a} (z)

10-

{\dot{ρ}}_{*} (z)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0902.2519

Formulas:

1-

ψ (x) + \ln (e^{1 / x} - 1)

2-

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t

3-

ψ (x) = \frac{Γ^{'} (x)}{Γ (x)}

4-

ψ^{(k)} (x)

5-

a \leq - \ln 2

6-

a - \ln (e^{1 / x} - 1) < ψ (x) < b - \ln (e^{1 / x} - 1)

7-

- γ = - 0.577 \dots

8-

ϕ (x) = ψ (x) + \ln (e^{1 / x} - 1)

9-

lim_{x \to \infty} ϕ (x) = 0 .

10-

{lim}_{x \to 0^{+}} ϕ (x) = - γ

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1803.06182

Formulas:

1-

{\dot{M}}_{thick disk} - {\dot{M}}_{wind} - {\dot{M}}_{thin disk} = 0,

2-

F = τ_{NIR} L / c

3-

ρ \frac{d v}{d t} = - ρ \nabla Φ - \nabla p + \frac{1}{4 π} (\nabla \times B) \times B + \frac{L ρ κ}{4 π R^{2} c}

4-

\frac{1}{4 π} (\nabla \times B) \times B \sim \frac{1}{4 π} (B \cdot \nabla) B .

5-

N \sim 5 \times 10^{20}

6-

N_{wind} \sim 5 \times 10^{21}

7-

(\frac{L_{UV}}{4 π R^{2} c}) \times (\frac{2 C_{UV}}{1 - C_{IR}}) \sim ρ v_{rot}^{2},

8-

C_{UV} = 1 - \cos (70^{\circ})

9-

C_{IR} = 1 - \cos (30^{\circ})

10-

ρ = v_{rot}^{2} / (R^{2} π G Q)

Formulas (html):

