Run 6249156

Paper: https://arxiv.org/abs/1610.10036

Formulas:

1-

B ≳ 10^{14} G

2-

𝐣 = (c / 4 π) \nabla \times 𝐁

3-

ψ = \int_{p}^{q} \frac{B_{ϕ}}{B r \sin θ} 𝑑 ℓ,

4-

R_{⋆} = 10 - 13

5-

𝐯_{rot} = 𝛀 \times 𝐫

6-

𝐄 = - 𝐯_{rot} \times 𝐁 / c

7-

ρ_{GJ} = \nabla \cdot 𝐄 / 4 π = - 𝛀 \cdot 𝐁 / 2 π c

8-

ω_{p}^{- 1} = {(m_{e} / 4 π e ρ)}^{1 / 2}

9-

λ_{p} = c / ω_{p}

10-

Σ \sim (j / c) λ_{p}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0705.1374

Formulas:

1-

R_{c} \sim \dot{m} \times R_{ISCO}

2-

R_{ISCO} < R < R_{c}

3-

4 π σ {(T_{in} / κ)}^{4} {(R_{in} / ξ)}^{2} \equiv 4 π σ T_{in}^{4} r_{in}^{2}

4-

R_{ISCO} \equiv 6 α G M / c^{2},

5-

1.5 \overset{<}{\sim} κ \overset{<}{\sim} 2.6

6-

R = (49 / 36) R_{in}

7-

R_{in} \equiv {(ξ^{1 / 2} κ)}^{2} r_{in} \approx {(1.09)}^{2} r_{in} .

8-

R_{in} \approx {(1.35)}^{2} r_{in}

9-

L_{disk} = 2 π d^{2} f_{bol} {(\cos i)}^{- 1},

10-

L_{disk} = \frac{G M \dot{M}}{2 R_{in}} \approx \frac{1}{12 α} \dot{M} c^{2} \equiv η \dot{M} c^{2} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1701.01637

Formulas:

1-

Z_{Mo, x x}^{*}

2-

Z_{Mo, z z}^{*}

3-

| Z_{Mo, x x}^{*} |

4-

| Z_{Mo, z z}^{*} |

5-

T i S_{2}

6-

M o S_{2}

7-

{(M S)}_{1 + x} {(T i S_{2})}_{2}

8-

M o S_{2}

9-

M o S e_{2}

10-

α - M o T e_{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0806.3942

Formulas:

1-

q (m) = c_{n} (m) m^{n} + c_{n - 1} (m) m^{n - 1} + \dots + c_{0} (m),

2-

L_{P} (m) := | m P \cap ℤ^{n} |

3-

{Ehr}_{P} (t) := \sum_{m \geq 0} L_{P} (m) t^{m} = \frac{δ_{0} + δ_{1} t + \dots + δ_{k (n + 1) - 1} t^{k (n + 1) - 1}}{{(1 - t^{k})}^{n + 1}},

4-

(δ_{0}, δ_{1}, \dots, δ_{k (n + 1) - 1})

5-

P^{\lor} := {u \in ℝ^{n} ∣ ⟨ u, v ⟩ \leq 1 for all v \in P}

6-

P = {(P^{\lor})}^{\lor}

7-

δ_{i} = δ_{k (n + 1) - 1 - i}

8-

m = l k + r

9-

l, r \in ℤ_{\geq 0}

10-

L_{P} (m) = L_{P, r} (l)

Formulas (html):

Correct Categorie: math

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0709.2076

Formulas:

1-

M_{*} \sim 10^{7 - 9} M_{⊙}

2-

M_{v i r} = 1.4 \times 10^{12} M_{⊙}

3-

Ω_{m} = 0.3, Ω_{Λ} = 0.7, h = 0.7,

4-

> 10^{5} L_{⊙}

5-

M_{*} \sim 10^{6} M_{⊙}

6-

10^{7} - 10^{8} M_{⊙}

7-

M_{*} \sim 10^{4} - 10^{8} M_{⊙}

8-

M_{*} \sim 10^{6} - 10^{9} M_{⊙}

9-

M_{*} \sim 10^{9} M_{⊙}

10-

M_{*} > 10^{8} M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1601.08009

Formulas:

1-

PG (2, 𝕂)

2-

PG (2, 𝕂)

3-

PG (2, 𝕂)

4-

PG (2, 𝕂)

5-

PG (2, 𝕂)

6-

PG (2, 𝕂)

7-

PG (2, 𝕂)

8-

PG (2, 𝕂)

9-

PG (2, 𝕂)

10-

(Λ_{1}, Λ_{2}, Λ_{3})

Formulas (html):

Correct Categorie: math

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0104106

Formulas:

1-

\sqrt{t a n β}

2-

_{c h a r g i n o}

3-

_{s n e u t r i n o}

4-

_{c h a r g i n o 1}

5-

_{n e u t r a l i n o 1}

6-

_{s l e p t o n R}

7-

_{s n e u t}

8-

_{c h a r g i n o 2}

9-

_{n e u t r a l i n o 2}

10-

_{s q u a r k}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0410502

Formulas:

1-

ρ_{a b} (T)

2-

Δ ρ_{a b}

3-

\ln (T_{c 0} / T_{c}) = Ψ (1 / 2 + 1 / 2 π T_{c} τ) - Ψ (1 / 2)

4-

τ^{- 1} \propto x_{d} \propto Δ ρ_{a b}

5-

T_{c} (Δ ρ_{a b})

6-

\ln (T_{c 0} / T_{c}) = (1 - χ) [Ψ (1 / 2 + 1 / 2 π T_{c} τ_{m}) - Ψ (1 / 2)] + χ [Ψ (1 / 2 + 1 / 4 π T_{c} τ_{n} + 1 / 4 π T_{c} τ_{m}) - Ψ (1 / 2)]

7-

χ = 1 - {⟨ Δ (𝐩) ⟩}_{F S}^{2} / {⟨ Δ^{2} (𝐩) ⟩}_{F S}

8-

T_{c} / T_{c 0}

9-

Δ ρ_{a b}

10-

α = τ_{m}^{- 1} / (τ_{n}^{- 1} + τ_{m}^{- 1})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0807.5095

Formulas:

1-

E_{N A - 1}

2-

E_{N B - 1}

3-

E_{N A - 1}

4-

E_{N A} with chemical potential

5-

μ_{N A} = E_{N A} - E_{N A - 1},

6-

E_{N B - 1}

7-

E_{N B} with chemical potential

8-

μ_{N B} = E_{N B} - E_{N B - 1} .

