Run 11314749

Paper: https://arxiv.org/abs/1403.1056

Formulas:

1-

S y m_{d}^{+}

2-

S y m_{d}^{+}

3-

\log_{𝑿} = \exp_{𝑿}^{- 1}

4-

S y m_{d}^{+}

5-

\exp_{𝑿} (𝒚)

6-

𝑿^{\frac{1}{2}} \exp (𝑿^{- \frac{1}{2}} 𝒚 𝑿^{- \frac{1}{2}}) 𝑿^{\frac{1}{2}}

7-

\log_{𝑿} (𝒀)

8-

𝑿^{\frac{1}{2}} \log (𝑿^{- \frac{1}{2}} 𝒀 𝑿^{- \frac{1}{2}}) 𝑿^{\frac{1}{2}}

9-

𝑿 = 𝑼 Σ 𝑼^{T}

10-

𝑼 \exp (Σ) 𝑼^{T}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0109373

Formulas:

1-

1.6 < \sim z < \sim 3.2

2-

N (HI) > \sim 2 \times 10^{17}

3-

N (HI) > \sim 10^{19}

4-

2 \times 10^{17} < \sim N (HI) < \sim 10^{19}

5-

N (HI) > \sim 10^{19}

6-

𝒫 (z) ≃ 0.001

7-

\sim 2 h_{100}^{- 1}

8-

N (HI) < \sim 10^{19}

9-

\sim 3.8 h_{100}^{- 1}

10-

\sim 6.7 h_{100}^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1708.02522

Formulas:

1-

Γ^{2} (E)

2-

Γ (E) = \sum_{i} \frac{α_{i} \cdot β_{i}}{E - ε_{i}}

3-

Γ^{2} (E)

4-

Γ (E) = γ_{1} / (E - ε_{1}) + γ_{2} / (E - ε_{2}) + γ_{3} / (E - ε_{3})

5-

γ_{i} = α_{i} β_{i}

6-

γ_{1} + γ_{2} + γ_{3}

7-

E_{0} = ε_{1} + \frac{1}{1 + \frac{γ_{3} (ε_{3} - ε_{2})}{γ_{1} (ε_{1} - ε_{2})}} \cdot (ε_{3} - ε_{1}) = ε_{1} + F_{1} F_{2}

8-

F_{s p l i t t i n g}

9-

F_{2} = ε_{3} - ε_{1}

10-

F_{1} = 1 / (1 + (γ_{3} / γ_{1}) \cdot ((ε_{3} - ε_{2}) / (ε_{1} - ε_{2})))

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-lat/0601002

Formulas:

1-

(d - 1) * N^{(d - 1)}

2-

2 (d - 1)

3-

2 (d - 1) N

4-

2 (d - 1) N^{3}

5-

\exp (- m^{2} / 2 σ_{m}^{2})

6-

P (m) / m^{3}

7-

β_{a d j} = 1.5

8-

m_{bin}^{3} = \frac{1}{4} \frac{(m_{2}^{4} - m_{1}^{4})}{(m_{2} - m_{1})} .

9-

β_{a d j} = 1.5

10-

U = 1 - < m^{4} > / (3 {< m^{2} >}^{2}),

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0110427

Formulas:

1-

v (x, y)

2-

u (x, y)

3-

y \sim x^{2} / d

4-

v_{s} (s, z)

5-

s_{i}, i = 1 - 4

6-

v_{s} (s_{i}, z)

7-

v_{s} (s, z) = v_{0} \exp (- α_{1} z / d - α_{2} {(z / d)}^{2})

8-

v_{s} (s, z) = v_{0} \exp (- α_{1} z / d)

9-

Q \propto {(W - α d)}^{5 / 2}

10-

W \times \int v_{s} (s, z) 𝑑 z

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1610.04757

Formulas:

1-

ω (\vec{q}) \neq ω (- \vec{q})

2-

E (- \hat{y})

3-

ϵ_{11} = ϵ_{22} = \frac{2 α ϵ_{i} ϵ_{h} + (1 - α) ϵ_{h} (ϵ_{i} + ϵ_{h})}{(1 - α) (ϵ_{i} + ϵ_{h}) + 2 α ϵ_{h}}

4-

ϵ_{33} = α ϵ_{i} + (1 - α) ϵ_{h},

5-

ϵ_{11} = ϵ_{22} \approx ϵ_{h} - 2 α ϵ_{h} \frac{ϵ_{h} - ϵ_{i}}{ϵ_{h} + ϵ_{i}} .