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xref="S2.E2.m1.4.4.1.2.cmml">+</mo><mpadded width="+1.7pt" id="S2.E2.m1.4.4.1.4" xref="S2.E2.m1.4.4.1.4.cmml"><mfrac id="S2.E2.m1.4.4.1.4a" xref="S2.E2.m1.4.4.1.4.cmml"><mrow id="S2.E2.m1.4.4.1.4.2" xref="S2.E2.m1.4.4.1.4.2.cmml"><mi id="S2.E2.m1.4.4.1.4.2.2" xref="S2.E2.m1.4.4.1.4.2.2.cmml">L</mi><mo id="S2.E2.m1.4.4.1.4.2.1" xref="S2.E2.m1.4.4.1.4.2.1.cmml">⁢</mo><mi id="S2.E2.m1.4.4.1.4.2.3" xref="S2.E2.m1.4.4.1.4.2.3.cmml">ρ</mi><mo id="S2.E2.m1.4.4.1.4.2.1a" xref="S2.E2.m1.4.4.1.4.2.1.cmml">⁢</mo><mi id="S2.E2.m1.4.4.1.4.2.4" xref="S2.E2.m1.4.4.1.4.2.4.cmml">κ</mi></mrow><mrow id="S2.E2.m1.4.4.1.4.3" xref="S2.E2.m1.4.4.1.4.3.cmml"><mn id="S2.E2.m1.4.4.1.4.3.2" xref="S2.E2.m1.4.4.1.4.3.2.cmml">4</mn><mo id="S2.E2.m1.4.4.1.4.3.1" xref="S2.E2.m1.4.4.1.4.3.1.cmml">⁢</mo><mi id="S2.E2.m1.4.4.1.4.3.3" xref="S2.E2.m1.4.4.1.4.3.3.cmml">π</mi><mo id="S2.E2.m1.4.4.1.4.3.1a" xref="S2.E2.m1.4.4.1.4.3.1.cmml">⁢</mo><msup id="S2.E2.m1.4.4.1.4.3.4" xref="S2.E2.m1.4.4.1.4.3.4.cmml"><mi id="S2.E2.m1.4.4.1.4.3.4.2" xref="S2.E2.m1.4.4.1.4.3.4.2.cmml">R</mi><mn id="S2.E2.m1.4.4.1.4.3.4.3" xref="S2.E2.m1.4.4.1.4.3.4.3.cmml">2</mn></msup><mo id="S2.E2.m1.4.4.1.4.3.1b" xref="S2.E2.m1.4.4.1.4.3.1.cmml">⁢</mo><mi id="S2.E2.m1.4.4.1.4.3.5" xref="S2.E2.m1.4.4.1.4.3.5.cmml">c</mi></mrow></mfrac></mpadded></mrow></mrow></math>, <math><mrow id="S2.E3.m1.5.5.1" xref="S2.E3.m1.5.5.1.1.cmml"><mrow id="S2.E3.m1.5.5.1.1" xref="S2.E3.m1.5.5.1.1.cmml"><mrow id="S2.E3.m1.5.5.1.1.1" xref="S2.E3.m1.5.5.1.1.1.cmml"><mrow id="S2.E3.m1.5.5.1.1.1.1" xref="S2.E3.m1.5.5.1.1.1.1.cmml"><mfrac id="S2.E3.m1.5.5.1.1.1.1.3" xref="S2.E3.m1.5.5.1.1.1.1.3.cmml"><mn id="S2.E3.m1.5.5.1.1.1.1.3.2" xref="S2.E3.m1.5.5.1.1.1.1.3.2.cmml">1</mn><mrow id="S2.E3.m1.5.5.1.1.1.1.3.3" xref="S2.E3.m1.5.5.1.1.1.1.3.3.cmml"><mn id="S2.E3.m1.5.5.1.1.1.1.3.3.2" xref="S2.E3.m1.5.5.1.1.1.1.3.3.2.cmml">4</mn><mo id="S2.E3.m1.5.5.1.1.1.1.3.3.1" xref="S2.E3.m1.5.5.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.E3.m1.5.5.1.1.1.1.3.3.3" xref="S2.E3.m1.5.5.1.1.1.1.3.3.3.cmml">π</mi></mrow></mfrac><mo id="S2.E3.m1.5.5.1.1.1.1.2" xref="S2.E3.m1.5.5.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.5.5.1.1.1.1.1.1" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.5.5.1.1.1.1.1.1.2" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.5.5.1.1.1.1.1.1.1" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.cmml"><mo id="S2.E3.m1.5.5.1.1.1.1.1.1.1.2" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.2.cmml">∇</mo><mo id="S2.E3.m1.5.5.1.1.1.1.1.1.1.1" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.1.cmml">×</mo><mi id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">B</mi></mrow><mo stretchy="false" id="S2.E3.m1.5.5.1.1.1.1.1.1.3" xref="S2.E3.m1.5.5.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.5.5.1.1.1.2" xref="S2.E3.m1.5.5.1.1.1.2.cmml">×</mo><mi id="S2.E3.m1.2.2.1.1" xref="S2.E3.m1.2.2.1.1.cmml">B</mi></mrow><mo id="S2.E3.m1.5.5.1.1.3" xref="S2.E3.m1.5.5.1.1.3.cmml">∼</mo><mrow id="S2.E3.m1.5.5.1.1.2" xref="S2.E3.m1.5.5.1.1.2.cmml"><mfrac id="S2.E3.m1.5.5.1.1.2.3" xref="S2.E3.m1.5.5.1.1.2.3.cmml"><mn id="S2.E3.m1.5.5.1.1.2.3.2" xref="S2.E3.m1.5.5.1.1.2.3.2.cmml">1</mn><mrow id="S2.E3.m1.5.5.1.1.2.3.3" xref="S2.E3.m1.5.5.1.1.2.3.3.cmml"><mn id="S2.E3.m1.5.5.1.1.2.3.3.2" xref="S2.E3.m1.5.5.1.1.2.3.3.2.cmml">4</mn><mo id="S2.E3.m1.5.5.1.1.2.3.3.1" xref="S2.E3.m1.5.5.1.1.2.3.3.1.cmml">⁢</mo><mi id="S2.E3.m1.5.5.1.1.2.3.3.3" xref="S2.E3.m1.5.5.1.1.2.3.3.3.cmml">π</mi></mrow></mfrac><mo id="S2.E3.m1.5.5.1.1.2.2" xref="S2.E3.m1.5.5.1.1.2.2.cmml">⁢</mo><mrow id="S2.E3.m1.5.5.1.1.2.1.1" xref="S2.E3.m1.5.5.1.1.2.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.5.5.1.1.2.1.1.2" xref="S2.E3.m1.5.5.1.1.2.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.5.5.1.1.2.1.1.1" xref="S2.E3.m1.5.5.1.1.2.1.1.1.cmml"><mi id="S2.E3.m1.3.3.1.1" xref="S2.E3.m1.3.3.1.1.cmml">B</mi><mo id="S2.E3.m1.5.5.1.1.2.1.1.1.1" xref="S2.E3.m1.5.5.1.1.2.1.1.1.1.cmml">⋅</mo><mo id="S2.E3.m1.5.5.1.1.2.1.1.1.2" xref="S2.E3.m1.5.5.1.1.2.1.1.1.2.cmml">∇</mo></mrow><mo stretchy="false" id="S2.E3.m1.5.5.1.1.2.1.1.3" xref="S2.E3.m1.5.5.1.1.2.1.1.1.cmml">)</mo></mrow><mo id="S2.E3.m1.5.5.1.1.2.2a" xref="S2.E3.m1.5.5.1.1.2.2.cmml">⁢</mo><mpadded width="+5pt" id="S2.E3.m1.4.4" xref="S2.E3.m1.4.4.2b.cmml"><mi id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml">B</mi></mpadded></mrow></mrow><mo id="S2.E3.m1.5.5.1.2" xref="S2.E3.m1.5.5.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.SS1.p8.1.m1.1.1" xref="S2.SS1.p8.1.m1.1.1.cmml"><mi id="S2.SS1.p8.1.m1.1.1.2" xref="S2.SS1.p8.1.m1.1.1.2.cmml">N</mi><mo id="S2.SS1.p8.1.m1.1.1.1" xref="S2.SS1.p8.1.m1.1.1.1.cmml">∼</mo><mrow id="S2.SS1.p8.1.m1.1.1.3" xref="S2.SS1.p8.1.m1.1.1.3.cmml"><mn id="S2.SS1.p8.1.m1.1.1.3.2" xref="S2.SS1.p8.1.m1.1.1.3.2.cmml">5</mn><mo id="S2.SS1.p8.1.m1.1.1.3.1" xref="S2.SS1.p8.1.m1.1.1.3.1.cmml">×</mo><msup id="S2.SS1.p8.1.m1.1.1.3.3" xref="S2.SS1.p8.1.m1.1.1.3.3.cmml"><mn id="S2.SS1.p8.1.m1.1.1.3.3.2" xref="S2.SS1.p8.1.m1.1.1.3.3.2.cmml">10</mn><mn id="S2.SS1.p8.1.m1.1.1.3.3.3" xref="S2.SS1.p8.1.m1.1.1.3.3.3.cmml">20</mn></msup></mrow></mrow></math>, <math><mrow id="S2.SS1.p8.6.m6.1.1" xref="S2.SS1.p8.6.m6.1.1.cmml"><msub id="S2.SS1.p8.6.m6.1.1.2" xref="S2.SS1.p8.6.m6.1.1.2.cmml"><mi id="S2.SS1.p8.6.m6.1.1.2.2" xref="S2.SS1.p8.6.m6.1.1.2.2.cmml">N</mi><mi id="S2.SS1.p8.6.m6.1.1.2.3" xref="S2.SS1.p8.6.m6.1.1.2.3.cmml">wind</mi></msub><mo id="S2.SS1.p8.6.m6.1.1.1" xref="S2.SS1.p8.6.m6.1.1.1.cmml">∼</mo><mrow id="S2.SS1.p8.6.m6.1.1.3" xref="S2.SS1.p8.6.m6.1.1.3.cmml"><mn id="S2.SS1.p8.6.m6.1.1.3.2" xref="S2.SS1.p8.6.m6.1.1.3.2.cmml">5</mn><mo id="S2.SS1.p8.6.m6.1.1.3.1" xref="S2.SS1.p8.6.m6.1.1.3.1.cmml">×</mo><msup id="S2.SS1.p8.6.m6.1.1.3.3" xref="S2.SS1.p8.6.m6.1.1.3.3.cmml"><mn id="S2.SS1.p8.6.m6.1.1.3.3.2" xref="S2.SS1.p8.6.m6.1.1.3.3.2.cmml">10</mn><mn id="S2.SS1.p8.6.m6.1.1.3.3.3" xref="S2.SS1.p8.6.m6.1.1.3.3.3.cmml">21</mn></msup></mrow></mrow></math>, <math><mrow id="S2.E4.m1.3.3.1" xref="S2.E4.m1.3.3.1.1.cmml"><mrow id="S2.E4.m1.3.3.1.1" xref="S2.E4.m1.3.3.1.1.cmml"><mrow id="S2.E4.m1.3.3.1.1.2" xref="S2.E4.m1.3.3.1.1.2.cmml"><mrow id="S2.E4.m1.3.3.1.1.2.2.2" xref="S2.E4.m1.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.3.3.1.1.2.2.2.1" xref="S2.E4.m1.1.1.cmml">(</mo><mfrac id="S2.E4.m1.1.1" xref="S2.E4.m1.1.1.cmml"><msub id="S2.E4.m1.1.1.2" xref="S2.E4.m1.1.1.2.cmml"><mi id="S2.E4.m1.1.1.2.2" xref="S2.E4.m1.1.1.2.2.cmml">L</mi><mi id="S2.E4.m1.1.1.2.3" xref="S2.E4.m1.1.1.2.3.cmml">UV</mi></msub><mrow id="S2.E4.m1.1.1.3" xref="S2.E4.m1.1.1.3.cmml"><mn id="S2.E4.m1.1.1.3.2" xref="S2.E4.m1.1.1.3.2.cmml">4</mn><mo id="S2.E4.m1.1.1.3.1" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E4.m1.1.1.3.3" xref="S2.E4.m1.1.1.3.3.cmml">π</mi><mo id="S2.E4.m1.1.1.3.1a" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><msup id="S2.E4.m1.1.1.3.4" xref="S2.E4.m1.1.1.3.4.cmml"><mi id="S2.E4.m1.1.1.3.4.2" xref="S2.E4.m1.1.1.3.4.2.cmml">R</mi><mn id="S2.E4.m1.1.1.3.4.3" xref="S2.E4.m1.1.1.3.4.3.cmml">2</mn></msup><mo id="S2.E4.m1.1.1.3.1b" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E4.m1.1.1.3.5" xref="S2.E4.m1.1.1.3.5.cmml">c</mi></mrow></mfrac><mo stretchy="false" id="S2.E4.m1.3.3.1.1.2.2.2.2" xref="S2.E4.m1.1.1.cmml">)</mo></mrow><mo id="S2.E4.m1.3.3.1.1.2.1" xref="S2.E4.m1.3.3.1.1.2.1.cmml">×</mo><mrow id="S2.E4.m1.3.3.1.1.2.3.2" xref="S2.E4.m1.2.2.cmml"><mo stretchy="false" id="S2.E4.m1.3.3.1.1.2.3.2.1" xref="S2.E4.m1.2.2.cmml">(</mo><mfrac id="S2.E4.m1.2.2" xref="S2.E4.m1.2.2.cmml"><mrow id="S2.E4.m1.2.2.2" xref="S2.E4.m1.2.2.2.cmml"><mn id="S2.E4.m1.2.2.2.2" xref="S2.E4.m1.2.2.2.2.cmml">2</mn><mo id="S2.E4.m1.2.2.2.1" xref="S2.E4.m1.2.2.2.1.cmml">⁢</mo><msub id="S2.E4.m1.2.2.2.3" xref="S2.E4.m1.2.2.2.3.cmml"><mi id="S2.E4.m1.2.2.2.3.2" xref="S2.E4.m1.2.2.2.3.2.cmml">C</mi><mi id="S2.E4.m1.2.2.2.3.3" xref="S2.E4.m1.2.2.2.3.3.cmml">UV</mi></msub></mrow><mrow id="S2.E4.m1.2.2.3" xref="S2.E4.m1.2.2.3.cmml"><mn id="S2.E4.m1.2.2.3.2" xref="S2.E4.m1.2.2.3.2.cmml">1</mn><mo id="S2.E4.m1.2.2.3.1" xref="S2.E4.m1.2.2.3.1.cmml">-</mo><msub id="S2.E4.m1.2.2.3.3" xref="S2.E4.m1.2.2.3.3.cmml"><mi id="S2.E4.m1.2.2.3.3.2" xref="S2.E4.m1.2.2.3.3.2.cmml">C</mi><mi id="S2.E4.m1.2.2.3.3.3" xref="S2.E4.m1.2.2.3.3.3.cmml">IR</mi></msub></mrow></mfrac><mo stretchy="false" id="S2.E4.m1.3.3.1.1.2.3.2.2" xref="S2.E4.m1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E4.m1.3.3.1.1.1" xref="S2.E4.m1.3.3.1.1.1.cmml">∼</mo><mrow id="S2.E4.m1.3.3.1.1.3" xref="S2.E4.m1.3.3.1.1.3.cmml"><mi id="S2.E4.m1.3.3.1.1.3.2" xref="S2.E4.m1.3.3.1.1.3.2.cmml">ρ</mi><mo id="S2.E4.m1.3.3.1.1.3.1" xref="S2.E4.m1.3.3.1.1.3.1.cmml">⁢</mo><mpadded width="+5pt" id="S2.E4.m1.3.3.1.1.3.3" xref="S2.E4.m1.3.3.1.1.3.3.cmml"><msubsup id="S2.E4.m1.3.3.1.1.3.3a" xref="S2.E4.m1.3.3.1.1.3.3.cmml"><mi id="S2.E4.m1.3.3.1.1.3.3.2.2" xref="S2.E4.m1.3.3.1.1.3.3.2.2.cmml">v</mi><mi id="S2.E4.m1.3.3.1.1.3.3.2.3" xref="S2.E4.m1.3.3.1.1.3.3.2.3.cmml">rot</mi><mn id="S2.E4.m1.3.3.1.1.3.3.3" xref="S2.E4.m1.3.3.1.1.3.3.3.cmml">2</mn></msubsup></mpadded></mrow></mrow><mo id="S2.E4.m1.3.3.1.2" xref="S2.E4.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p9.5.m5.2.2" xref="S2.SS1.p9.5.m5.2.2.cmml"><msub id="S2.SS1.p9.5.m5.2.2.3" xref="S2.SS1.p9.5.m5.2.2.3.cmml"><mi id="S2.SS1.p9.5.m5.2.2.3.2" xref="S2.SS1.p9.5.m5.2.2.3.2.cmml">C</mi><mi id="S2.SS1.p9.5.m5.2.2.3.3" xref="S2.SS1.p9.5.m5.2.2.3.3.cmml">UV</mi></msub><mo id="S2.SS1.p9.5.m5.2.2.2" xref="S2.SS1.p9.5.m5.2.2.2.cmml">=</mo><mrow id="S2.SS1.p9.5.m5.2.2.1" xref="S2.SS1.p9.5.m5.2.2.1.cmml"><mn id="S2.SS1.p9.5.m5.2.2.1.3" xref="S2.SS1.p9.5.m5.2.2.1.3.cmml">1</mn><mo id="S2.SS1.p9.5.m5.2.2.1.2" xref="S2.SS1.p9.5.m5.2.2.1.2.cmml">-</mo><mrow id="S2.SS1.p9.5.m5.2.2.1.1.1" xref="S2.SS1.p9.5.m5.2.2.1.1.2.cmml"><mi id="S2.SS1.p9.5.m5.1.1" xref="S2.SS1.p9.5.m5.1.1.cmml">cos</mi><mo id="S2.SS1.p9.5.m5.2.2.1.1.1a" xref="S2.SS1.p9.5.m5.2.2.1.1.2.cmml">⁡</mo><mrow id="S2.SS1.p9.5.m5.2.2.1.1.1.1" xref="S2.SS1.p9.5.m5.2.2.1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p9.5.m5.2.2.1.1.1.1.2" xref="S2.SS1.p9.5.m5.2.2.1.1.2.cmml">(</mo><msup id="S2.SS1.p9.5.m5.2.2.1.1.1.1.1" xref="S2.SS1.p9.5.m5.2.2.1.1.1.1.1.cmml"><mn id="S2.SS1.p9.5.m5.2.2.1.1.1.1.1.2" xref="S2.SS1.p9.5.m5.2.2.1.1.1.1.1.2.cmml">70</mn><mo id="S2.SS1.p9.5.m5.2.2.1.1.1.1.1.3" xref="S2.SS1.p9.5.m5.2.2.1.1.1.1.1.3.cmml">∘</mo></msup><mo stretchy="false" id="S2.SS1.p9.5.m5.2.2.1.1.1.1.3" xref="S2.SS1.p9.5.m5.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p9.6.m6.2.2" xref="S2.SS1.p9.6.m6.2.2.cmml"><msub id="S2.SS1.p9.6.m6.2.2.3" xref="S2.SS1.p9.6.m6.2.2.3.cmml"><mi id="S2.SS1.p9.6.m6.2.2.3.2" xref="S2.SS1.p9.6.m6.2.2.3.2.cmml">C</mi><mi id="S2.SS1.p9.6.m6.2.2.3.3" xref="S2.SS1.p9.6.m6.2.2.3.3.cmml">IR</mi></msub><mo id="S2.SS1.p9.6.m6.2.2.2" xref="S2.SS1.p9.6.m6.2.2.2.cmml">=</mo><mrow id="S2.SS1.p9.6.m6.2.2.1" xref="S2.SS1.p9.6.m6.2.2.1.cmml"><mn id="S2.SS1.p9.6.m6.2.2.1.3" xref="S2.SS1.p9.6.m6.2.2.1.3.cmml">1</mn><mo id="S2.SS1.p9.6.m6.2.2.1.2" xref="S2.SS1.p9.6.m6.2.2.1.2.cmml">-</mo><mrow id="S2.SS1.p9.6.m6.2.2.1.1.1" xref="S2.SS1.p9.6.m6.2.2.1.1.2.cmml"><mi id="S2.SS1.p9.6.m6.1.1" xref="S2.SS1.p9.6.m6.1.1.cmml">cos</mi><mo id="S2.SS1.p9.6.m6.2.2.1.1.1a" xref="S2.SS1.p9.6.m6.2.2.1.1.2.cmml">⁡</mo><mrow id="S2.SS1.p9.6.m6.2.2.1.1.1.1" xref="S2.SS1.p9.6.m6.2.2.1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p9.6.m6.2.2.1.1.1.1.2" xref="S2.SS1.p9.6.m6.2.2.1.1.2.cmml">(</mo><msup id="S2.SS1.p9.6.m6.2.2.1.1.1.1.1" xref="S2.SS1.p9.6.m6.2.2.1.1.1.1.1.cmml"><mn id="S2.SS1.p9.6.m6.2.2.1.1.1.1.1.2" xref="S2.SS1.p9.6.m6.2.2.1.1.1.1.1.2.cmml">30</mn><mo id="S2.SS1.p9.6.m6.2.2.1.1.1.1.1.3" xref="S2.SS1.p9.6.m6.2.2.1.1.1.1.1.3.cmml">∘</mo></msup><mo stretchy="false" id="S2.SS1.p9.6.m6.2.2.1.1.1.1.3" xref="S2.SS1.p9.6.m6.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p9.7.m7.1.1" xref="S2.SS1.p9.7.m7.1.1.cmml"><mi id="S2.SS1.p9.7.m7.1.1.3" xref="S2.SS1.p9.7.m7.1.1.3.cmml">ρ</mi><mo id="S2.SS1.p9.7.m7.1.1.2" xref="S2.SS1.p9.7.m7.1.1.2.cmml">=</mo><mrow id="S2.SS1.p9.7.m7.1.1.1" xref="S2.SS1.p9.7.m7.1.1.1.cmml"><msubsup id="S2.SS1.p9.7.m7.1.1.1.3" xref="S2.SS1.p9.7.m7.1.1.1.3.cmml"><mi id="S2.SS1.p9.7.m7.1.1.1.3.2.2" xref="S2.SS1.p9.7.m7.1.1.1.3.2.2.cmml">v</mi><mi id="S2.SS1.p9.7.m7.1.1.1.3.2.3" xref="S2.SS1.p9.7.m7.1.1.1.3.2.3.cmml">rot</mi><mn id="S2.SS1.p9.7.m7.1.1.1.3.3" xref="S2.SS1.p9.7.m7.1.1.1.3.3.cmml">2</mn></msubsup><mo id="S2.SS1.p9.7.m7.1.1.1.2" xref="S2.SS1.p9.7.m7.1.1.1.2.cmml">/</mo><mrow id="S2.SS1.p9.7.m7.1.1.1.1.1" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p9.7.m7.1.1.1.1.1.2" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p9.7.m7.1.1.1.1.1.1" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.cmml"><msup id="S2.SS1.p9.7.m7.1.1.1.1.1.1.2" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p9.7.m7.1.1.1.1.1.1.2.2" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.2.2.cmml">R</mi><mn id="S2.SS1.p9.7.m7.1.1.1.1.1.1.2.3" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.SS1.p9.7.m7.1.1.1.1.1.1.1" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p9.7.m7.1.1.1.1.1.1.3" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.3.cmml">π</mi><mo id="S2.SS1.p9.7.m7.1.1.1.1.1.1.1a" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p9.7.m7.1.1.1.1.1.1.4" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.4.cmml">G</mi><mo id="S2.SS1.p9.7.m7.1.1.1.1.1.1.1b" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p9.7.m7.1.1.1.1.1.1.5" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.5.cmml">Q</mi></mrow><mo stretchy="false" id="S2.SS1.p9.7.m7.1.1.1.1.1.3" xref="S2.SS1.p9.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1403.8052

Formulas:

1-

Z_{S^{3}} (m)

2-

{| Z_{S^{3}} (m) |}^{2}

3-

F (m) = - \log | Z_{S^{3}} (m) |

4-

\frac{{| Z_{H S^{4}} |}^{2}}{Z_{S_{4}}} = e^{a_{3} \frac{R^{3}}{ϵ^{3}} + a_{1} \frac{R}{ϵ} - 2 F_{\partial}}

5-

S U {(2)}_{R}

6-

F_{\partial} (m)

7-

\frac{{| Z_{S^{4}}^{A B} |}^{2}}{Z_{S_{4}}^{A} Z_{S_{4}}^{B}} = e^{a_{3} \frac{R^{3}}{ϵ^{3}} + a_{1} \frac{R}{ϵ} - 2 F_{\partial}}

8-

F = - \log Z = \frac{1}{2} \sum_{n} d_{n} \log \frac{λ_{n}}{R^{2} μ^{2}}

9-

f (s) = \frac{1}{2} \sum_{n} \frac{d_{n}}{λ_{n}^{s}}

10-

F = {[- \partial_{s} f (s) - f (s) \log R^{2} μ^{2}] |}_{s = 0}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/nucl-ex/0611019