9-

μ_{N A} = μ_{N B}

10-

μ_{N A} = μ_{S}

Formulas (html):

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xref="p7.7.m7.1.1.3.1.cmml">-</mo><mn id="p7.7.m7.1.1.3.3" xref="p7.7.m7.1.1.3.3.cmml">1</mn></mrow></msub></math>, <math><msub id="S0.Ex1.m1.1.1" xref="S0.Ex1.m1.1.1.cmml"><mi id="S0.Ex1.m1.1.1.2" xref="S0.Ex1.m1.1.1.2.cmml">E</mi><mrow id="S0.Ex1.m1.1.1.3" xref="S0.Ex1.m1.1.1.3.cmml"><mrow id="S0.Ex1.m1.1.1.3.2" xref="S0.Ex1.m1.1.1.3.2.cmml"><mi id="S0.Ex1.m1.1.1.3.2.2" xref="S0.Ex1.m1.1.1.3.2.2.cmml">N</mi><mo id="S0.Ex1.m1.1.1.3.2.1" xref="S0.Ex1.m1.1.1.3.2.1.cmml">⁢</mo><mtext id="S0.Ex1.m1.1.1.3.2.3" xref="S0.Ex1.m1.1.1.3.2.3a.cmml">A</mtext></mrow><mo id="S0.Ex1.m1.1.1.3.1" xref="S0.Ex1.m1.1.1.3.1.cmml">-</mo><mn id="S0.Ex1.m1.1.1.3.3" xref="S0.Ex1.m1.1.1.3.3.cmml">1</mn></mrow></msub></math>, <math><mrow id="S0.Ex1.m3.1.1" xref="S0.Ex1.m3.1.1.cmml"><msub id="S0.Ex1.m3.1.1.2" xref="S0.Ex1.m3.1.1.2.cmml"><mi id="S0.Ex1.m3.1.1.2.2" xref="S0.Ex1.m3.1.1.2.2.cmml">E</mi><mrow id="S0.Ex1.m3.1.1.2.3" xref="S0.Ex1.m3.1.1.2.3.cmml"><mi id="S0.Ex1.m3.1.1.2.3.2" 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id="S0.Ex2.m3.1.1.1.1.3" xref="S0.Ex2.m3.1.1.1.1.3.cmml"><msub id="S0.Ex2.m3.1.1.1.1.3.2" xref="S0.Ex2.m3.1.1.1.1.3.2.cmml"><mi id="S0.Ex2.m3.1.1.1.1.3.2.2" xref="S0.Ex2.m3.1.1.1.1.3.2.2.cmml">E</mi><mrow id="S0.Ex2.m3.1.1.1.1.3.2.3" xref="S0.Ex2.m3.1.1.1.1.3.2.3.cmml"><mi id="S0.Ex2.m3.1.1.1.1.3.2.3.2" xref="S0.Ex2.m3.1.1.1.1.3.2.3.2.cmml">N</mi><mo id="S0.Ex2.m3.1.1.1.1.3.2.3.1" xref="S0.Ex2.m3.1.1.1.1.3.2.3.1.cmml">⁢</mo><mtext id="S0.Ex2.m3.1.1.1.1.3.2.3.3" xref="S0.Ex2.m3.1.1.1.1.3.2.3.3a.cmml">A</mtext></mrow></msub><mo id="S0.Ex2.m3.1.1.1.1.3.1" xref="S0.Ex2.m3.1.1.1.1.3.1.cmml">-</mo><msub id="S0.Ex2.m3.1.1.1.1.3.3" xref="S0.Ex2.m3.1.1.1.1.3.3.cmml"><mi id="S0.Ex2.m3.1.1.1.1.3.3.2" xref="S0.Ex2.m3.1.1.1.1.3.3.2.cmml">E</mi><mrow id="S0.Ex2.m3.1.1.1.1.3.3.3" xref="S0.Ex2.m3.1.1.1.1.3.3.3.cmml"><mrow id="S0.Ex2.m3.1.1.1.1.3.3.3.2" xref="S0.Ex2.m3.1.1.1.1.3.3.3.2.cmml"><mi id="S0.Ex2.m3.1.1.1.1.3.3.3.2.2" xref="S0.Ex2.m3.1.1.1.1.3.3.3.2.2.cmml">N</mi><mo id="S0.Ex2.m3.1.1.1.1.3.3.3.2.1" xref="S0.Ex2.m3.1.1.1.1.3.3.3.2.1.cmml">⁢</mo><mtext id="S0.Ex2.m3.1.1.1.1.3.3.3.2.3" xref="S0.Ex2.m3.1.1.1.1.3.3.3.2.3a.cmml">A</mtext></mrow><mo id="S0.Ex2.m3.1.1.1.1.3.3.3.1" xref="S0.Ex2.m3.1.1.1.1.3.3.3.1.cmml">-</mo><mn id="S0.Ex2.m3.1.1.1.1.3.3.3.3" xref="S0.Ex2.m3.1.1.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S0.Ex2.m3.1.1.1.2" xref="S0.Ex2.m3.1.1.1.1.cmml">,</mo></mrow></math>, <math><msub id="S0.Ex3.m1.1.1" xref="S0.Ex3.m1.1.1.cmml"><mi id="S0.Ex3.m1.1.1.2" xref="S0.Ex3.m1.1.1.2.cmml">E</mi><mrow id="S0.Ex3.m1.1.1.3" xref="S0.Ex3.m1.1.1.3.cmml"><mrow id="S0.Ex3.m1.1.1.3.2" xref="S0.Ex3.m1.1.1.3.2.cmml"><mi id="S0.Ex3.m1.1.1.3.2.2" xref="S0.Ex3.m1.1.1.3.2.2.cmml">N</mi><mo id="S0.Ex3.m1.1.1.3.2.1" xref="S0.Ex3.m1.1.1.3.2.1.cmml">⁢</mo><mtext id="S0.Ex3.m1.1.1.3.2.3" xref="S0.Ex3.m1.1.1.3.2.3a.cmml">B</mtext></mrow><mo id="S0.Ex3.m1.1.1.3.1" xref="S0.Ex3.m1.1.1.3.1.cmml">-</mo><mn id="S0.Ex3.m1.1.1.3.3" xref="S0.Ex3.m1.1.1.3.3.cmml">1</mn></mrow></msub></math>, <math><mrow id="S0.Ex3.m3.1.1" xref="S0.Ex3.m3.1.1.cmml"><msub id="S0.Ex3.m3.1.1.2" xref="S0.Ex3.m3.1.1.2.cmml"><mi id="S0.Ex3.m3.1.1.2.2" xref="S0.Ex3.m3.1.1.2.2.cmml">E</mi><mrow id="S0.Ex3.m3.1.1.2.3" xref="S0.Ex3.m3.1.1.2.3.cmml"><mi id="S0.Ex3.m3.1.1.2.3.2" xref="S0.Ex3.m3.1.1.2.3.2.cmml">N</mi><mo id="S0.Ex3.m3.1.1.2.3.1" xref="S0.Ex3.m3.1.1.2.3.1.cmml">⁢</mo><mtext id="S0.Ex3.m3.1.1.2.3.3" xref="S0.Ex3.m3.1.1.2.3.3a.cmml">B</mtext></mrow></msub><mo id="S0.Ex3.m3.1.1.1" xref="S0.Ex3.m3.1.1.1.cmml">⁢</mo><mtext id="S0.Ex3.m3.1.1.3" xref="S0.Ex3.m3.1.1.3a.cmml"> with chemical