Formulas:

1-

N_{p a r t}

2-

Y_{s a m e} (Δ ϕ)

3-

2.5 < p_{T} < 4

4-

1 < p_{T} < 2.5

5-

Y_{m i x e d} (Δ ϕ)

6-

C (Δ ϕ) \equiv \frac{Y_{s a m e} (Δ ϕ)}{Y_{m i x e d} (Δ ϕ)} \times \frac{\int Y_{m i x e d} (Δ ϕ) 𝑑 Δ ϕ}{\int Y_{s a m e} (Δ ϕ) 𝑑 Δ ϕ}

7-

1 + 2 v_{2} c o s (2 (ϕ - Φ_{R P}))

8-

J (Δ ϕ)

9-

C (Δ ϕ) = b_{0} (1 + 2 ⟨ v_{2}^{a s s o c} ⟩ ⟨ v_{2}^{t r i g g} ⟩ \cos (2 Δ ϕ)) + J (Δ ϕ)

10-

3 < | η | < 4

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: nucl-ex

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0211248

Formulas:

1-

d Δ σ \equiv \frac{1}{2} [d σ^{+ +} - d σ^{+ -}],

2-

p p \to π X

3-

\sum_{a, b, c} \int 𝑑 x_{a} \int 𝑑 x_{b} \int 𝑑 z_{c} Δ f_{a} (x_{a}, μ_{F}) Δ f_{b} (x_{b}, μ_{F}) D_{c}^{π} (z_{c}, μ_{F}^{'})

4-

\times d Δ {\hat{σ}}_{a b}^{c} (x_{a} P_{A}, x_{b} P_{B}, P_{π} / z_{c}, μ_{R}, μ_{F}, μ_{F}^{'}),

5-

a + b \to c + X

6-

d Δ {\hat{σ}}_{a b}^{c}

7-

Δ f_{a, b}

8-

A_{L L}^{π} = \frac{d Δ σ}{d σ} = \frac{d σ^{+ +} - d σ^{+ -}}{d σ^{+ +} + d σ^{+ -}}

9-

p p \to π^{0} X

10-

d Δ {\hat{σ}}_{a b}^{c}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1110.6214

Formulas:

1-

ℋ^{I} = 𝟏_{I} ℋ 𝟏_{I}

2-

ℤ {[q_{s}]}_{s \in S}

3-

(G, P_{I})

4-

{(q_{s})}_{s \in S}

5-

ℤ {[q_{s}]}_{s \in S}

6-

{T_{w} ∣ w \in W}

7-

W_{I} = ⟨ {s ∣ s \in I} ⟩

8-

ℋ^{I} = 𝟏_{I} ℋ 𝟏_{I}, where 𝟏_{I} = \sum_{w \in W_{I}} T_{w} .

9-

| S \ I | = 1

10-

| S \ I | = 1

Formulas (html):

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<math><mrow id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml"><mi id="id6.6.m6.1.1.3" xref="id6.6.m6.1.1.3.cmml">ℤ</mi><mo id="id6.6.m6.1.1.2" xref="id6.6.m6.1.1.2.cmml">⁢</mo><msub id="id6.6.m6.1.1.1" xref="id6.6.m6.1.1.1.cmml"><mrow id="id6.6.m6.1.1.1.1.1" xref="id6.6.m6.1.1.1.1.2.cmml"><mo stretchy="false" id="id6.6.m6.1.1.1.1.1.2" xref="id6.6.m6.1.1.1.1.2.1.cmml">[</mo><msub id="id6.6.m6.1.1.1.1.1.1" xref="id6.6.m6.1.1.1.1.1.1.cmml"><mi id="id6.6.m6.1.1.1.1.1.1.2" xref="id6.6.m6.1.1.1.1.1.1.2.cmml">q</mi><mi id="id6.6.m6.1.1.1.1.1.1.3" xref="id6.6.m6.1.1.1.1.1.1.3.cmml">s</mi></msub><mo stretchy="false" id="id6.6.m6.1.1.1.1.1.3" xref="id6.6.m6.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="id6.6.m6.1.1.1.3" xref="id6.6.m6.1.1.1.3.cmml"><mi id="id6.6.m6.1.1.1.3.2" xref="id6.6.m6.1.1.1.3.2.cmml">s</mi><mo id="id6.6.m6.1.1.1.3.1" xref="id6.6.m6.1.1.1.3.1.cmml">∈</mo><mi id="id6.6.m6.1.1.1.3.3" xref="id6.6.m6.1.1.1.3.3.cmml">S</mi></mrow></msub></mrow></math>, <math><mrow id="Sx1.p1.8.m8.2.2.1" xref="Sx1.p1.8.m8.2.2.2.cmml"><mo stretchy="false" id="Sx1.p1.8.m8.2.2.1.2" xref="Sx1.p1.8.m8.2.2.2.cmml">(</mo><mi id="Sx1.p1.8.m8.1.1" xref="Sx1.p1.8.m8.1.1.cmml">G</mi><mo id="Sx1.p1.8.m8.2.2.1.3" xref="Sx1.p1.8.m8.2.2.2.cmml">,</mo><msub id="Sx1.p1.8.m8.2.2.1.1" xref="Sx1.p1.8.m8.2.2.1.1.cmml"><mi id="Sx1.p1.8.m8.2.2.1.1.2" xref="Sx1.p1.8.m8.2.2.1.1.2.cmml">P</mi><mi id="Sx1.p1.8.m8.2.2.1.1.3" xref="Sx1.p1.8.m8.2.2.1.1.3.cmml">I</mi></msub><mo stretchy="false" id="Sx1.p1.8.m8.2.2.1.4" xref="Sx1.p1.8.m8.2.2.2.cmml">)</mo></mrow></math>, <math><msub id="Sx1.p2.2.m2.1.1" xref="Sx1.p2.2.m2.1.1.cmml"><mrow id="Sx1.p2.2.m2.1.1.1.1" xref="Sx1.p2.2.m2.1.1.1.1.1.cmml"><mo stretchy="false" id="Sx1.p2.2.m2.1.1.1.1.2" xref="Sx1.p2.2.m2.1.1.1.1.1.cmml">(</mo><msub id="Sx1.p2.2.m2.1.1.1.1.1" xref="Sx1.p2.2.m2.1.1.1.1.1.cmml"><mi id="Sx1.p2.2.m2.1.1.1.1.1.2" xref="Sx1.p2.2.m2.1.1.1.1.1.2.cmml">q</mi><mi id="Sx1.p2.2.m2.1.1.1.1.1.3" xref="Sx1.p2.2.m2.1.1.1.1.1.3.cmml">s</mi></msub><mo stretchy="false" id="Sx1.p2.2.m2.1.1.1.1.3" xref="Sx1.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="Sx1.p2.2.m2.1.1.3" xref="Sx1.p2.2.m2.1.1.3.cmml"><mi id="Sx1.p2.2.m2.1.1.3.2" xref="Sx1.p2.2.m2.1.1.3.2.cmml">s</mi><mo id="Sx1.p2.2.m2.1.1.3.1" xref="Sx1.p2.2.m2.1.1.3.1.cmml">∈</mo><mi id="Sx1.p2.2.m2.1.1.3.3" xref="Sx1.p2.2.m2.1.1.3.3.cmml">S</mi></mrow></msub></math>, <math><mrow id="Sx1.p2.7.m7.1.1" xref="Sx1.p2.7.m7.1.1.cmml"><mi id="Sx1.p2.7.m7.1.1.3" xref="Sx1.p2.7.m7.1.1.3.cmml">ℤ</mi><mo id="Sx1.p2.7.m7.1.1.2" xref="Sx1.p2.7.m7.1.1.2.cmml">⁢</mo><msub id="Sx1.p2.7.m7.1.1.1" xref="Sx1.p2.7.m7.1.1.1.cmml"><mrow id="Sx1.p2.7.m7.1.1.1.1.1" xref="Sx1.p2.7.m7.1.1.1.1.2.cmml"><mo stretchy="false" id="Sx1.p2.7.m7.1.1.1.1.1.2" xref="Sx1.p2.7.m7.1.1.1.1.2.1.cmml">[</mo><msub id="Sx1.p2.7.m7.1.1.1.1.1.1" xref="Sx1.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="Sx1.p2.7.m7.1.1.1.1.1.1.2" xref="Sx1.p2.7.m7.1.1.1.1.1.1.2.cmml">q</mi><mi id="Sx1.p2.7.m7.1.1.1.1.1.1.3" xref="Sx1.p2.7.m7.1.1.1.1.1.1.3.cmml">s</mi></msub><mo stretchy="false" id="Sx1.p2.7.m7.1.1.1.1.1.3" xref="Sx1.p2.7.m7.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="Sx1.p2.7.m7.1.1.1.3" xref="Sx1.p2.7.m7.1.1.1.3.cmml"><mi id="Sx1.p2.7.m7.1.1.1.3.2" xref="Sx1.p2.7.m7.1.1.1.3.2.cmml">s</mi><mo id="Sx1.p2.7.m7.1.1.1.3.1" xref="Sx1.p2.7.m7.1.1.1.3.1.cmml">∈</mo><mi id="Sx1.p2.7.m7.1.1.1.3.3" xref="Sx1.p2.7.m7.1.1.1.3.3.cmml">S</mi></mrow></msub></mrow></math>, <math><mrow id="Sx1.p2.9.m9.2.2.2" xref="Sx1.p2.9.m9.2.2.3.cmml"><mo stretchy="false" id="Sx1.p2.9.m9.2.2.2.3" xref="Sx1.p2.9.m9.2.2.3.1.cmml">{</mo><msub id="Sx1.p2.9.m9.1.1.1.1" xref="Sx1.p2.9.m9.1.1.1.1.cmml"><mi id="Sx1.p2.9.m9.1.1.1.1.2" xref="Sx1.p2.9.m9.1.1.1.1.2.cmml">T</mi><mi id="Sx1.p2.9.m9.1.1.1.1.3" xref="Sx1.p2.9.m9.1.1.1.1.3.cmml">w</mi></msub><mo id="Sx1.p2.9.m9.2.2.2.4" xref="Sx1.p2.9.m9.2.2.3.1.cmml">∣</mo><mrow id="Sx1.p2.9.m9.2.2.2.2" xref="Sx1.p2.9.m9.2.2.2.2.cmml"><mi id="Sx1.p2.9.m9.2.2.2.2.2" xref="Sx1.p2.9.m9.2.2.2.2.2.cmml">w</mi><mo id="Sx1.p2.9.m9.2.2.2.2.1" xref="Sx1.p2.9.m9.2.2.2.2.1.cmml">∈</mo><mi id="Sx1.p2.9.m9.2.2.2.2.3" xref="Sx1.p2.9.m9.2.2.2.2.3.cmml">W</mi></mrow><mo stretchy="false" id="Sx1.p2.9.m9.2.2.2.5" xref="Sx1.p2.9.m9.2.2.3.1.cmml">}</mo></mrow></math>, <math><mrow id="Sx1.p2.11.m11.2.2" xref="Sx1.p2.11.m11.2.2.cmml"><msub id="Sx1.p2.11.m11.2.2.3" xref="Sx1.p2.11.m11.2.2.3.cmml"><mi id="Sx1.p2.11.m11.2.2.3.2" xref="Sx1.p2.11.m11.2.2.3.2.cmml">W</mi><mi id="Sx1.p2.11.m11.2.2.3.3" xref="Sx1.p2.11.m11.2.2.3.3.cmml">I</mi></msub><mo id="Sx1.p2.11.m11.2.2.2" xref="Sx1.p2.11.m11.2.2.2.cmml">=</mo><mrow id="Sx1.p2.11.m11.2.2.1.1" xref="Sx1.p2.11.m11.2.2.1.2.cmml"><mo stretchy="false" id="Sx1.p2.11.m11.2.2.1.1.2" xref="Sx1.p2.11.m11.2.2.1.2.1.cmml">⟨</mo><mrow id="Sx1.p2.11.m11.2.2.1.1.1.1" xref="Sx1.p2.11.m11.2.2.1.1.1.2.cmml"><mo stretchy="false" id="Sx1.p2.11.m11.2.2.1.1.1.1.2" xref="Sx1.p2.11.m11.2.2.1.1.1.2.1.cmml">{</mo><mi id="Sx1.p2.11.m11.1.1" xref="Sx1.p2.11.m11.1.1.cmml">s</mi><mo id="Sx1.p2.11.m11.2.2.1.1.1.1.3" xref="Sx1.p2.11.m11.2.2.1.1.1.2.1.cmml">∣</mo><mrow id="Sx1.p2.11.m11.2.2.1.1.1.1.1" xref="Sx1.p2.11.m11.2.2.1.1.1.1.1.cmml"><mi id="Sx1.p2.11.m11.2.2.1.1.1.1.1.2" xref="Sx1.p2.11.m11.2.2.1.1.1.1.1.2.cmml">s</mi><mo id="Sx1.p2.11.m11.2.2.1.1.1.1.1.1" xref="Sx1.p2.11.m11.2.2.1.1.1.1.1.1.cmml">∈</mo><mi id="Sx1.p2.11.m11.2.2.1.1.1.1.1.3" xref="Sx1.p2.11.m11.2.2.1.1.1.1.1.3.cmml">I</mi></mrow><mo stretchy="false" id="Sx1.p2.11.m11.2.2.1.1.1.1.4" xref="Sx1.p2.11.m11.2.2.1.1.1.2.1.cmml">}</mo></mrow><mo stretchy="false" id="Sx1.p2.11.m11.2.2.1.1.3" xref="Sx1.p2.11.m11.2.2.1.2.1.cmml">⟩</mo></mrow></mrow></math>, <math><mrow id="Sx1.Ex1.m1.2.2.1"><mrow id="Sx1.Ex1.m1.2.2.1.1.2" xref="Sx1.Ex1.m1.2.2.1.1.3.cmml"><mrow id="Sx1.Ex1.m1.2.2.1.1.1.1" xref="Sx1.Ex1.m1.2.2.1.1.1.1.cmml"><msup id="Sx1.Ex1.m1.2.2.1.1.1.1.3" xref="Sx1.Ex1.m1.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathscript" id="Sx1.Ex1.m1.2.2.1.1.1.1.3.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.3.2.cmml">ℋ</mi><mi id="Sx1.Ex1.m1.2.2.1.1.1.1.3.3" xref="Sx1.Ex1.m1.2.2.1.1.1.1.3.3.cmml">I</mi></msup><mo id="Sx1.Ex1.m1.2.2.1.1.1.1.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.2.cmml">=</mo><mrow id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.2.cmml"><mrow id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.cmml"><msub id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mn id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">𝟏</mn><mi id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.3" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">I</mi></msub><mo id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.1" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathscript" id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.3" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.3.cmml">ℋ</mi><mo id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.1a" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4.cmml"><mn id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4.2.cmml">𝟏</mn><mi id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4.3" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.1.4.3.cmml">I</mi></msub></mrow><mo rspace="22.5pt" id="Sx1.Ex1.m1.2.2.1.1.1.1.1.1.2" xref="Sx1.Ex1.m1.2.2.1.1.1.1.1.2.cmml">,</mo><mtext id="Sx1.Ex1.m1.1.1" xref="Sx1.Ex1.m1.1.1a.cmml">where</mtext></mrow></mrow><mo mathvariant="italic" separator="true" id="Sx1.Ex1.m1.2.2.1.1.2.3" xref="Sx1.Ex1.m1.2.2.1.1.3a.cmml">  </mo><mrow id="Sx1.Ex1.m1.2.2.1.1.2.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.cmml"><msub id="Sx1.Ex1.m1.2.2.1.1.2.2.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.2.cmml"><mn id="Sx1.Ex1.m1.2.2.1.1.2.2.2.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.2.2.cmml">𝟏</mn><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.2.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.2.3.cmml">I</mi></msub><mo id="Sx1.Ex1.m1.2.2.1.1.2.2.1" xref="Sx1.Ex1.m1.2.2.1.1.2.2.1.cmml">=</mo><mrow id="Sx1.Ex1.m1.2.2.1.1.2.2.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.cmml"><munder id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.2.cmml">∑</mo><mrow id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.cmml"><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.2.cmml">w</mi><mo id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.1" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.1.cmml">∈</mo><msub id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3.cmml"><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3.2.cmml">W</mi><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.1.3.3.3.cmml">I</mi></msub></mrow></munder><msub id="Sx1.Ex1.m1.2.2.1.1.2.2.3.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.2.cmml"><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.3.2.2" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.2.2.cmml">T</mi><mi id="Sx1.Ex1.m1.2.2.1.1.2.2.3.2.3" xref="Sx1.Ex1.m1.2.2.1.1.2.2.3.2.3.cmml">w</mi></msub></mrow></mrow></mrow><mo id="Sx1.Ex1.m1.2.2.1.2">.</mo></mrow></math>, <math><mrow id="Sx1.p3.3.m3.1.1" xref="Sx1.p3.3.m3.1.1.cmml"><mrow id="Sx1.p3.3.m3.1.1.1.1" xref="Sx1.p3.3.m3.1.1.1.2.cmml"><mo stretchy="false" id="Sx1.p3.3.m3.1.1.1.1.2" xref="Sx1.p3.3.m3.1.1.1.2.1.cmml">|</mo><mrow id="Sx1.p3.3.m3.1.1.1.1.1" xref="Sx1.p3.3.m3.1.1.1.1.1.cmml"><mi id="Sx1.p3.3.m3.1.1.1.1.1.2" xref="Sx1.p3.3.m3.1.1.1.1.1.2.cmml">S</mi><mo id="Sx1.p3.3.m3.1.1.1.1.1.1" xref="Sx1.p3.3.m3.1.1.1.1.1.1.cmml">\</mo><mi id="Sx1.p3.3.m3.1.1.1.1.1.3" xref="Sx1.p3.3.m3.1.1.1.1.1.3.cmml">I</mi></mrow><mo stretchy="false" id="Sx1.p3.3.m3.1.1.1.1.3" xref="Sx1.p3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="Sx1.p3.3.m3.1.1.2" xref="Sx1.p3.3.m3.1.1.2.cmml">=</mo><mn id="Sx1.p3.3.m3.1.1.3" xref="Sx1.p3.3.m3.1.1.3.cmml">1</mn></mrow></math>, <math><mrow id="Sx1.p4.1.m1.1.1" xref="Sx1.p4.1.m1.1.1.cmml"><mrow id="Sx1.p4.1.m1.1.1.1.1" xref="Sx1.p4.1.m1.1.1.1.2.cmml"><mo stretchy="false" id="Sx1.p4.1.m1.1.1.1.1.2" xref="Sx1.p4.1.m1.1.1.1.2.1.cmml">|</mo><mrow id="Sx1.p4.1.m1.1.1.1.1.1" xref="Sx1.p4.1.m1.1.1.1.1.1.cmml"><mi id="Sx1.p4.1.m1.1.1.1.1.1.2" xref="Sx1.p4.1.m1.1.1.1.1.1.2.cmml">S</mi><mo id="Sx1.p4.1.m1.1.1.1.1.1.1" xref="Sx1.p4.1.m1.1.1.1.1.1.1.cmml">\</mo><mi id="Sx1.p4.1.m1.1.1.1.1.1.3" xref="Sx1.p4.1.m1.1.1.1.1.1.3.cmml">I</mi></mrow><mo stretchy="false" id="Sx1.p4.1.m1.1.1.1.1.3" xref="Sx1.p4.1.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="Sx1.p4.1.m1.1.1.2" xref="Sx1.p4.1.m1.1.1.2.cmml">=</mo><mn id="Sx1.p4.1.m1.1.1.3" xref="Sx1.p4.1.m1.1.1.3.cmml">1</mn></mrow></math>

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1304.3296

Formulas:

1-

Λ_{Q C D}

2-

α_{T} (p^{2}) = \frac{g_{0}^{2} (a^{- 1})}{4 π} F^{2} (p^{2}, a^{- 1}) G (p^{2}, a^{- 1}) .

3-

α_{T} (p^{2}) = α_{T}^{pert} (p^{2}) (1 + 9 R (α_{T}^{pert} (p^{2}), α_{T}^{pert} (μ^{2})) {(\frac{α (p^{2})}{α (μ^{2})})}^{\frac{27}{100}} \frac{g^{2} (μ^{2}) {⟨ A^{2} ⟩}_{R, μ^{2}}}{4 (N_{c}^{2} - 1) p^{2}} + \dots)

4-

Δ_{μ ν}^{a b} (q)

5-

⟨ A_{μ}^{a} (q) A_{ν}^{b} (- q) ⟩ = δ^{a b} (δ_{μ ν} - \frac{q_{μ} q_{ν}}{q^{2}}) \frac{G (q^{2})}{q^{2}},

6-

F^{a b} (q^{2})

7-

\frac{1}{V} ⟨ \sum_{x, y} \exp [i q \cdot (x - y)] {(M^{- 1})}_{x y}^{a b} ⟩ = δ^{a b} \frac{F (q^{2})}{q^{2}};

8-

M^{a b} \equiv \partial_{μ} D_{μ}^{a b}

9-

{(L / a)}^{3} \times T / a

10-

Z_{3}^{Latt} (p^{2}, a^{- 1}) = G (p^{2}, a^{- 1}) + c_{a 2 p 2} a^{2} p^{2},

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/0904.0100

Formulas:

1-

κ = 2 ω c^{- 1} Im \sqrt{ϵ_{M} (ω)} .

2-

ϵ_{M} (ω) = 1 - ω_{p}^{2} / ω^{2} + ϵ^{(2)} (ω) .

3-

ω_{p}^{2} = n_{e} e^{2} / (ϵ_{0} m_{e})

4-

ϵ^{(2)} (ω)

5-

ϵ^{(2)} (ω) = \frac{n_{i}}{6 π^{2} m_{e}^{2} ω^{4}} \int 𝑑 q q^{6} \frac{V_{q}^{2}}{{| ε (q, ω) |}^{2}} S (q) [ε (q, ω) - ε (q, 0)] .

6-

ε (q, ω)

7-

S_{OCP} (q)

8-

S_{WDM} (q) = S_{OCP} (q) / [1 + f_{v} (q) S_{OCP} (q)],

9-

f_{v} (q)

10-

- Z e^{2} / (4 π ϵ_{0} r)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1307.5158

Formulas:

1-

H = T (p) + V (r),

2-

H_{0} = \frac{𝒑^{2}}{2 μ} + ρ P (r) with P (r) = sgn (λ) r^{λ}

3-

0 \neq λ > - 2

4-

E_{0} = \frac{λ + 2}{2 λ} {(| λ | ρ)}^{2 / (λ + 2)} {(\frac{Q^{2}}{μ})}^{λ / (λ + 2)},

5-

Q = n + l + \frac{D - 1}{2}

6-

Q = 2 n + l + \frac{D}{2}

7-

2 {(- α_{n} / 3)}^{3 / 2}

8-

E = T (p_{0}) + V (r_{0}),

9-

r_{0} p_{0} = Q,

10-

p_{0} T^{'} (p_{0}) = r_{0} V^{'} (r_{0}) .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/1404.5444

Formulas:

1-

i γ^{μ} \partial_{μ} ψ - m ψ_{c} = 0

2-

ψ = (\binom{ψ_{1}}{ψ_{2}})

3-

i \partial_{t} ψ - σ_{x} p_{x} ψ + i m σ_{y} ψ^{*} = 0 .

4-

ψ_{c} = - i σ_{z} σ_{y} ψ^{*}

5-

σ_{x}, σ_{y}, σ_{z}

6-

ψ = ψ_{+} + i ψ_{-},

7-

- i σ_{z} σ_{y} ψ_{\pm}^{*} = ψ_{\pm}

8-

i \partial_{t} ψ_{\pm} - σ_{x} p_{x} ψ_{\pm} \mp m σ_{z} ψ_{\pm} = 0 .

9-

⟨ σ_{z} ⟩ = \sum_{n} {| ψ_{1, n} |}^{2} - {| ψ_{2, n} |}^{2}

10-

κ = 0.064 m m^{- 1}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0610051

Formulas:

1-

n_{pk} (B)

2-

ω_{μ} = γ_{μ} B

3-

P (t) = \sum_{i} \cos [γ_{μ} B (r_{i}) t],

4-

B (r_{i})

5-

r_{i} = (x_{i}, y_{i})

6-

B (r_{i})

7-

\exp (- σ^{2} t^{2} / 2)

8-

P (t) = \sum_{i} \exp (- Λ e^{- {(r_{i} / ξ_{a b})}^{2}} t) \cos [γ_{μ} B (r_{i}) t] .

9-

H_{c 2} (p)

10-

p_{c} (H)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/9709053

Formulas:

1-

y = f (x)

2-

y = f (x)

3-

y = f (x)

4-

y = f (x)

5-

(n, k, d)

6-

d / n > 10 ϵ

7-

H = H_{A} \otimes H_{B} \otimes H_{C}

8-

| u_{b} ⟩ = {| b ⟩}_{A} \otimes {| v ⟩}_{B}

9-

D \in {A, B}

10-

U ({| 0 ⟩}_{A} \otimes {| v ⟩}_{B})

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9902050

Formulas:

1-

⟹ U (1) \times S U (2) \times S U (3)

2-

{1 = e_{0}, e_{1}, \dots, e_{7}}

3-

{1 = q_{0}, q_{1}, q_{2}, q_{3}}

4-

𝒞 ℒ (p, q)

5-

1, e_{L a}, e_{L a b}, e_{L a b c}

6-

1, e_{R a}, e_{R a b}, e_{R a b c}

7-

e_{L a b} [x] \equiv e_{a} (e_{b} x)

8-

e_{R a b} [x] \equiv (x e_{a}) e_{b}

9-

{1 = q_{L 0}, q_{L 1}, q_{L 2}, q_{L 3}}

10-

{1 = q_{R 0}, q_{R 1}, q_{R 2}, q_{R 3}}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1005.4727

Formulas:

1-

\sim 10^{10} M_{⊙}

2-

\sim 100 M_{⊙} {yr}^{- 1}

3-

\sim 10^{- 8} {yr}^{- 1}

4-

0.089 - 0.136 M_{⊙} {yr}^{- 1} {Mpc}^{- 3}

5-

U_{n} G ℛ

6-

h \equiv H_{0} / 100 = 0.7

7-

B z K \equiv (z - K) - (B - z) > - 0.2

8-

B z K > - 0.2 ⋂ z - K > 2.5

9-

S_{24} > 80 μ Jy

10-

L_{T I R} \sim 4.1 \times 10^{11} - 3.6 \times 10^{12} L_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0403044

Formulas:

1-

(1 / S) d N_{ch} / d y

2-

d N_{ch} / d y

3-

v_{2}^{meas} / v_{2}^{hydro}

4-

v_{2} (η)

5-

d N_{ch} / d η

6-

d N_{ch} / d η

7-

v_{2} (η)

8-

v_{2} (η)