potential</mtext></mrow></math>, <math><mrow id="S0.Ex4.m3.1.1.1" xref="S0.Ex4.m3.1.1.1.1.cmml"><mrow id="S0.Ex4.m3.1.1.1.1" xref="S0.Ex4.m3.1.1.1.1.cmml"><msub id="S0.Ex4.m3.1.1.1.1.2" xref="S0.Ex4.m3.1.1.1.1.2.cmml"><mi id="S0.Ex4.m3.1.1.1.1.2.2" xref="S0.Ex4.m3.1.1.1.1.2.2.cmml">μ</mi><mrow id="S0.Ex4.m3.1.1.1.1.2.3" xref="S0.Ex4.m3.1.1.1.1.2.3.cmml"><mi id="S0.Ex4.m3.1.1.1.1.2.3.2" xref="S0.Ex4.m3.1.1.1.1.2.3.2.cmml">N</mi><mo id="S0.Ex4.m3.1.1.1.1.2.3.1" xref="S0.Ex4.m3.1.1.1.1.2.3.1.cmml">⁢</mo><mtext id="S0.Ex4.m3.1.1.1.1.2.3.3" xref="S0.Ex4.m3.1.1.1.1.2.3.3a.cmml">B</mtext></mrow></msub><mo id="S0.Ex4.m3.1.1.1.1.1" xref="S0.Ex4.m3.1.1.1.1.1.cmml">=</mo><mrow id="S0.Ex4.m3.1.1.1.1.3" xref="S0.Ex4.m3.1.1.1.1.3.cmml"><msub id="S0.Ex4.m3.1.1.1.1.3.2" xref="S0.Ex4.m3.1.1.1.1.3.2.cmml"><mi id="S0.Ex4.m3.1.1.1.1.3.2.2" xref="S0.Ex4.m3.1.1.1.1.3.2.2.cmml">E</mi><mrow id="S0.Ex4.m3.1.1.1.1.3.2.3" xref="S0.Ex4.m3.1.1.1.1.3.2.3.cmml"><mi id="S0.Ex4.m3.1.1.1.1.3.2.3.2" xref="S0.Ex4.m3.1.1.1.1.3.2.3.2.cmml">N</mi><mo id="S0.Ex4.m3.1.1.1.1.3.2.3.1" xref="S0.Ex4.m3.1.1.1.1.3.2.3.1.cmml">⁢</mo><mtext id="S0.Ex4.m3.1.1.1.1.3.2.3.3" xref="S0.Ex4.m3.1.1.1.1.3.2.3.3a.cmml">B</mtext></mrow></msub><mo id="S0.Ex4.m3.1.1.1.1.3.1" xref="S0.Ex4.m3.1.1.1.1.3.1.cmml">-</mo><msub id="S0.Ex4.m3.1.1.1.1.3.3" xref="S0.Ex4.m3.1.1.1.1.3.3.cmml"><mi id="S0.Ex4.m3.1.1.1.1.3.3.2" xref="S0.Ex4.m3.1.1.1.1.3.3.2.cmml">E</mi><mrow id="S0.Ex4.m3.1.1.1.1.3.3.3" xref="S0.Ex4.m3.1.1.1.1.3.3.3.cmml"><mrow id="S0.Ex4.m3.1.1.1.1.3.3.3.2" xref="S0.Ex4.m3.1.1.1.1.3.3.3.2.cmml"><mi id="S0.Ex4.m3.1.1.1.1.3.3.3.2.2" xref="S0.Ex4.m3.1.1.1.1.3.3.3.2.2.cmml">N</mi><mo id="S0.Ex4.m3.1.1.1.1.3.3.3.2.1" xref="S0.Ex4.m3.1.1.1.1.3.3.3.2.1.cmml">⁢</mo><mtext id="S0.Ex4.m3.1.1.1.1.3.3.3.2.3" xref="S0.Ex4.m3.1.1.1.1.3.3.3.2.3a.cmml">B</mtext></mrow><mo id="S0.Ex4.m3.1.1.1.1.3.3.3.1" xref="S0.Ex4.m3.1.1.1.1.3.3.3.1.cmml">-</mo><mn id="S0.Ex4.m3.1.1.1.1.3.3.3.3" xref="S0.Ex4.m3.1.1.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S0.Ex4.m3.1.1.1.2" xref="S0.Ex4.m3.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="p7.14.m6.1.1" xref="p7.14.m6.1.1.cmml"><msub id="p7.14.m6.1.1.2" xref="p7.14.m6.1.1.2.cmml"><mi id="p7.14.m6.1.1.2.2" xref="p7.14.m6.1.1.2.2.cmml">μ</mi><mrow id="p7.14.m6.1.1.2.3" xref="p7.14.m6.1.1.2.3.cmml"><mi id="p7.14.m6.1.1.2.3.2" xref="p7.14.m6.1.1.2.3.2.cmml">N</mi><mo id="p7.14.m6.1.1.2.3.1" xref="p7.14.m6.1.1.2.3.1.cmml">⁢</mo><mtext id="p7.14.m6.1.1.2.3.3" xref="p7.14.m6.1.1.2.3.3a.cmml">A</mtext></mrow></msub><mo id="p7.14.m6.1.1.1" xref="p7.14.m6.1.1.1.cmml">=</mo><msub id="p7.14.m6.1.1.3" xref="p7.14.m6.1.1.3.cmml"><mi id="p7.14.m6.1.1.3.2" xref="p7.14.m6.1.1.3.2.cmml">μ</mi><mrow id="p7.14.m6.1.1.3.3" xref="p7.14.m6.1.1.3.3.cmml"><mi id="p7.14.m6.1.1.3.3.2" xref="p7.14.m6.1.1.3.3.2.cmml">N</mi><mo id="p7.14.m6.1.1.3.3.1" xref="p7.14.m6.1.1.3.3.1.cmml">⁢</mo><mtext id="p7.14.m6.1.1.3.3.3" xref="p7.14.m6.1.1.3.3.3a.cmml">B</mtext></mrow></msub></mrow></math>, <math><mrow id="p8.7.m7.1.1" xref="p8.7.m7.1.1.cmml"><msub id="p8.7.m7.1.1.2" xref="p8.7.m7.1.1.2.cmml"><mi id="p8.7.m7.1.1.2.2" xref="p8.7.m7.1.1.2.2.cmml">μ</mi><mrow id="p8.7.m7.1.1.2.3" xref="p8.7.m7.1.1.2.3.cmml"><mi id="p8.7.m7.1.1.2.3.2" xref="p8.7.m7.1.1.2.3.2.cmml">N</mi><mo id="p8.7.m7.1.1.2.3.1" xref="p8.7.m7.1.1.2.3.1.cmml">⁢</mo><mtext id="p8.7.m7.1.1.2.3.3" xref="p8.7.m7.1.1.2.3.3a.cmml">A</mtext></mrow></msub><mo id="p8.7.m7.1.1.1" xref="p8.7.m7.1.1.1.cmml">=</mo><msub id="p8.7.m7.1.1.3" xref="p8.7.m7.1.1.3.cmml"><mi id="p8.7.m7.1.1.3.2" xref="p8.7.m7.1.1.3.2.cmml">μ</mi><mtext id="p8.7.m7.1.1.3.3" xref="p8.7.m7.1.1.3.3a.cmml">S</mtext></msub></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/math/9807036