9-

| η_{s} | = \frac{1}{2} \ln \frac{t + z}{t - z}

10-

e (η_{s}; τ_{0}) = ⟨ e (𝒓, η_{s}; τ_{0}) ⟩

Formulas (html):

<math><mrow id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml"><mrow id="S1.p1.5.m5.1.1.1" xref="S1.p1.5.m5.1.1.1.cmml"><mrow id="S1.p1.5.m5.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.cmml"><mrow id="S1.p1.5.m5.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p1.5.m5.1.1.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.5.m5.1.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.cmml"><mn id="S1.p1.5.m5.1.1.1.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.p1.5.m5.1.1.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.p1.5.m5.1.1.1.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.1.1.3.cmml">S</mi></mrow><mo rspace="4.2pt" stretchy="false" id="S1.p1.5.m5.1.1.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p1.5.m5.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.2.cmml">⁢</mo><mi id="S1.p1.5.m5.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.3.cmml">d</mi><mo id="S1.p1.5.m5.1.1.1.1.2a" xref="S1.p1.5.m5.1.1.1.1.2.cmml">⁢</mo><msub id="S1.p1.5.m5.1.1.1.1.4" xref="S1.p1.5.m5.1.1.1.1.4.cmml"><mi id="S1.p1.5.m5.1.1.1.1.4.2" xref="S1.p1.5.m5.1.1.1.1.4.2.cmml">N</mi><mi id="S1.p1.5.m5.1.1.1.1.4.3" xref="S1.p1.5.m5.1.1.1.1.4.3.cmml">ch</mi></msub></mrow><mo id="S1.p1.5.m5.1.1.1.2" xref="S1.p1.5.m5.1.1.1.2.cmml">/</mo><mi id="S1.p1.5.m5.1.1.1.3" xref="S1.p1.5.m5.1.1.1.3.cmml">d</mi></mrow><mo id="S1.p1.5.m5.1.1.2" xref="S1.p1.5.m5.1.1.2.cmml">⁢</mo><mi id="S1.p1.5.m5.1.1.3" xref="S1.p1.5.m5.1.1.3.cmml">y</mi></mrow></math>, <math><mrow id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml"><mrow id="S1.p1.6.m6.1.1.2" xref="S1.p1.6.m6.1.1.2.cmml"><mrow id="S1.p1.6.m6.1.1.2.2" xref="S1.p1.6.m6.1.1.2.2.cmml"><mi id="S1.p1.6.m6.1.1.2.2.2" xref="S1.p1.6.m6.1.1.2.2.2.cmml">d</mi><mo id="S1.p1.6.m6.1.1.2.2.1" xref="S1.p1.6.m6.1.1.2.2.1.cmml">⁢</mo><msub id="S1.p1.6.m6.1.1.2.2.3" xref="S1.p1.6.m6.1.1.2.2.3.cmml"><mi id="S1.p1.6.m6.1.1.2.2.3.2" xref="S1.p1.6.m6.1.1.2.2.3.2.cmml">N</mi><mi id="S1.p1.6.m6.1.1.2.2.3.3" xref="S1.p1.6.m6.1.1.2.2.3.3.cmml">ch</mi></msub></mrow><mo id="S1.p1.6.m6.1.1.2.1" xref="S1.p1.6.m6.1.1.2.1.cmml">/</mo><mi id="S1.p1.6.m6.1.1.2.3" xref="S1.p1.6.m6.1.1.2.3.cmml">d</mi></mrow><mo id="S1.p1.6.m6.1.1.1" xref="S1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p1.6.m6.1.1.3" xref="S1.p1.6.m6.1.1.3.cmml">y</mi></mrow></math>, <math><mrow id="S1.p1.11.m11.1.1" xref="S1.p1.11.m11.1.1.cmml"><msubsup id="S1.p1.11.m11.1.1.2" xref="S1.p1.11.m11.1.1.2.cmml"><mi id="S1.p1.11.m11.1.1.2.2.2" xref="S1.p1.11.m11.1.1.2.2.2.cmml">v</mi><mn id="S1.p1.11.m11.1.1.2.2.3" xref="S1.p1.11.m11.1.1.2.2.3.cmml">2</mn><mi id="S1.p1.11.m11.1.1.2.3" xref="S1.p1.11.m11.1.1.2.3.cmml">meas</mi></msubsup><mo id="S1.p1.11.m11.1.1.1" xref="S1.p1.11.m11.1.1.1.cmml">/</mo><msubsup id="S1.p1.11.m11.1.1.3" xref="S1.p1.11.m11.1.1.3.cmml"><mi id="S1.p1.11.m11.1.1.3.2.2" xref="S1.p1.11.m11.1.1.3.2.2.cmml">v</mi><mn id="S1.p1.11.m11.1.1.3.2.3" xref="S1.p1.11.m11.1.1.3.2.3.cmml">2</mn><mi id="S1.p1.11.m11.1.1.3.3" xref="S1.p1.11.m11.1.1.3.3.cmml">hydro</mi></msubsup></mrow></math>, <math><mrow id="S1.p2.8.m8.1.2" xref="S1.p2.8.m8.1.2.cmml"><msub id="S1.p2.8.m8.1.2.2" xref="S1.p2.8.m8.1.2.2.cmml"><mi id="S1.p2.8.m8.1.2.2.2" xref="S1.p2.8.m8.1.2.2.2.cmml">v</mi><mn id="S1.p2.8.m8.1.2.2.3" xref="S1.p2.8.m8.1.2.2.3.cmml">2</mn></msub><mo id="S1.p2.8.m8.1.2.1" xref="S1.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.p2.8.m8.1.2.3.2" xref="S1.p2.8.m8.1.2.cmml"><mo stretchy="false" id="S1.p2.8.m8.1.2.3.2.1" xref="S1.p2.8.m8.1.2.cmml">(</mo><mi id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml">η</mi><mo stretchy="false" id="S1.p2.8.m8.1.2.3.2.2" xref="S1.p2.8.m8.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p2.9.m9.1.1" xref="S1.p2.9.m9.1.1.cmml"><mrow id="S1.p2.9.m9.1.1.2" xref="S1.p2.9.m9.1.1.2.cmml"><mrow id="S1.p2.9.m9.1.1.2.2" xref="S1.p2.9.m9.1.1.2.2.cmml"><mi id="S1.p2.9.m9.1.1.2.2.2" xref="S1.p2.9.m9.1.1.2.2.2.cmml">d</mi><mo id="S1.p2.9.m9.1.1.2.2.1" xref="S1.p2.9.m9.1.1.2.2.1.cmml">⁢</mo><msub id="S1.p2.9.m9.1.1.2.2.3" xref="S1.p2.9.m9.1.1.2.2.3.cmml"><mi id="S1.p2.9.m9.1.1.2.2.3.2" xref="S1.p2.9.m9.1.1.2.2.3.2.cmml">N</mi><mi id="S1.p2.9.m9.1.1.2.2.3.3" xref="S1.p2.9.m9.1.1.2.2.3.3.cmml">ch</mi></msub></mrow><mo id="S1.p2.9.m9.1.1.2.1" xref="S1.p2.9.m9.1.1.2.1.cmml">/</mo><mi id="S1.p2.9.m9.1.1.2.3" xref="S1.p2.9.m9.1.1.2.3.cmml">d</mi></mrow><mo id="S1.p2.9.m9.1.1.1" xref="S1.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S1.p2.9.m9.1.1.3" xref="S1.p2.9.m9.1.1.3.cmml">η</mi></mrow></math>, <math><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mrow id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml"><mrow id="S1.p3.1.m1.1.1.2.2" xref="S1.p3.1.m1.1.1.2.2.cmml"><mi id="S1.p3.1.m1.1.1.2.2.2" xref="S1.p3.1.m1.1.1.2.2.2.cmml">d</mi><mo id="S1.p3.1.m1.1.1.2.2.1" xref="S1.p3.1.m1.1.1.2.2.1.cmml">⁢</mo><msub id="S1.p3.1.m1.1.1.2.2.3" xref="S1.p3.1.m1.1.1.2.2.3.cmml"><mi id="S1.p3.1.m1.1.1.2.2.3.2" xref="S1.p3.1.m1.1.1.2.2.3.2.cmml">N</mi><mi id="S1.p3.1.m1.1.1.2.2.3.3" xref="S1.p3.1.m1.1.1.2.2.3.3.cmml">ch</mi></msub></mrow><mo id="S1.p3.1.m1.1.1.2.1" xref="S1.p3.1.m1.1.1.2.1.cmml">/</mo><mi id="S1.p3.1.m1.1.1.2.3" xref="S1.p3.1.m1.1.1.2.3.cmml">d</mi></mrow><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">η</mi></mrow></math>, <math><mrow id="S1.p3.2.m2.1.2" xref="S1.p3.2.m2.1.2.cmml"><msub id="S1.p3.2.m2.1.2.2" xref="S1.p3.2.m2.1.2.2.cmml"><mi id="S1.p3.2.m2.1.2.2.2" xref="S1.p3.2.m2.1.2.2.2.cmml">v</mi><mn id="S1.p3.2.m2.1.2.2.3" xref="S1.p3.2.m2.1.2.2.3.cmml">2</mn></msub><mo id="S1.p3.2.m2.1.2.1" xref="S1.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.1.2.3.2" xref="S1.p3.2.m2.1.2.cmml"><mo stretchy="false" id="S1.p3.2.m2.1.2.3.2.1" xref="S1.p3.2.m2.1.2.cmml">(</mo><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">η</mi><mo stretchy="false" id="S1.p3.2.m2.1.2.3.2.2" xref="S1.p3.2.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p3.3.m3.1.2" xref="S1.p3.3.m3.1.2.cmml"><msub id="S1.p3.3.m3.1.2.2" xref="S1.p3.3.m3.1.2.2.cmml"><mi id="S1.p3.3.m3.1.2.2.2" xref="S1.p3.3.m3.1.2.2.2.cmml">v</mi><mn id="S1.p3.3.m3.1.2.2.3" xref="S1.p3.3.m3.1.2.2.3.cmml">2</mn></msub><mo id="S1.p3.3.m3.1.2.1" xref="S1.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p3.3.m3.1.2.3.2" xref="S1.p3.3.m3.1.2.cmml"><mo stretchy="false" id="S1.p3.3.m3.1.2.3.2.1" xref="S1.p3.3.m3.1.2.cmml">(</mo><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">η</mi><mo stretchy="false" id="S1.p3.3.m3.1.2.3.2.2" xref="S1.p3.3.m3.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml"><mrow id="S1.p3.6.m6.1.1.1.1" xref="S1.p3.6.m6.1.1.1.2.cmml"><mo stretchy="false" id="S1.p3.6.m6.1.1.1.1.2" xref="S1.p3.6.m6.1.1.1.2.1.cmml">|</mo><msub id="S1.p3.6.m6.1.1.1.1.1" xref="S1.p3.6.m6.1.1.1.1.1.cmml"><mi id="S1.p3.6.m6.1.1.1.1.1.2" xref="S1.p3.6.m6.1.1.1.1.1.2.cmml">η</mi><mi id="S1.p3.6.m6.1.1.1.1.1.3" xref="S1.p3.6.m6.1.1.1.1.1.3.cmml">s</mi></msub><mo rspace="4.2pt" stretchy="false" id="S1.p3.6.m6.1.1.1.1.3" xref="S1.p3.6.m6.1.1.1.2.1.cmml">|</mo></mrow><mo rspace="4.2pt" id="S1.p3.6.m6.1.1.2" xref="S1.p3.6.m6.1.1.2.cmml">=</mo><mrow id="S1.p3.6.m6.1.1.3" xref="S1.p3.6.m6.1.1.3.cmml"><mfrac id="S1.p3.6.m6.1.1.3.2" xref="S1.p3.6.m6.1.1.3.2.cmml"><mn id="S1.p3.6.m6.1.1.3.2.2" xref="S1.p3.6.m6.1.1.3.2.2.cmml">1</mn><mn id="S1.p3.6.m6.1.1.3.2.3" xref="S1.p3.6.m6.1.1.3.2.3.cmml">2</mn></mfrac><mo id="S1.p3.6.m6.1.1.3.1" xref="S1.p3.6.m6.1.1.3.1.cmml">⁢</mo><mrow id="S1.p3.6.m6.1.1.3.3" xref="S1.p3.6.m6.1.1.3.3.cmml"><mi id="S1.p3.6.m6.1.1.3.3.1" xref="S1.p3.6.m6.1.1.3.3.1.cmml">ln</mi><mo id="S1.p3.6.m6.1.1.3.3a" xref="S1.p3.6.m6.1.1.3.3.cmml">⁡</mo><mfrac id="S1.p3.6.m6.1.1.3.3.2" xref="S1.p3.6.m6.1.1.3.3.2.cmml"><mrow id="S1.p3.6.m6.1.1.3.3.2.2" xref="S1.p3.6.m6.1.1.3.3.2.2.cmml"><mi id="S1.p3.6.m6.1.1.3.3.2.2.2" xref="S1.p3.6.m6.1.1.3.3.2.2.2.cmml">t</mi><mo id="S1.p3.6.m6.1.1.3.3.2.2.1" xref="S1.p3.6.m6.1.1.3.3.2.2.1.cmml">+</mo><mi id="S1.p3.6.m6.1.1.3.3.2.2.3" xref="S1.p3.6.m6.1.1.3.3.2.2.3.cmml">z</mi></mrow><mrow id="S1.p3.6.m6.1.1.3.3.2.3" xref="S1.p3.6.m6.1.1.3.3.2.3.cmml"><mi id="S1.p3.6.m6.1.1.3.3.2.3.2" xref="S1.p3.6.m6.1.1.3.3.2.3.2.cmml">t</mi><mo id="S1.p3.6.m6.1.1.3.3.2.3.1" xref="S1.p3.6.m6.1.1.3.3.2.3.1.cmml">-</mo><mi id="S1.p3.6.m6.1.1.3.3.2.3.3" xref="S1.p3.6.m6.1.1.3.3.2.3.3.cmml">z</mi></mrow></mfrac></mrow></mrow></mrow></math>, <math><mrow id="S1.p3.9.m9.4.4" xref="S1.p3.9.m9.4.4.cmml"><mrow id="S1.p3.9.m9.3.3.2" xref="S1.p3.9.m9.3.3.2.cmml"><mi id="S1.p3.9.m9.3.3.2.4" xref="S1.p3.9.m9.3.3.2.4.cmml">e</mi><mo id="S1.p3.9.m9.3.3.2.3" xref="S1.p3.9.m9.3.3.2.3.cmml">⁢</mo><mrow id="S1.p3.9.m9.3.3.2.2.2" xref="S1.p3.9.m9.3.3.2.2.3.cmml"><mo stretchy="false" id="S1.p3.9.m9.3.3.2.2.2.3" xref="S1.p3.9.m9.3.3.2.2.3.cmml">(</mo><msub id="S1.p3.9.m9.2.2.1.1.1.1" xref="S1.p3.9.m9.2.2.1.1.1.1.cmml"><mi id="S1.p3.9.m9.2.2.1.1.1.1.2" xref="S1.p3.9.m9.2.2.1.1.1.1.2.cmml">η</mi><mi id="S1.p3.9.m9.2.2.1.1.1.1.3" xref="S1.p3.9.m9.2.2.1.1.1.1.3.cmml">s</mi></msub><mo id="S1.p3.9.m9.3.3.2.2.2.4" xref="S1.p3.9.m9.3.3.2.2.3.cmml">;</mo><msub id="S1.p3.9.m9.3.3.2.2.2.2" xref="S1.p3.9.m9.3.3.2.2.2.2.cmml"><mi id="S1.p3.9.m9.3.3.2.2.2.2.2" xref="S1.p3.9.m9.3.3.2.2.2.2.2.cmml">τ</mi><mn id="S1.p3.9.m9.3.3.2.2.2.2.3" xref="S1.p3.9.m9.3.3.2.2.2.2.3.cmml">0</mn></msub><mo stretchy="false" id="S1.p3.9.m9.3.3.2.2.2.5" xref="S1.p3.9.m9.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S1.p3.9.m9.4.4.4" xref="S1.p3.9.m9.4.4.4.cmml">=</mo><mrow id="S1.p3.9.m9.4.4.3.1" xref="S1.p3.9.m9.4.4.3.2.cmml"><mo stretchy="false" id="S1.p3.9.m9.4.4.3.1.2" xref="S1.p3.9.m9.4.4.3.2.1.cmml">⟨</mo><mrow id="S1.p3.9.m9.4.4.3.1.1" xref="S1.p3.9.m9.4.4.3.1.1.cmml"><mi id="S1.p3.9.m9.4.4.3.1.1.4" xref="S1.p3.9.m9.4.4.3.1.1.4.cmml">e</mi><mo id="S1.p3.9.m9.4.4.3.1.1.3" xref="S1.p3.9.m9.4.4.3.1.1.3.cmml">⁢</mo><mrow id="S1.p3.9.m9.4.4.3.1.1.2.2" xref="S1.p3.9.m9.4.4.3.1.1.2.3.cmml"><mo stretchy="false" id="S1.p3.9.m9.4.4.3.1.1.2.2.3" xref="S1.p3.9.m9.4.4.3.1.1.2.3.cmml">(</mo><mi id="S1.p3.9.m9.1.1" xref="S1.p3.9.m9.1.1.cmml">𝒓</mi><mo id="S1.p3.9.m9.4.4.3.1.1.2.2.4" xref="S1.p3.9.m9.4.4.3.1.1.2.3.cmml">,</mo><msub id="S1.p3.9.m9.4.4.3.1.1.1.1.1" xref="S1.p3.9.m9.4.4.3.1.1.1.1.1.cmml"><mi id="S1.p3.9.m9.4.4.3.1.1.1.1.1.2" xref="S1.p3.9.m9.4.4.3.1.1.1.1.1.2.cmml">η</mi><mi id="S1.p3.9.m9.4.4.3.1.1.1.1.1.3" xref="S1.p3.9.m9.4.4.3.1.1.1.1.1.3.cmml">s</mi></msub><mo id="S1.p3.9.m9.4.4.3.1.1.2.2.5" xref="S1.p3.9.m9.4.4.3.1.1.2.3.cmml">;</mo><msub id="S1.p3.9.m9.4.4.3.1.1.2.2.2" xref="S1.p3.9.m9.4.4.3.1.1.2.2.2.cmml"><mi id="S1.p3.9.m9.4.4.3.1.1.2.2.2.2" xref="S1.p3.9.m9.4.4.3.1.1.2.2.2.2.cmml">τ</mi><mn id="S1.p3.9.m9.4.4.3.1.1.2.2.2.3" xref="S1.p3.9.m9.4.4.3.1.1.2.2.2.3.cmml">0</mn></msub><mo stretchy="false" id="S1.p3.9.m9.4.4.3.1.1.2.2.6" xref="S1.p3.9.m9.4.4.3.1.1.2.3.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S1.p3.9.m9.4.4.3.1.3" xref="S1.p3.9.m9.4.4.3.2.1.cmml">⟩</mo></mrow></mrow></math>

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/hep-ex/0208008

Formulas:

1-

B_{s}^{0} – \bar{B_{s}^{0}}

2-

Δ m_{s} > 14.4

3-

B_{d}^{0} – \bar{B_{d}^{0}}

4-

B_{s}^{0} – \bar{B_{s}^{0}}

5-

\frac{G_{F}^{2}}{6 π^{2}} m_{B_{d}} m_{t}^{2} F (m_{t}^{2} / m_{W}^{2}) f_{B_{d}}^{2} B_{B_{d}}

6-

\times η_{Q C D} {| V_{t b}^{*} V_{t d} |}^{2}

7-

Δ m_{d} = 0.503 \pm 0.006

8-

| V_{t d} |

9-

f_{B_{d}} \sqrt{B_{B_{d}}}

10-

\frac{Δ m_{s}}{Δ m_{d}} = \frac{m_{B_{s}}}{m_{B_{d}}} ξ^{2} {| \frac{V_{t s}}{V_{t d}} |}^{2},

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-lat/0610041

Formulas:

1-

m_{q} = 0.1 m_{s}

2-

β = 6 / g^{2}

3-

\exp (- F_{1} (r, T) / T + C) = \frac{1}{3} ⟨ T r W (\vec{r}) W^{†} (0) ⟩,

4-

W (\vec{r})

5-

F_{1} (r, T)

6-

F_{1} (r, T)

7-

V (r) = - \frac{0.385}{r} + \frac{1.263}{r_{0}^{2}} r .

8-

F_{1} (r, T)

9-

F_{1} (r, T)

10-

m_{q} = 0.1 m_{s}

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/1810.01442

Formulas:

1-

P_{RX} (α) = P_{TX} G_{TX} (α) G_{RX} (α) {(\frac{λ}{4 π d})}^{γ},

2-

G_{TX} (α) \leq 1

3-

G_{RX} (α) \leq 1

4-

G_{TX} (α)

5-

G_{RX} (α)

6-

G_{RX} (α)

7-

G_{TX} (α)

8-

G_{RX} (α) G_{TX} (α)

9-

\cos^{2} (α)

10-

0.5 \sin (2 α)

Formulas (html):

Correct Categorie: eess

Guessed Categorie: eess

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9905564

Formulas:

1-

g_{1} (x)

2-

h_{1} (x)

3-

g_{1}^{p} (x)

4-

g_{1}^{n} (x)

5-

h_{1} (x)

6-

g_{1} (x)

7-

h_{1} (x)

8-

l \sim 1 / M x

9-

g_{1} (x)

10-

g_{1}^{p} (x)

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1110.6779

Formulas:

1-

R (n, k)

2-

{1, 2, \dots, n}

3-

R (n, k)

4-

[n] = {1, 2, \dots, n}

5-

π = π (1) π (2) \dots π (n) \in 𝒮_{n}

6-

π (i - 1) < π (i) > π (i + 1)

7-

π (i - 1) > π (i) < π (i + 1)

8-

i \in {2, 3, \dots, n - 1}

9-

R (n, k)

10-

R (n, k) = k R (n - 1, k) + 2 R (n - 1, k - 1) + (n - k) R (n - 1, k - 2)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0205186

Formulas:

1-

10^{- 4} - 10^{- 6}

2-

⟨ n | \hat{O} | m ⟩ = O_{0} δ_{m n} + o (\exp (- L))

3-

(a b), (c d) \dots

4-

Φ = (n_{s} + \frac{1}{2}) Φ_{0}

5-

E_{J} = \frac{ℏ}{2 e} I_{c}

6-

E_{C} = \frac{e 2}{2 C}

7-

ℒ = \sum_{(i j)} \frac{1}{16 E_{C}} (\dot{ϕ_{i}} - \dot{ϕ_{j}}) 2 + E_{J} \cos (ϕ_{i} - ϕ_{j} - a_{i j})

8-

ϕ_{i j} = ϕ_{i} - ϕ_{j} - a_{i j}

9-

\sum_{Γ} ϕ_{i j} = 2 π Φ_{Γ} / Φ_{0} + 2 π n

10-

\frac{Φ_{0}}{2} (\frac{1}{2} + m)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0806.1846

Formulas:

1-

(1 + β k_{i})

2-

(1 + β k_{i})

3-

1 + β k_{i}

4-

< n (t) >

5-

β = 0.06 < β_{c}, β = 0.061 \approx β_{c}

6-

β = 0.1 > β_{c}

7-

δ_{ℓ} = h d_{ℓ, j} + (1 - h) n_{ℓ}

8-

(1 + β k_{i})

9-

δ_{ℓ} = h d_{ℓ, j} + (1 - h) \frac{n_{ℓ}}{1 + β k_{ℓ}} .

10-

{S P : ℓ, j}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/0505050

Formulas:

1-

T_{H} = \frac{ℏ}{8 π M}

2-

β ℏ = \frac{ℏ}{k_{B} T_{H}}

3-

F_{G H} = - k_{B} T_{H} \ln Z_{G H}

4-

(t_{i}, x_{i})

5-

{(Δ x)}^{2} / Δ t = ℏ / m

6-

(t_{i}, x_{i})

7-

t_{i + 1} > t_{i}

8-

Δ x, Δ t \to 0

9-

{(Δ x)}^{2} / Δ t = ℏ / m

10-

D = ℏ / 2 m

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1208.5860

Formulas:

1-

Δ V (i)

2-

i = A, B, C

3-

Δ V (i)

4-

Δ V (i)

5-

+ \frac{α_{gt}}{1 - α_{sc}}

6-

α_{j} = C_{j} / C

7-

Δ V (i)

8-

Δ E = 2 | t |

9-

Δ V (i)

10-

Δ V (A)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0105500

Formulas:

1-

0.1 < (r / h^{- 1} Mpc) < 10

2-

M_{b_{J}} - 5 \log_{10} h = - 19.7

3-

r_{0} = 4.9 \pm 0.3 h^{- 1} Mpc

4-

M_{b_{J}} - 5 \log_{10} h = - 18

5-

- 37^{\circ} .5 < δ < - 22^{\circ} .5

6-

21^{h} 40^{m} < α < 3^{h} 30^{m}

7-

- 7^{\circ} .5 < δ < 2^{\circ} .5

8-

9^{h} 50^{m} < α < 14^{h} 50^{m}

9-

0.5 \leq σ / (h^{- 1} Mpc) \leq 10

10-

N_{g a l}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1805.11413

Formulas:

1-

⟨ 𝐮^{'} \cdot \nabla \bar{𝐮} \cdot 𝐮^{'} ⟩

2-

\nabla . 𝒖 = 0, \frac{\partial 𝒖}{\partial t} + 𝒖 . \nabla 𝒖 = - \nabla p + ν Δ 𝒖 + 𝒇_{𝒖},

3-

\frac{\partial ϕ}{\partial t} + 𝒖 . \nabla ϕ = D \nabla^{2} ϕ + [\underset{Mean Gradient}{\underset{⏟}{f (𝒖)}} or \underset{linear or RA}{\underset{⏟}{f (ϕ)}}],

4-

S c = ν / 𝒟 = 1

5-

N = N_{A} + N_{B}

6-

X_{A} = N_{A} / N

7-

X_{B} = N_{B} / N

8-

ϕ \in [0, 1]

9-

X_{B} = 1 - ϕ

10-

ϕ \in [ϕ_{l}, ϕ_{u}]

Formulas (html):