Formulas:

1-

m \geq 2 n - 1

2-

m \geq 2 n - 1

3-

{1, 2, \dots, n}

4-

{1, 2, \dots, n}

5-

m = 2 n - 1

6-

{1, 2, \dots, k}

7-

m \geq 2 n - 1

8-

m - (n - 1)

9-

1, 2, \dots, n

10-

2, 3, \dots, n, 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1901.09814

Formulas:

1-

A \subseteq {0, \dots, k}^{n}

2-

A \subseteq {0, 1}^{n}

3-

x \in {0, \dots, n}^{k}

4-

| x | = \sum_{i = 1}^{k} x_{i}

5-

A \subseteq {0, 1}^{n}

6-

{x \in {0, 1}^{n} : | x | = r}

7-

x \leq_{c o l e x} y

8-

\max (X Δ Y) \in Y

9-

X = {i : x_{i} = 1}

10-

Y = {i : y_{i} = 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/nucl-ex/0602003

Formulas:

1-

E t r a n s

2-

E t r a n s

3-

E t r a n s

4-

E t r a n s

5-

v_{p a r} - v_{p e r}

6-

E t r a n s

7-

E t r a n s

8-

E t r a n s

9-

E t r a n s

10-

E t r a n s

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1305.1119

Formulas:

1-

\sim 10^{''} - 40^{''}

2-

145 / 63 = 0.12 \pm 0.05

3-

n_{cr} (H_{2}) > 10^{7}

4-

d T_{k} = a T_{k}^{- b}

5-

ζ_{CR} ≳ 10^{- 15}

6-

ζ_{X, CR} > 10^{- 13}

7-

{}^{3}P_{2} - {}^{3}P_{1}

8-

{}^{2}P_{3 / 2} - {}^{2}P_{1 / 2}

9-

{}^{3}P_{1} - {}^{3}P_{2}

10-

{}^{3}P_{1} - {}^{3}P_{0}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0211604

Formulas:

1-

h_{o b s} = Δ l / l

2-

h_{o b s}

3-

h_{l i m}

4-

h_{o b s}

5-

h_{o b s} = 1.4 \times 10^{- 20} (\frac{1 M p c}{r}) (\frac{10^{3} H z}{ν_{0}}) {(\frac{E_{g w}}{1 M_{⊙} c^{2}})}^{1 / 2} {(\frac{1 s}{τ_{g}})}^{1 / 2}

6-

\frac{h_{o b s}}{h_{⋆}} \propto \frac{| G (α, δ) |}{r}

7-

G (α, δ)

8-

h_{⋆} = 10^{- 20}

9-

h_{⋆} = 10^{- 23}

10-

h_{⋆} = 10^{- 17}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/0706.0088

Formulas:

1-

Δ f = ℓ (ℓ + 1) f

2-

S O (3)

3-

f (x_{1}, x_{2}, x_{3})

4-

f = (5 \cos^{3} θ - 3 \cos θ) / 2

5-

x_{3} = 0, x_{3} = \pm \sqrt{3 / 5}

6-

ϕ = (1 + \sqrt{5}) / 2

7-

(0, \pm ϕ, \pm 1)

8-

(\pm 1, 0, \pm ϕ)

9-

(\pm ϕ, \pm 1, 0)

10-

f = x_{1} x_{2} x_{3}

Formulas (html):

Correct Categorie: math

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/0706.2737

Formulas:

1-

d z / d t

2-

d z / d t

3-

r_{c} = \frac{M_{Pl}}{2 M_{5}^{3}},

4-

H^{2} = {[\sqrt{\frac{ρ}{3 M_{Pl}^{2}} + \frac{1}{4 r_{c}^{2}}} + \frac{1}{2 r_{c}}]}^{2} - \frac{k}{a {(t)}^{2}},

5-

k = 0, \pm 1

6-

Ω_{r_{c}} = \frac{1}{4 r_{c}^{2} H_{0}^{2}},

7-

H {(z)}^{2} / H_{0}^{2} = Ω_{k} {(1 + z)}^{2} + {[\sqrt{Ω_{r_{c}}} + \sqrt{Ω_{r_{c}} + Ω_{m} {(1 + z)}^{3}}]}^{2},

8-

H_{0} = 100 h

9-

Ω_{k} + {[\sqrt{Ω_{r_{c}}} + \sqrt{Ω_{r_{c}} + Ω_{m}}]}^{2} = 1 .

10-

q = - \ddot{a} / a H^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1112.1402

Formulas:

1-

δ v (O vi)

2-

δ v (C iv)

3-

δ v (O vi)

4-

δ v (C iv)

5-

0 \leq | Δ v (O vi - C iv) | \leq 48

6-

N (O vi) / N (H i)

7-

N (C iv) / N (H i)

8-

(1.0 \pm 0.2) \times 10^{- 7}

9-

N (O vi) > 10^{13.7}

10-

δ (\equiv n_{H} / {\bar{n}}_{H}) ≪ 10

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1812.00481

Formulas:

1-

ℒ (f (x; 𝒜, θ_{𝒜}), y)

2-

f (x; 𝒜, θ_{𝒜})

3-

\begin{matrix} 𝒜, θ_{𝒜} = & \arg \min_{𝒜, θ_{𝒜}} \frac{1}{N} \sum_{i = 1}^{N} ℒ (f (x_{i}; 𝒜, θ_{𝒜}), y_{i}) \\ s.t. c (𝒜) \leq 𝒞 \end{matrix}

4-

𝒰 (θ_{𝒜})

5-

𝒰 (θ_{𝒜}) \leq c (𝒜)

6-

r (θ_{𝒜}) = 𝒰 (θ_{𝒜}) / c (𝒜)

7-

c (θ_{𝒜^{'}}) = 𝒰 (θ_{𝒜})

8-

r (θ_{𝒜^{'}})

9-

θ_{𝒜} = 𝒫 (𝒜, ℒ, 𝒟)

10-

r (θ_{𝒜})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0503292

Formulas:

1-

\sim 9 \overset{\circ}{.} 5 \times 5 \overset{\circ}{.} 5

2-

α_{J2000} = 18^{h} 07^{m} 00 \overset{s}{.} 38

3-

δ_{J2000} = - 29 ° 24' 30 \overset{''}{.} 8

4-

F \propto e^{- t / 19.2 d}

5-

\sim 2 \times 10^{- 9}

6-

Δ χ^{2} = 2.71

7-

\exp (- τ_{T} / \cos θ)

8-

k T_{e} \approx 20

9-

A_{seed} \sim 86 {(D / 8 kpc)}^{2}

10-

N_{H} (10^{22} {cm}^{- 2})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/2001.09441

Formulas:

1-

c_{0} (T)

2-

{Ric}_{g} = c T

3-

c > c_{0} (T)

4-

{tr}_{g} T = 1

5-

S : ℳ_{K} \to ℝ

6-

{tr}_{g} T = 1

7-

{tr}_{g} T = 1

8-

𝔤 = 𝔞 \oplus 𝔷 (𝔨) \oplus 𝔨_{1} \oplus \dots \oplus 𝔨_{r},

9-

g = {α Q |}_{𝔞} + h + {α_{1} Q |}_{𝔨_{1}} + \dots + {α_{r} Q |}_{𝔨_{r}} .

10-

α, α_{1}, \dots, α_{r}

Formulas (html):

Correct Categorie: math

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9503088

Formulas:

1-

H = - \sum_{i = 1, σ}^{N_{a}} t_{i, i + 1} e^{i \frac{2 π Φ}{N_{a} Φ_{0}}} (c_{i, σ}^{†} c_{i + 1, σ} + c_{i + 1, σ}^{†} c_{i, σ}) + 𝒱 [n_{i, σ}]

2-

𝒱 [n_{i, σ}]

3-

n_{i, σ} = c_{i, σ}^{†} c_{i, σ}

4-

t_{i, i + 1}

5-

𝒱 \equiv 𝒱_{1} = U \sum_{i} n_{i, ↑} n_{i, ↓}

6-

𝒱 = 𝒱_{1} + V \sum_{i, σ} n_{i, σ} n_{i + 1, σ}

7-

t_{i, i + 1} = t e^{i δ_{i}}

8-

- π < δ_{i} \leq π

9-

𝒰 = \exp {- i \sum_{j = 1}^{N_{a}} \sum_{n = 1, σ}^{N_{a}} \sum_{l}^{n - 1} (δ_{l} + \frac{2 π Φ}{N_{a} Φ_{0}}) c_{j, σ}^{†} c_{j, σ}}

10-

d_{j, σ} = 𝒰 c_{j, σ} 𝒰^{†} = \exp {i (\sum_{l = 1}^{j - 1} δ_{l} + (j - 1) \frac{2 π Φ}{N_{a} Φ_{0}})} c_{j, σ} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1307.4205

Formulas:

1-

(R, ϕ, z)

2-

V (R, z)

3-

V_{d} (R, z) = - \frac{G M_{d}}{\sqrt{R^{2} + {(α + \sqrt{h^{2} + z^{2}})}^{2}}} .

4-

V_{n} (R, z) = - \frac{G M_{n}}{\sqrt{R^{2} + z^{2} + c_{n}^{2}}},

5-

2.325 \times 10^{7} M_{⊙}

6-

θ (R) = \sqrt{R {| \frac{\partial V}{\partial R} |}_{z = 0}} .

7-

θ (R) = R \sqrt{\frac{M_{d}}{{(R^{2} + {(α + h)}^{2})}^{3 / 2}} + \frac{M_{n}}{{(R^{2} + c_{n}^{2})}^{3 / 2}}} .

8-

V (R, z)

9-

V_{eff} (R, z) = V (R, z) + \frac{L_{z}^{2}}{2 R^{2}} .

10-

L_{z}^{2} / (2 R^{2})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0711.1013

Formulas:

1-

4 < 1 / λ < 6.5

2-

_{a v e r a g e}

3-

_{a v e r a g e}

4-

α_{F U V}

5-

F_{ν} \propto ν^{- α}

6-

α_{U V O}

7-

α_{U V O}

8-

α_{F U V}

9-

α_{F U V}

10-

α_{U V O}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0207545

Formulas:

1-

V_{c} \sim E I_{c} / n_{c}^{2}

2-

V_{c} \sim E I_{c} / n_{h}^{2} \cdot k T_{c}^{2} / k T_{h}^{2}

3-

V_{h} \sim E I_{h} / n_{h}^{2}

4-

V_{c} / V_{h} = E I_{c} / E I_{h} {(\frac{k T_{c}}{k T_{h}})}^{2} .

5-

V_{h} = A^{3 / 2}

6-

V_{h} \cdot n_{h}^{2}

7-

10'' \times 10''

8-

4'' \times 4''

9-

\sim 10''

10-

S \equiv T / n^{2 / 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/physics/0506084

Formulas:

1-

ρ^{'} (x)

2-

w_{\max} / ⟨ w ⟩

3-

w = ρ (x) / ρ^{^{'}} (x)

4-

w = ρ (x) / ρ^{'} (x)

5-

w_{\max} / ⟨ w ⟩

6-

g (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}), \dots, g_{n} (x_{n})

7-

\sum_{i = 1}^{k} R_{i}^{'}

8-

R = R^{'} ⟨ w ⟩

9-

0 < α_{i} < 1

10-

R^{'} = \int ρ^{'} (x) 𝑑 x

Formulas (html):

Correct Categorie: physics

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0407144

Formulas:

1-

{YBa}_{2} {Cu}_{3} O_{7 - δ}

2-

{Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}

3-

{Nd}_{1.85} {Ce}_{0.15} {CuO}_{4 - δ}

4-

V = f (I)

5-

f (I) = - f (- I)

6-

⟨ V ⟩ = \int_{- \infty}^{\infty} f (I) P (I - I_{0}) 𝑑 I,

7-

P (I - I_{0})

8-

P (I - I_{0})

9-

P (I - I_{0})

10-

σ_{I}^{2} = \int_{- \infty}^{\infty} {(I - I_{0})}^{2} P (I - I_{0}) 𝑑 I

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1712.10067

Formulas:

1-

= A x + B u,

2-

= C x + D u,

3-

A \in ℂ^{n \times n}

4-

B \in ℂ^{n \times m}

5-

C \in ℂ^{p \times n}

6-

D \in ℂ^{p \times m}

7-

E \in ℂ^{n \times n}

8-

Δ \in ℂ^{m \times p}

9-

E \dot{x} = A x + B Δ y

10-

y = {(I - D Δ)}^{- 1} C x

Formulas (html):

Correct Categorie: math

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1112.1328

Formulas:

1-

1 + z = a (t_{0}) / a (t_{1}),

2-

\sim 10^{14} M_{⊙}

3-

Δ ν / ν = - Δ λ / λ = \frac{Δ U}{c^{2}} = \frac{G M_{1}}{R_{1} c^{2}} - \frac{G M_{0}}{R_{0} c^{2}},

4-

Δ t = 1 + Δ U / c^{2},

5-

{(\frac{1}{a} \frac{d a}{d t})}^{2} = \frac{8 π G}{3} ρ - \frac{k c^{2}}{a^{2}} .

6-

\frac{d τ_{2}}{d τ_{2}} = \frac{\sqrt{g_{00} (2) d t}}{\sqrt{g_{00} (1) d t}} ≃ 1 + Φ_{2} - Φ_{1} = 1 - \frac{G M}{R_{2} c^{2}} + \frac{G M}{R_{1} c^{2}} .

7-

\nabla \cdot 𝐠 = 4 π G (ρ + 3 p)

8-

Δ m_{1} = - h ν_{1}

9-

Δ m_{2} = h ν_{2}

10-

Δ m_{1} + Φ_{1} Δ m_{1} + Δ m_{2} + Φ_{2} Δ m_{2} = 0

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1603.01414

Formulas:

1-

E_{ℓ} (𝒓_{j}, ω)

2-

𝚪 = (\begin{matrix} 𝚪_{1, 1} & 𝚪_{1, 2} \\ 𝚪_{2, 1} & 𝚪_{2, 2} \end{matrix}),

3-

Γ_{j, k} = Γ_{k, j}^{†}

4-

Γ_{j, k} = ⟨ 𝑬 (𝒓_{j}) 𝑬^{†} (𝒓_{k}) ⟩, 𝑬 (𝒓_{j}) = (\begin{matrix} E_{x} (𝒓_{j}) \\ E_{y} (𝒓_{j}) \end{matrix})

5-

E_{1} = E_{x} (𝒓_{1})

6-

E_{2} = E_{y} (𝒓_{1})

7-

E_{3} = E_{x} (𝒓_{2})

8-

E_{4} = E_{y} (𝒓_{2})

9-

Γ_{j, k} = ⟨ E_{j} E_{k}^{*} ⟩

10-

j, k = 1, \dots, 4

Formulas (html):

Correct Categorie: physics

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0901.4798

Formulas:

1-

(n / k) \cdot b

2-

P_{i} = (i, s_{i})

3-

0 \leq i \leq (k - 1)

4-

f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{k - 1} x^{k - 1}

5-

0 \leq i \leq (k - 1)

6-

D_{i} = f (i + k - 1)

7-

(k, D_{1})

8-

(k + 1, D_{2})

9-

(k + n - 1, D_{n})

10-

s_{i} = f (i)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0501494

Formulas:

1-

Δ T (θ, φ) = \sum_{ℓ = 2}^{\infty} \sum_{m = - ℓ}^{ℓ} | a_{ℓ m} | e^{i ϕ_{ℓ m}} Y_{ℓ m} (θ, φ),

2-

| a_{ℓ m} |

3-

[0, 2 π]

4-

C_{ℓ} = \frac{1}{2 ℓ + 1} \sum_{m = - ℓ}^{ℓ} {| a_{ℓ m} |}^{2}

5-

D_{ℓ} = \frac{ℓ (ℓ + 1)}{2 π} \frac{C_{ℓ}}{T_{0}} .