<math><mrow id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.2.cmml"><mo stretchy="false" id="S1.p3.1.m1.1.1.1.2" xref="S1.p3.1.m1.1.1.2.1.cmml">⟨</mo><mrow id="S1.p3.1.m1.1.1.1.1" xref="S1.p3.1.m1.1.1.1.1.cmml"><msup id="S1.p3.1.m1.1.1.1.1.2" xref="S1.p3.1.m1.1.1.1.1.2.cmml"><mi id="S1.p3.1.m1.1.1.1.1.2.2" xref="S1.p3.1.m1.1.1.1.1.2.2.cmml">𝐮</mi><mo id="S1.p3.1.m1.1.1.1.1.2.3" xref="S1.p3.1.m1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S1.p3.1.m1.1.1.1.1.1" xref="S1.p3.1.m1.1.1.1.1.1.cmml">⋅</mo><mrow id="S1.p3.1.m1.1.1.1.1.3" xref="S1.p3.1.m1.1.1.1.1.3.cmml"><mo id="S1.p3.1.m1.1.1.1.1.3.1" xref="S1.p3.1.m1.1.1.1.1.3.1.cmml">∇</mo><mo id="S1.p3.1.m1.1.1.1.1.3a" xref="S1.p3.1.m1.1.1.1.1.3.cmml">⁡</mo><mrow id="S1.p3.1.m1.1.1.1.1.3.2" xref="S1.p3.1.m1.1.1.1.1.3.2.cmml"><mover accent="true" id="S1.p3.1.m1.1.1.1.1.3.2.2" xref="S1.p3.1.m1.1.1.1.1.3.2.2.cmml"><mi id="S1.p3.1.m1.1.1.1.1.3.2.2.2" xref="S1.p3.1.m1.1.1.1.1.3.2.2.2.cmml">𝐮</mi><mo id="S1.p3.1.m1.1.1.1.1.3.2.2.1" xref="S1.p3.1.m1.1.1.1.1.3.2.2.1.cmml">¯</mo></mover><mo id="S1.p3.1.m1.1.1.1.1.3.2.1" xref="S1.p3.1.m1.1.1.1.1.3.2.1.cmml">⋅</mo><msup id="S1.p3.1.m1.1.1.1.1.3.2.3" xref="S1.p3.1.m1.1.1.1.1.3.2.3.cmml"><mi id="S1.p3.1.m1.1.1.1.1.3.2.3.2" xref="S1.p3.1.m1.1.1.1.1.3.2.3.2.cmml">𝐮</mi><mo id="S1.p3.1.m1.1.1.1.1.3.2.3.3" xref="S1.p3.1.m1.1.1.1.1.3.2.3.3.cmml">′</mo></msup></mrow></mrow></mrow><mo stretchy="false" id="S1.p3.1.m1.1.1.1.3" xref="S1.p3.1.m1.1.1.2.1.cmml">⟩</mo></mrow></math>, <math><mrow id="S2.E1.m1.3.3.1"><mrow id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.3.cmml"><mo id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">∇</mo><mo id="S2.E1.m1.3.3.1.1.2.3" xref="S2.E1.m1.3.3.1.1.3a.cmml">.</mo><mrow id="S2.E1.m1.3.3.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.3.cmml">𝒖</mi><mo id="S2.E1.m1.3.3.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.2.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.2.cmml"><mn id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">0</mn><mo rspace="12.5pt" id="S2.E1.m1.3.3.1.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.2.cmml">,</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.cmml"><mfrac id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.cmml"><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.cmml"><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.1.cmml">∂</mo><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2a" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">⁡</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.2.2.cmml">𝒖</mi></mrow><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.cmml"><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.1.cmml">∂</mo><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3a" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.cmml">⁡</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.2.3.2.cmml">t</mi></mrow></mfrac><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.3.cmml">𝒖</mi></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.2.4" xref="S2.E1.m1.3.3.1.1.3a.cmml">.</mo><mrow id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml"><mrow id="S2.E1.m1.3.3.1.1.2.2.2" xref="S2.E1.m1.3.3.1.1.2.2.2.cmml"><mo id="S2.E1.m1.3.3.1.1.2.2.2.1" xref="S2.E1.m1.3.3.1.1.2.2.2.1.cmml">∇</mo><mo id="S2.E1.m1.3.3.1.1.2.2.2a" xref="S2.E1.m1.3.3.1.1.2.2.2.cmml">⁡</mo><mi id="S2.E1.m1.3.3.1.1.2.2.2.2" xref="S2.E1.m1.3.3.1.1.2.2.2.2.cmml">𝒖</mi></mrow><mo id="S2.E1.m1.3.3.1.1.2.2.1" xref="S2.E1.m1.3.3.1.1.2.2.1.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.2.2.3" xref="S2.E1.m1.3.3.1.1.2.2.3.cmml"><mrow id="S2.E1.m1.3.3.1.1.2.2.3.2" xref="S2.E1.m1.3.3.1.1.2.2.3.2.cmml"><mo id="S2.E1.m1.3.3.1.1.2.2.3.2.1" xref="S2.E1.m1.3.3.1.1.2.2.3.2.1.cmml">-</mo><mrow id="S2.E1.m1.3.3.1.1.2.2.3.2.2" xref="S2.E1.m1.3.3.1.1.2.2.3.2.2.cmml"><mo id="S2.E1.m1.3.3.1.1.2.2.3.2.2.1" xref="S2.E1.m1.3.3.1.1.2.2.3.2.2.1.cmml">∇</mo><mo id="S2.E1.m1.3.3.1.1.2.2.3.2.2a" xref="S2.E1.m1.3.3.1.1.2.2.3.2.2.cmml">⁡</mo><mi id="S2.E1.m1.3.3.1.1.2.2.3.2.2.2" xref="S2.E1.m1.3.3.1.1.2.2.3.2.2.2.cmml">p</mi></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.2.2.3.1" xref="S2.E1.m1.3.3.1.1.2.2.3.1.cmml">+</mo><mrow id="S2.E1.m1.3.3.1.1.2.2.3.3" xref="S2.E1.m1.3.3.1.1.2.2.3.3.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2.3.3.2" xref="S2.E1.m1.3.3.1.1.2.2.3.3.2.cmml">ν</mi><mo id="S2.E1.m1.3.3.1.1.2.2.3.3.1" xref="S2.E1.m1.3.3.1.1.2.2.3.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.E1.m1.3.3.1.1.2.2.3.3.3" xref="S2.E1.m1.3.3.1.1.2.2.3.3.3.cmml">Δ</mi><mo id="S2.E1.m1.3.3.1.1.2.2.3.3.1a" xref="S2.E1.m1.3.3.1.1.2.2.3.3.1.cmml">⁢</mo><mi id="S2.E1.m1.3.3.1.1.2.2.3.3.4" xref="S2.E1.m1.3.3.1.1.2.2.3.3.4.cmml">𝒖</mi></mrow><mo id="S2.E1.m1.3.3.1.1.2.2.3.1a" xref="S2.E1.m1.3.3.1.1.2.2.3.1.cmml">+</mo><msub id="S2.E1.m1.3.3.1.1.2.2.3.4" xref="S2.E1.m1.3.3.1.1.2.2.3.4.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2.3.4.2" xref="S2.E1.m1.3.3.1.1.2.2.3.4.2.cmml">𝒇</mi><mi id="S2.E1.m1.3.3.1.1.2.2.3.4.3" xref="S2.E1.m1.3.3.1.1.2.2.3.4.3.cmml">𝒖</mi></msub></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.1.2">,</mo></mrow></math>, <math><mrow id="S2.E2.m1.3.3.1"><mrow id="S2.E2.m1.3.3.1.1.2" xref="S2.E2.m1.3.3.1.1.3.cmml"><mrow id="S2.E2.m1.3.3.1.1.1.1" xref="S2.E2.m1.3.3.1.1.1.1.cmml"><mfrac id="S2.E2.m1.3.3.1.1.1.1.2" xref="S2.E2.m1.3.3.1.1.1.1.2.cmml"><mrow id="S2.E2.m1.3.3.1.1.1.1.2.2" xref="S2.E2.m1.3.3.1.1.1.1.2.2.cmml"><mo id="S2.E2.m1.3.3.1.1.1.1.2.2.1" xref="S2.E2.m1.3.3.1.1.1.1.2.2.1.cmml">∂</mo><mo id="S2.E2.m1.3.3.1.1.1.1.2.2a" xref="S2.E2.m1.3.3.1.1.1.1.2.2.cmml">⁡</mo><mi id="S2.E2.m1.3.3.1.1.1.1.2.2.2" xref="S2.E2.m1.3.3.1.1.1.1.2.2.2.cmml">ϕ</mi></mrow><mrow id="S2.E2.m1.3.3.1.1.1.1.2.3" xref="S2.E2.m1.3.3.1.1.1.1.2.3.cmml"><mo id="S2.E2.m1.3.3.1.1.1.1.2.3.1" xref="S2.E2.m1.3.3.1.1.1.1.2.3.1.cmml">∂</mo><mo id="S2.E2.m1.3.3.1.1.1.1.2.3a" xref="S2.E2.m1.3.3.1.1.1.1.2.3.cmml">⁡</mo><mi id="S2.E2.m1.3.3.1.1.1.1.2.3.2" xref="S2.E2.m1.3.3.1.1.1.1.2.3.2.cmml">t</mi></mrow></mfrac><mo id="S2.E2.m1.3.3.1.1.1.1.1" xref="S2.E2.m1.3.3.1.1.1.1.1.cmml">+</mo><mi id="S2.E2.m1.3.3.1.1.1.1.3" xref="S2.E2.m1.3.3.1.1.1.1.3.cmml">𝒖</mi></mrow><mo id="S2.E2.m1.3.3.1.1.2.3" xref="S2.E2.m1.3.3.1.1.3a.cmml">.</mo><mrow id="S2.E2.m1.3.3.1.1.2.2" xref="S2.E2.m1.3.3.1.1.2.2.cmml"><mrow id="S2.E2.m1.3.3.1.1.2.2.3" xref="S2.E2.m1.3.3.1.1.2.2.3.cmml"><mo id="S2.E2.m1.3.3.1.1.2.2.3.1" xref="S2.E2.m1.3.3.1.1.2.2.3.1.cmml">∇</mo><mo id="S2.E2.m1.3.3.1.1.2.2.3a" xref="S2.E2.m1.3.3.1.1.2.2.3.cmml">⁡</mo><mi id="S2.E2.m1.3.3.1.1.2.2.3.2" xref="S2.E2.m1.3.3.1.1.2.2.3.2.cmml">ϕ</mi></mrow><mo id="S2.E2.m1.3.3.1.1.2.2.2" xref="S2.E2.m1.3.3.1.1.2.2.2.cmml">=</mo><mrow id="S2.E2.m1.3.3.1.1.2.2.1" xref="S2.E2.m1.3.3.1.1.2.2.1.cmml"><mrow id="S2.E2.m1.3.3.1.1.2.2.1.3" xref="S2.E2.m1.3.3.1.1.2.2.1.3.cmml"><mpadded width="+1.7pt" id="S2.E2.m1.3.3.1.1.2.2.1.3.2" xref="S2.E2.m1.3.3.1.1.2.2.1.3.2.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.1.3.2a" xref="S2.E2.m1.3.3.1.1.2.2.1.3.2.cmml">D</mi></mpadded><mo id="S2.E2.m1.3.3.1.1.2.2.1.3.1" xref="S2.E2.m1.3.3.1.1.2.2.1.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.3.1.1.2.2.1.3.3" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.cmml"><msup id="S2.E2.m1.3.3.1.1.2.2.1.3.3.1" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.1.cmml"><mo id="S2.E2.m1.3.3.1.1.2.2.1.3.3.1.2" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.1.2.cmml">∇</mo><mn id="S2.E2.m1.3.3.1.1.2.2.1.3.3.1.3" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.1.3.cmml">2</mn></msup><mo id="S2.E2.m1.3.3.1.1.2.2.1.3.3a" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.cmml">⁡</mo><mi id="S2.E2.m1.3.3.1.1.2.2.1.3.3.2" xref="S2.E2.m1.3.3.1.1.2.2.1.3.3.2.cmml">ϕ</mi></mrow></mrow><mo id="S2.E2.m1.3.3.1.1.2.2.1.2" xref="S2.E2.m1.3.3.1.1.2.2.1.2.cmml">+</mo><mrow id="S2.E2.m1.3.3.1.1.2.2.1.1.1" xref="S2.E2.m1.3.3.1.1.2.2.1.1.2.cmml"><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.cmml"><munder id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.cmml"><mpadded lspace="3.3pt" width="+3.3pt" id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml"><munder accentunder="true" id="S2.E2.m1.1.1a" xref="S2.E2.m1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml">f</mi><mo movablelimits="false" id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.4.2" xref="S2.E2.m1.1.1.1.cmml"><mo movablelimits="false" stretchy="false" id="S2.E2.m1.1.1.1.4.2.1" xref="S2.E2.m1.1.1.1.cmml">(</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">𝒖</mi><mo movablelimits="false" stretchy="false" id="S2.E2.m1.1.1.1.4.2.2" xref="S2.E2.m1.1.1.1.cmml">)</mo></mrow></mrow><mo movablelimits="false" id="S2.E2.m1.1.1.2" xref="S2.E2.m1.1.1.2.cmml">⏟</mo></munder></mpadded><mrow id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.cmml"><mpadded width="+1.7pt" id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.2.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.2a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.2.cmml">Mean</mi></mpadded><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.1" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.3" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.2.2.3.cmml">Gradient</mi></mrow></munder><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.1" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mpadded width="+5pt" id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.3" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.3.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.3a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.3.cmml">or</mi></mpadded><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.1a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mpadded width="+3.3pt" id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.cmml"><munder id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.cmml"><munder accentunder="true" id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><mrow id="S2.E2.m1.2.2.1" xref="S2.E2.m1.2.2.1.cmml"><mi id="S2.E2.m1.2.2.1.3" xref="S2.E2.m1.2.2.1.3.cmml">f</mi><mo movablelimits="false" id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.2.2.1.4.2" xref="S2.E2.m1.2.2.1.cmml"><mo movablelimits="false" stretchy="false" id="S2.E2.m1.2.2.1.4.2.1" xref="S2.E2.m1.2.2.1.cmml">(</mo><mi id="S2.E2.m1.2.2.1.1" xref="S2.E2.m1.2.2.1.1.cmml">ϕ</mi><mo movablelimits="false" stretchy="false" id="S2.E2.m1.2.2.1.4.2.2" xref="S2.E2.m1.2.2.1.cmml">)</mo></mrow></mrow><mo movablelimits="false" id="S2.E2.m1.2.2.2" xref="S2.E2.m1.2.2.2.cmml">⏟</mo></munder><mrow id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.cmml"><mpadded width="+1.7pt" id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.2" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.2.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.2a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.2.cmml">linear</mi></mpadded><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.1" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.3" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.3.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.3a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.3.cmml">or</mi></mpadded><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.1a" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.1.cmml">⁢</mo><mi id="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.4" xref="S2.E2.m1.3.3.1.1.2.2.1.1.1.1.4.2.4.cmml">RA</mi></mrow></munder></mpadded></mrow><mo id="S2.E2.m1.3.3.1.1.2.2.1.1.1.3" xref="S2.E2.m1.3.3.1.1.2.2.1.1.2.1.cmml">]</mo></mrow></mrow></mrow></mrow><mo id="S2.E2.m1.3.3.1.2">,</mo></mrow></math>, <math><mrow id="S2.p3.5.m4.1.1" xref="S2.p3.5.m4.1.1.cmml"><mrow id="S2.p3.5.m4.1.1.2" xref="S2.p3.5.m4.1.1.2.cmml"><mi id="S2.p3.5.m4.1.1.2.2" xref="S2.p3.5.m4.1.1.2.2.cmml">S</mi><mo id="S2.p3.5.m4.1.1.2.1" xref="S2.p3.5.m4.1.1.2.1.cmml">⁢</mo><mi id="S2.p3.5.m4.1.1.2.3" xref="S2.p3.5.m4.1.1.2.3.cmml">c</mi></mrow><mo id="S2.p3.5.m4.1.1.3" xref="S2.p3.5.m4.1.1.3.cmml">=</mo><mrow id="S2.p3.5.m4.1.1.4" xref="S2.p3.5.m4.1.1.4.cmml"><mi id="S2.p3.5.m4.1.1.4.2" xref="S2.p3.5.m4.1.1.4.2.cmml">ν</mi><mo id="S2.p3.5.m4.1.1.4.1" xref="S2.p3.5.m4.1.1.4.1.cmml">/</mo><mi class="ltx_font_mathcaligraphic" id="S2.p3.5.m4.1.1.4.3" xref="S2.p3.5.m4.1.1.4.3.cmml">𝒟</mi></mrow><mo id="S2.p3.5.m4.1.1.5" xref="S2.p3.5.m4.1.1.5.cmml">=</mo><mn id="S2.p3.5.m4.1.1.6" xref="S2.p3.5.m4.1.1.6.cmml">1</mn></mrow></math>, <math><mrow id="S2.p5.7.m7.1.1" xref="S2.p5.7.m7.1.1.cmml"><mi id="S2.p5.7.m7.1.1.2" xref="S2.p5.7.m7.1.1.2.cmml">N</mi><mo id="S2.p5.7.m7.1.1.1" xref="S2.p5.7.m7.1.1.1.cmml">=</mo><mrow id="S2.p5.7.m7.1.1.3" xref="S2.p5.7.m7.1.1.3.cmml"><msub id="S2.p5.7.m7.1.1.3.2" xref="S2.p5.7.m7.1.1.3.2.cmml"><mi id="S2.p5.7.m7.1.1.3.2.2" xref="S2.p5.7.m7.1.1.3.2.2.cmml">N</mi><mi id="S2.p5.7.m7.1.1.3.2.3" xref="S2.p5.7.m7.1.1.3.2.3.cmml">A</mi></msub><mo id="S2.p5.7.m7.1.1.3.1" xref="S2.p5.7.m7.1.1.3.1.cmml">+</mo><msub id="S2.p5.7.m7.1.1.3.3" xref="S2.p5.7.m7.1.1.3.3.cmml"><mi id="S2.p5.7.m7.1.1.3.3.2" xref="S2.p5.7.m7.1.1.3.3.2.cmml">N</mi><mi id="S2.p5.7.m7.1.1.3.3.3" xref="S2.p5.7.m7.1.1.3.3.3.cmml">B</mi></msub></mrow></mrow></math>, <math><mrow id="S2.p5.8.m8.1.1" xref="S2.p5.8.m8.1.1.cmml"><msub id="S2.p5.8.m8.1.1.2" xref="S2.p5.8.m8.1.1.2.cmml"><mi id="S2.p5.8.m8.1.1.2.2" xref="S2.p5.8.m8.1.1.2.2.cmml">X</mi><mi id="S2.p5.8.m8.1.1.2.3" xref="S2.p5.8.m8.1.1.2.3.cmml">A</mi></msub><mo id="S2.p5.8.m8.1.1.1" xref="S2.p5.8.m8.1.1.1.cmml">=</mo><mrow id="S2.p5.8.m8.1.1.3" xref="S2.p5.8.m8.1.1.3.cmml"><msub id="S2.p5.8.m8.1.1.3.2" xref="S2.p5.8.m8.1.1.3.2.cmml"><mi id="S2.p5.8.m8.1.1.3.2.2" xref="S2.p5.8.m8.1.1.3.2.2.cmml">N</mi><mi id="S2.p5.8.m8.1.1.3.2.3" xref="S2.p5.8.m8.1.1.3.2.3.cmml">A</mi></msub><mo id="S2.p5.8.m8.1.1.3.1" xref="S2.p5.8.m8.1.1.3.1.cmml">/</mo><mi id="S2.p5.8.m8.1.1.3.3" xref="S2.p5.8.m8.1.1.3.3.cmml">N</mi></mrow></mrow></math>, <math><mrow id="S2.p5.9.m9.1.1" xref="S2.p5.9.m9.1.1.cmml"><msub id="S2.p5.9.m9.1.1.2" xref="S2.p5.9.m9.1.1.2.cmml"><mi id="S2.p5.9.m9.1.1.2.2" xref="S2.p5.9.m9.1.1.2.2.cmml">X</mi><mi id="S2.p5.9.m9.1.1.2.3" xref="S2.p5.9.m9.1.1.2.3.cmml">B</mi></msub><mo id="S2.p5.9.m9.1.1.1" xref="S2.p5.9.m9.1.1.1.cmml">=</mo><mrow id="S2.p5.9.m9.1.1.3" xref="S2.p5.9.m9.1.1.3.cmml"><msub id="S2.p5.9.m9.1.1.3.2" xref="S2.p5.9.m9.1.1.3.2.cmml"><mi id="S2.p5.9.m9.1.1.3.2.2" xref="S2.p5.9.m9.1.1.3.2.2.cmml">N</mi><mi id="S2.p5.9.m9.1.1.3.2.3" xref="S2.p5.9.m9.1.1.3.2.3.cmml">B</mi></msub><mo id="S2.p5.9.m9.1.1.3.1" xref="S2.p5.9.m9.1.1.3.1.cmml">/</mo><mi id="S2.p5.9.m9.1.1.3.3" xref="S2.p5.9.m9.1.1.3.3.cmml">N</mi></mrow></mrow></math>, <math><mrow id="S2.p5.11.m11.2.3" xref="S2.p5.11.m11.2.3.cmml"><mi id="S2.p5.11.m11.2.3.2" xref="S2.p5.11.m11.2.3.2.cmml">ϕ</mi><mo id="S2.p5.11.m11.2.3.1" xref="S2.p5.11.m11.2.3.1.cmml">∈</mo><mrow id="S2.p5.11.m11.2.3.3.2" xref="S2.p5.11.m11.2.3.3.1.cmml"><mo stretchy="false" id="S2.p5.11.m11.2.3.3.2.1" xref="S2.p5.11.m11.2.3.3.1.cmml">[</mo><mn id="S2.p5.11.m11.1.1" xref="S2.p5.11.m11.1.1.cmml">0</mn><mo id="S2.p5.11.m11.2.3.3.2.2" xref="S2.p5.11.m11.2.3.3.1.cmml">,</mo><mn id="S2.p5.11.m11.2.2" xref="S2.p5.11.m11.2.2.cmml">1</mn><mo stretchy="false" id="S2.p5.11.m11.2.3.3.2.3" xref="S2.p5.11.m11.2.3.3.1.cmml">]</mo></mrow></mrow></math>, <math><mrow id="S2.p5.13.m13.1.1" xref="S2.p5.13.m13.1.1.cmml"><msub id="S2.p5.13.m13.1.1.2" xref="S2.p5.13.m13.1.1.2.cmml"><mi id="S2.p5.13.m13.1.1.2.2" xref="S2.p5.13.m13.1.1.2.2.cmml">X</mi><mi id="S2.p5.13.m13.1.1.2.3" xref="S2.p5.13.m13.1.1.2.3.cmml">B</mi></msub><mo id="S2.p5.13.m13.1.1.1" xref="S2.p5.13.m13.1.1.1.cmml">=</mo><mrow id="S2.p5.13.m13.1.1.3" xref="S2.p5.13.m13.1.1.3.cmml"><mn id="S2.p5.13.m13.1.1.3.2" xref="S2.p5.13.m13.1.1.3.2.cmml">1</mn><mo id="S2.p5.13.m13.1.1.3.1" xref="S2.p5.13.m13.1.1.3.1.cmml">-</mo><mi id="S2.p5.13.m13.1.1.3.3" xref="S2.p5.13.m13.1.1.3.3.cmml">ϕ</mi></mrow></mrow></math>, <math><mrow id="S2.p5.16.m16.2.2" xref="S2.p5.16.m16.2.2.cmml"><mi id="S2.p5.16.m16.2.2.4" xref="S2.p5.16.m16.2.2.4.cmml">ϕ</mi><mo id="S2.p5.16.m16.2.2.3" xref="S2.p5.16.m16.2.2.3.cmml">∈</mo><mrow id="S2.p5.16.m16.2.2.2.2" xref="S2.p5.16.m16.2.2.2.3.cmml"><mo stretchy="false" id="S2.p5.16.m16.2.2.2.2.3" xref="S2.p5.16.m16.2.2.2.3.cmml">[</mo><msub id="S2.p5.16.m16.1.1.1.1.1" xref="S2.p5.16.m16.1.1.1.1.1.cmml"><mi id="S2.p5.16.m16.1.1.1.1.1.2" xref="S2.p5.16.m16.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S2.p5.16.m16.1.1.1.1.1.3" xref="S2.p5.16.m16.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S2.p5.16.m16.2.2.2.2.4" xref="S2.p5.16.m16.2.2.2.3.cmml">,</mo><msub id="S2.p5.16.m16.2.2.2.2.2" xref="S2.p5.16.m16.2.2.2.2.2.cmml"><mi id="S2.p5.16.m16.2.2.2.2.2.2" xref="S2.p5.16.m16.2.2.2.2.2.2.cmml">ϕ</mi><mi id="S2.p5.16.m16.2.2.2.2.2.3" xref="S2.p5.16.m16.2.2.2.2.2.3.cmml">u</mi></msub><mo stretchy="false" id="S2.p5.16.m16.2.2.2.2.5" xref="S2.p5.16.m16.2.2.2.3.cmml">]</mo></mrow></mrow></math>