6-

a_{ℓ m}^{^{'}} = B_{ℓ} a_{ℓ m},

7-

B_{ℓ} = \exp [- σ^{2} ℓ (ℓ + 1) / 2]

8-

σ = FWHM / \sqrt{8 \ln 2}

9-

a_{ℓ m}^{^{''}} = \frac{a_{ℓ m}}{B_{ℓ} + α},

10-

| a_{ℓ m} |

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0107016

Formulas:

1-

ℋ_{A_{1} \dots A_{N}}

2-

U = U_{1} \otimes \dots \otimes U_{N}

3-

ϱ = U σ U^{†}

4-

ℋ_{A_{1} \dots A_{N}} = ℋ_{A_{1}} \otimes \dots \otimes ℋ_{A_{N}} = ℋ_{X} \otimes ℋ_{Y} .

5-

ϱ = U_{X} \otimes U_{Y} σ U_{X}^{†} \otimes U_{Y}^{†} .

6-

ϱ_{X} = {Tr}_{Y} ϱ

7-

σ_{X} = {Tr}_{Y} σ

8-

ϱ_{Y} = {Tr}_{X} ϱ

9-

σ_{Y} = {Tr}_{X} σ

10-

e i g (ϱ)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1501.03960

Formulas:

1-

w = - 1 + h (t), | h (t) | ≪ 1 .

2-

a (t) = e^{f (t)}

3-

w = \frac{p}{ρ} = - \frac{1}{3} - \frac{2}{3} \frac{a \ddot{a}}{{\dot{a}}^{2}} \Rightarrow f (t) = \frac{2}{3} \int \frac{d t}{\int h (t) 𝑑 t + k} + C .

4-

a (t) \to e^{C} a (t)

5-

ρ = {(\frac{\dot{a}}{a})}^{2} = {\dot{f}}^{2} \Rightarrow \sqrt{ρ} = \frac{2}{3 \int h + k},

6-

h (t) = α {(- t)}^{p}

7-

w (0) = - 1

8-

\sqrt{ρ (t)} = - \frac{2 (p + 1)}{3 α} \frac{1}{k + {(- t)}^{p + 1}} .

9-

ρ \sim {(t - t_{0})}^{- 2}

10-

β = 2 (1 + p) / 3 α p

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1704.03805

Formulas:

1-

X = {x_{1}, \dots, x_{n}}

2-

Y = {y_{1}, \dots, y_{m}}

3-

p, q, X, Y

4-

MMD [ℱ, p, q] := sup_{f \in ℱ} (E_{x \sim p} [f (x)] - E_{y \sim q} [f (y)])

5-

MMD [ℱ, X, Y] := sup_{f \in ℱ} (\frac{1}{n} \sum_{i = 1}^{n} f (x_{i}) - \frac{1}{m} \sum_{i = 1}^{m} f (y_{i}))

6-

MMD [ℱ, X, Y] = 0

7-

\begin{matrix} MSML (X, Y, Z) = {∥ E_{x \sim p} [f (x)] - E_{y \sim p} [f (y)] ∥}_{2} \\ + {[α - {∥ E_{x \sim p} [f (x)] - E_{z \sim q} [f (z)] ∥}_{2}]}_{+} \end{matrix}

8-

m a x (\cdot, 0)

9-

ω (x) = p (x) / \hat{p} (x)

10-

\begin{matrix} \underset{ω (x) \geq 0}{m i n i m i z e} {∥ E_{p} [f (x)] - E_{\hat{p}} [ω (x) f (x)] ∥}_{2} \\ s . t . E_{\hat{p}} [ω (x)] = 1 \end{matrix}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0008160

Formulas:

1-

1 H z - 40 H z

2-

χ_{V q} (ω)

3-

ν = ω / 2 π

4-

δ V (ω)

5-

δ q (ω)

6-

χ_{V q} (ω) = δ V (ω) / δ q (ω)

7-

S (ω) = \frac{2 k_{B} T}{π ω} 𝐼𝑚 [χ_{V q} (ω)]

8-

S (ω) = < {| V (ω) |}^{2} >

9-

𝐼𝑚 [χ_{V q} (ω)]

10-

χ_{V q} (ω)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1204.3964

Formulas:

1-

3 s^{2} {}^{1}S_{0}

2-

3 s 3 p {}^{3}P_{0}

3-

3 s 3 p {}^{1}P_{1}

4-

n s^{2} {}^{1}S_{0}

5-

n s n p {}^{3}P_{0}

6-

3 s^{2} {}^{1}S_{0}

7-

α_{0} = 11.666 (4)

8-

α_{0} = 11.669 (9)

9-

3 s^{2} {}^{1}S_{0}

10-

3 p^{2} {}^{1}D_{2}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1507.01023

Formulas:

1-

O (n / e^{Θ (\sqrt{\log n})})

2-

O (n \cdot \frac{{(\log \log n)}^{2}}{\log n})

3-

| A | \leq 2 n / 3

4-

| B | \leq 2 n / 3

5-

| C | \leq c \sqrt{n}

6-

G_{i} = A_{i} \cup B_{i} \cup C_{i}

7-

| A_{i} | \leq 2 | G_{i} | / 3

8-

| B_{i} | \leq 2 | G_{i} | / 3

9-

| C_{i} | \leq c \sqrt{| G_{i} |}

10-

c \sqrt{| G_{i} |}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/2005.14602

Formulas:

1-

S I_{1} I_{2}

2-

S I_{1} I_{2}

3-

S I_{1} I_{2}

4-

\begin{matrix} S^{'} & = - β_{1} S I_{1} - β_{2} S I_{2} + f_{0} S \\ I_{1}^{'} & = β_{1} S I_{1} - θ I_{1} + f_{1} I_{1} \\ I_{2}^{'} & = β_{2} S I_{2} + θ I_{1} + f_{2} I_{2} \end{matrix}

5-

f_{0} = μ_{0} - δ_{0}

6-

f_{1} = μ_{1} - δ_{1}

7-

f_{2} = μ_{2} - δ_{2}

8-

S I_{1} I_{2}

9-

S + I_{1} + I_{2}

10-

\begin{matrix} θ - f_{1} > 0 \\ f_{2} < 0, \end{matrix}

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/9408308

Formulas:

1-

m_{n e w} > M_{Z} / 2

2-

\begin{matrix} S^{'} & = & S + 4 s_{w}^{2} c_{w}^{2} V + 4 (c_{w}^{2} - s_{w}^{2}) X & = & 0.0084 (ϵ_{3} - ϵ_{3}^{S M}) \\ T^{'} & = & T + V & = & 0.0078 (ϵ_{1} - ϵ_{1}^{S M}) \end{matrix}

3-

m_{t o p} = 150 G e V

4-

m_{H} = 300 G e V

5-

\begin{matrix} - 0.49 < S^{'} < - 0.036 \\ - 0.13 < T^{'} < 0.32 \end{matrix}

6-

Z b \bar{b}

7-

B = 2.0 S^{'} - 4.8 T^{'} + 436 \frac{δ Γ_{b}}{Γ_{0}} = (ϵ_{b} - ϵ_{b}^{S M}) \times 10^{3}

8-

Γ_{0} = 379.4 M e V

9-

m_{t o p} = 150 G e V

10-

m_{H} = 300 G e V

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: cs

Result: incorrect

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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xref="S1.E1.m1.8.8.2.3.cmml">⁢</mo><mrow id="S1.E1.m1.8.8.2.2.1" xref="S1.E1.m1.8.8.2.2.1.1.cmml"><mo stretchy="false" id="S1.E1.m1.8.8.2.2.1.2" xref="S1.E1.m1.8.8.2.2.1.1.cmml">(</mo><mrow id="S1.E1.m1.8.8.2.2.1.1" xref="S1.E1.m1.8.8.2.2.1.1.cmml"><mi id="S1.E1.m1.8.8.2.2.1.1.2" xref="S1.E1.m1.8.8.2.2.1.1.2.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S1.E1.m1.8.8.2.2.1.1.1" xref="S1.E1.m1.8.8.2.2.1.1.1.cmml">|</mo><mi mathvariant="normal" id="S1.E1.m1.8.8.2.2.1.1.3" xref="S1.E1.m1.8.8.2.2.1.1.3.cmml">Θ</mi></mrow><mo stretchy="false" id="S1.E1.m1.8.8.2.2.1.3" xref="S1.E1.m1.8.8.2.2.1.1.cmml">)</mo></mrow><mo id="S1.E1.m1.8.8.2.3c" xref="S1.E1.m1.8.8.2.3.cmml">⁢</mo><mi id="S1.E1.m1.8.8.2.6" xref="S1.E1.m1.8.8.2.6.cmml">P</mi><mo id="S1.E1.m1.8.8.2.3d" xref="S1.E1.m1.8.8.2.3.cmml">⁢</mo><mrow id="S1.E1.m1.8.8.2.7.2" xref="S1.E1.m1.8.8.2.cmml"><mo stretchy="false" id="S1.E1.m1.8.8.2.7.2.1" xref="S1.E1.m1.8.8.2.cmml">(</mo><mi mathvariant="normal" id="S1.E1.m1.6.6" xref="S1.E1.m1.6.6.cmml">Θ</mi><mo stretchy="false" id="S1.E1.m1.8.8.2.7.2.2" xref="S1.E1.m1.8.8.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.E2.m1.2.3" xref="S1.E2.m1.2.3.cmml"><mi id="S1.E2.m1.2.3.2" xref="S1.E2.m1.2.3.2.cmml">P</mi><mo id="S1.E2.m1.2.3.1" xref="S1.E2.m1.2.3.1.cmml">⁢</mo><mrow id="S1.E2.m1.2.3.3.2" xref="S1.E2.m1.2.3.3.1.cmml"><mo stretchy="false" id="S1.E2.m1.2.3.3.2.1" xref="S1.E2.m1.2.3.3.1.cmml">(</mo><mi id="S1.E2.m1.1.1" xref="S1.E2.m1.1.1.cmml">X</mi><mo id="S1.E2.m1.2.3.3.2.2" xref="S1.E2.m1.2.3.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E2.m1.2.2" xref="S1.E2.m1.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E2.m1.2.3.3.2.3" xref="S1.E2.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.E2.m2.3.4" xref="S1.E2.m2.3.4.cmml"><mi id="S1.E2.m2.3.4.2" xref="S1.E2.m2.3.4.2.cmml"/><mo id="S1.E2.m2.3.4.1" xref="S1.E2.m2.3.4.1.cmml">=</mo><mstyle displaystyle="true" id="S1.E2.m2.3.4.3" xref="S1.E2.m2.3.4.3.cmml"><mrow 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" 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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.m1.124.124.9.116.28.21.21.1.2a" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.3" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.19.19.19.19.12.12" xref="S2.E5.m1.19.19.19.19.12.12.cmml">log</mi><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.3a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.20.20.20.20.13.13" xref="S2.E5.m1.20.20.20.20.13.13.cmml">P</mi></mrow><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2b" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.4" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.21.21.21.21.14.14" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mi id="S2.E5.m1.22.22.22.22.15.15" xref="S2.E5.m1.22.22.22.22.15.15.cmml">X</mi><mo id="S2.E5.m1.23.23.23.23.16.16" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.24.24.24.24.17.17" xref="S2.E5.m1.24.24.24.24.17.17.cmml">Θ</mi><mo rspace="4.2pt" stretchy="false" id="S2.E5.m1.25.25.25.25.18.18" 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id="S2.E5.m1.66.66.66.13.13.13.2" xref="S2.E5.m1.66.66.66.13.13.13.2.cmml"><mi id="S2.E5.m1.66.66.66.13.13.13.2.4" xref="S2.E5.m1.66.66.66.13.13.13.2.4.cmml">P</mi><mo id="S2.E5.m1.66.66.66.13.13.13.2.3" xref="S2.E5.m1.66.66.66.13.13.13.2.3.cmml">⁢</mo><mrow id="S2.E5.m1.66.66.66.13.13.13.2.2.1" xref="S2.E5.m1.66.66.66.13.13.13.2.2.2.cmml"><mo stretchy="false" id="S2.E5.m1.66.66.66.13.13.13.2.2.1.2" xref="S2.E5.m1.66.66.66.13.13.13.2.2.2.cmml">(</mo><mi id="S2.E5.m1.66.66.66.13.13.13.1.1" xref="S2.E5.m1.66.66.66.13.13.13.1.1.cmml">X</mi><mo id="S2.E5.m1.66.66.66.13.13.13.2.2.1.3" xref="S2.E5.m1.66.66.66.13.13.13.2.2.2.cmml">,</mo><mrow id="S2.E5.m1.66.66.66.13.13.13.2.2.1.1" xref="S2.E5.m1.66.66.66.13.13.13.2.2.1.1.cmml"><mi id="S2.E5.m1.66.66.66.13.13.13.2.2.1.1.2" xref="S2.E5.m1.66.66.66.13.13.13.2.2.1.1.2.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.66.66.66.13.13.13.2.2.1.1.1" xref="S2.E5.m1.66.66.66.13.13.13.2.2.1.1.1.cmml">|</mo><mi 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Correct Categorie: stat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1609.03025

Formulas:

1-

ρ_{i} = (1 - ϵ) | vac ⟩ ⟨ vac | + ϵ η_{i}

2-

O (ϵ^{2})

3-

= | ψ_{1} ⟩ ⟨ ψ_{1} |,

4-

= \frac{1}{2} | ψ_{+} ⟩ ⟨ ψ_{+} | + \frac{1}{2} | ψ_{-} ⟩ ⟨ ψ_{-} |,

5-

\equiv \int_{- \infty}^{\infty} d x \int_{- \infty}^{\infty} d y ψ (x, y) | x, y ⟩,