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0506348

Formulas:

1-

V (t) = A_{0} + \sum_{i = 1}^{n} A_{i} \cos (i ω t + ϕ_{i})

2-

ω = 2 π / P

3-

R_{i j} = A_{i} / A_{j}

4-

ϕ_{i j} = j ϕ_{i} - i ϕ_{j}

5-

{[Fe / H]}_{ZW}

6-

{[Fe / H]}_{CG}

7-

[Fe / H] = a P^{2} + b P + c ϕ_{31}^{2} + d ϕ_{31} + e P ϕ_{31} + f,

8-

{[Fe / H]}_{ZW}

9-

52.466 P^{2} - 30.075 P + 0.131 ϕ_{31}^{2}

10-

+ 0.982 ϕ_{31} - 4.198 ϕ_{31} P + 2.424

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2004.01527

Formulas:

1-

Δ T_{a d} \approx - 10

2-

| Δ T_{a d} |

3-

Δ T_{a d}

4-

Δ T_{a d}

5-

Δ T_{a d}

6-

d M (T) / d T

7-

\frac{(A_{s} + A_{f}) - (M_{s} + M_{f})}{2}

8-

d M (T) / d T

9-

Δ S_{M} (T)

10-

Δ S_{M} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1905.13158

Formulas:

1-

| β e^{i φ} ⟩

2-

ϱ = \int 𝑑 ϕ g_{σ} (ϕ) | β e^{i ϕ} ⟩ ⟨ β e^{i ϕ} |,

3-

g_{σ} (ϕ) = {(2 π σ^{2})}^{- \frac{1}{2}} \exp {- \frac{1}{2} ϕ^{2} / σ^{2}}

4-

x = a + a^{†}

5-

y = i (a^{†} - a)

6-

[a, a^{†}] = 1

7-

\hat{φ} = \tan^{- 1} \frac{⟨ y ⟩}{⟨ x ⟩} .

8-

var [\hat{φ}] =

9-

{(\frac{⟨ y ⟩}{{⟨ x ⟩}^{2} + {⟨ y ⟩}^{2}})}^{2} var [x]

10-

+ {(\frac{⟨ x ⟩}{{⟨ x ⟩}^{2} + {⟨ y ⟩}^{2}})}^{2} var [y] .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Run 16446907 (Agent672)