6-

\equiv \int_{- \infty}^{\infty} d x \int_{- \infty}^{\infty} d y ψ (x \pm d / 2, y) | x, y ⟩,

7-

ψ (x, y)

8-

{E (z)}

9-

\int d μ (z) E (z) = 𝟙

10-

E_{1} = \int_{z \in Ω_{1}} d μ (z) E (z)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1909.10106

Formulas:

1-

N (t) \in ℕ

2-

𝒩 (t) = {1, 2, \dots, N (t)}

3-

t_{7}^{f} = t_{7}^{3}

4-

# 1, # 2,

5-

i \in 𝒩 (t)

6-

\begin{matrix} {\dot{p}}_{i} & = v_{i} (t) \\ {\dot{s}}_{i} & = ξ_{i} \cdot (v_{k} (t) - v_{i} (t)) \\ {\dot{v}}_{i} & = u_{i} (t) \end{matrix}

7-

p_{i} (t) \in 𝒫_{i}

8-

v_{i} (t) \in 𝒱_{i}

9-

u_{i} (t) \in 𝒰_{i}

10-

s_{i} (t) \in 𝒮_{i}

Formulas (html):

Correct Categorie: math

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0204529

Formulas:

1-

\sim 5 \times 10^{3} km s^{- 1}

2-

m ≲ 10^{- 4} M_{⊙}

3-

H_{0} = 65 km \sec^{- 1} {Mpc}^{- 1}

4-

\sim 5000 km \sec^{- 1}

5-

N_{cl} ≲ 10^{7 - 8}

6-

N_{cl} ≳ 10^{7 - 8}

7-

20 km s^{- 1}

8-

\sim 100 km s^{- 1}

9-

R = \frac{r_{sch}}{2 f_{g}} {(\frac{c}{σ_{cl}})}^{2},

10-

r_{sch} = 2 G M_{bh} / c^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/2004.13188

Formulas:

1-

w e i g h t \propto v o l u m e

2-

w e i g h t = v o l u m e \times d e n s i t y

3-

f o o d e n e r g y \propto f o o d v o l u m e

4-

f o o d e n e r g y = f o o d v o l u m e \times u n i t v o l u m e e n e r g y

5-

(𝐱, y, z)

6-

= \sum_{i = 1}^{n} - {\hat{y}}^{(i)} l o g [f_{c}^{(i)} (x)]

7-

= | z - f_{r} (x) |

8-

= \sum_{i = 1}^{m} {(p_{c}^{(i)} - p_{r}^{(i)})}^{2}

9-

ℒ_{o v e r a l l} = ℒ_{c} + ℒ_{r} + ℒ_{p s}

10-

ℛ^{224 \times 224 \times 3} \to ℛ^{1 \times 1 \times 1}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/0803.2660

Formulas:

1-

\frac{1}{2} | N_{B} - N_{A} |

2-

\frac{1}{2} | N_{B} - N_{A} |

3-

Δ E_{p c} = E (F M) - E (P M)

4-

Δ E_{a c} = E (A F) - E (P M)

5-

Δ E_{p c} = E (F M) - E (P M)

6-

Δ E_{a c} = E (A F) - E (P M)

7-

| Δ E_{p c} |

8-

| Δ E_{a c} |

9-

T_{c} = \frac{2 Δ}{3 k_{B}}

10-

L = 8 \sqrt{3} a

Formulas (html):

Correct Categorie: physics

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/1210.8384

Formulas:

1-

23^{0} 10^{'} N

2-

89^{0} 10^{'} E

3-

30^{0} 27^{'} N

4-

78^{0} 06^{'} E

5-

17^{0} 10^{'} N

6-

70^{0} 20^{'} E

7-

L o g a p e r t u r e i n i n c h e s \times 5 + {9.2}^{''}

8-

L o g 3 \times 5 + 9.2

9-

L o g 6.25 \times 5 + 9.2

10-

N o v a A q u i l a e 1918

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/2008.07930

Formulas:

1-

\vec{x}, (\vec{f_{a}}, \vec{f_{b}}) \in ℝ^{k^{2}}

2-

g (\vec{x})

3-

= (\vec{f_{a}} \vec{x}) (\vec{f_{b}} \vec{x})

4-

= (\sum_{i = 0}^{n} f_{a}^{i} x^{i}) (\sum_{j = 0}^{n} f_{b}^{j} x^{j})

5-

= \sum_{i = 0}^{n} \sum_{j = 0}^{n} w^{i j} x^{j} x^{i},

6-

n = k^{2} - 1

7-

g (\vec{x})

8-

q \cdot d_{o u t}

9-

d_{o u t}

10-

d_{o u t}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/9902288

Formulas:

1-

g^{2} {(φ - M_{i})}^{2} χ_{i}^{2}

2-

g (φ - M_{i}) \bar{ψ} ψ

3-

ℒ [ϕ, χ_{j k}, {\bar{ψ}}_{j k}, ψ_{j k}] = \frac{1}{2} {(\partial_{μ} ϕ)}^{2} - \frac{m^{2}}{2} ϕ^{2} - \frac{λ}{4!} ϕ^{4}

4-

+ \sum_{j = 1}^{N_{M}} \sum_{k = 1}^{N_{χ}} {\frac{1}{2} {(\partial_{μ} χ_{j k})}^{2} - \frac{f_{j k}}{4!} χ_{j k}^{4} - \frac{g_{j k}^{2}}{2} {(ϕ - M_{j})}^{2} χ_{j k}^{2}}

5-

+ \sum_{j = 1}^{N_{M}} \sum_{k = 1}^{N_{ψ}} {i {\bar{ψ}}_{j k} \partial ψ_{j k} - h_{j k} (ϕ - M_{j}) {\bar{ψ}}_{j k} ψ_{j k}},

6-

f_{j k}, g_{j k}^{2}, h_{j k}

7-

f_{j k} = f

8-

g_{j k} = h_{j k} = g

9-

N_{ψ} = N_{χ} / 4

10-

m_{χ_{j k}} (ϕ, T, {M})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/2005.10811

Formulas:

1-

e^{- i H t}

2-

P e^{- i H t}

3-

Q H P = - H

4-

Q e^{- i H t} P e^{- i H t} = e^{i H t} e^{- i H t} = 𝕀 𝕕

5-

𝒮 = {C_{1}, \dots, C_{m}}

6-

| 00 \dots 0 ⟩

7-

| 00 \dots 0 ⟩

8-

p : {0, 1}^{n} \to [0, 1]

9-

1 - p (0^{n})

10-

p (0^{n}) = 1 / 2

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1012.4634

Formulas:

1-

\sum_{v \in V (G)} \max {0, d^{+} (v) - d^{-} (v)}

2-

d^{+} (v)

3-

d^{-} (v)

4-

V (G) \times V (H)

5-

b d \in E (H)

6-

a b \in E (G)

7-

1 = b (P_{m}) \leq b (G) \leq b (K_{m}) = ⌊ \frac{m^{2}}{4} ⌋,

8-

b (P_{m} \times H) \leq b (G \times H) \leq b (K_{m} \times H) .

9-

w_{0} (v)

10-

α = (α_{1}, α_{2}, \dots, α_{n})

Formulas (html):

Correct Categorie: math

Guessed Categorie: gr-qc

Result: incorrect

Run 6249156 (TestAgent)