6-

α = α (T, H)

7-

(α_{\infty} - α (T, H))

8-

ϵ_{h} \to ϵ_{h} [1 - 3 (α_{\infty} - α (T, H))]

9-

| ϵ_{i} | ≫ ϵ_{h}

10-

ϵ_{11} = ϵ_{22} \approx ϵ_{h} (1 + 3 α_{\infty} - α)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1504.01980

Formulas:

1-

ν \equiv n h / (e B) = 1

2-

n = (1.79 + 0.89 V_{G}) \times 10^{11}

3-

n = (1.79 + 0.69 V_{G}) \times 10^{11}

4-

R_{xx} (t) = R_{0} + Δ R_{xx} \exp (- t / τ) .

5-

Δ R_{xx} < 0

6-

Δ R_{xx} (t_{rest}) = Δ R_{\infty} + Δ R \exp (- t_{rest} / τ_{1}) .

7-

1 / T_{1} \propto | ν - 1 |

8-

| ν - 1 | \approx 0.18

9-

| ν - 1 | > 0.18

10-

Δ n (x, y)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0008273

Formulas:

1-

f = 1.1 \times 10^{- 13}

2-

\approx 15, 000 - 20, 000

3-

(1.8 - 9.3) \times 10^{7}

4-

r_{in} = 3 R_{s}

5-

M_{BH} \approx 5 \times 10^{7}

6-

n_{crit} = 1.6 \times 10^{3}

7-

n_{crit} = 1.6 \times 10^{6}

8-

n_{e} \propto r^{- 2}

9-

U = 10^{- 3} - 10^{- 3.5}

10-

n_{e} ≳ 2 \times 10^{4}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1308.6046

Formulas:

1-

Δ Ω \sim 10^{- 3} Ω

2-

q = - \frac{r}{Ω} \frac{d Ω}{d r}

3-

q = O (0.1)

4-

𝒖 \cdot (\nabla \times 𝒖) \neq 0

5-

(𝑽 \cdot \nabla) 𝑽 = - \frac{1}{ρ} \nabla P - \nabla ψ,

6-

𝑽 = 𝒆_{ϕ} r \sin θ Ω,

7-

Ω^{2} ≪ G M / R^{3}

8-

s = c_{v} \ln (P) - c_{p} \ln (ρ)

9-

N^{2} = \frac{g}{c_{p}} \frac{\partial s}{\partial r} > 0 .

10-

r \sin θ \frac{\partial Ω^{2}}{\partial z} = - \frac{1}{ρ^{2}} {(\nabla ρ \times \nabla P)}_{ϕ},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1407.6472

Formulas:

1-

χ_{c} = \arcsin 1 / n

2-

n = n_{1} / n_{2}

3-

χ_{in}^{eff} = χ_{in}

4-

χ_{in}^{eff} < χ_{in}

5-

n = n_{1} / n_{2}

6-

χ_{out} = χ_{in} + Δ χ_{FF}

7-

Δ χ_{F F}

8-

χ_{B} = \arctan 1 / n < χ_{c}

9-

E_{I} (α) = \sum_{m} e_{I} (m) e^{i m α} J_{m} (n k R)

10-

k = 2 π / λ

Formulas (html):

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xref="p5.2.m2.1.1.3.3.3.cmml">F</mi></mrow></msub></mrow></math>, <math><mrow id="p6.2.m2.1.1" xref="p6.2.m2.1.1.cmml"><msub id="p6.2.m2.1.1.2" xref="p6.2.m2.1.1.2.cmml"><mi id="p6.2.m2.1.1.2.2" xref="p6.2.m2.1.1.2.2.cmml">χ</mi><mi id="p6.2.m2.1.1.2.3" xref="p6.2.m2.1.1.2.3.cmml">B</mi></msub><mo id="p6.2.m2.1.1.3" xref="p6.2.m2.1.1.3.cmml">=</mo><mrow id="p6.2.m2.1.1.4" xref="p6.2.m2.1.1.4.cmml"><mi id="p6.2.m2.1.1.4.1" xref="p6.2.m2.1.1.4.1.cmml">arctan</mi><mo id="p6.2.m2.1.1.4a" xref="p6.2.m2.1.1.4.cmml">⁡</mo><mrow id="p6.2.m2.1.1.4.2" xref="p6.2.m2.1.1.4.2.cmml"><mn id="p6.2.m2.1.1.4.2.2" xref="p6.2.m2.1.1.4.2.2.cmml">1</mn><mo id="p6.2.m2.1.1.4.2.1" xref="p6.2.m2.1.1.4.2.1.cmml">/</mo><mi id="p6.2.m2.1.1.4.2.3" xref="p6.2.m2.1.1.4.2.3.cmml">n</mi></mrow></mrow><mo id="p6.2.m2.1.1.5" xref="p6.2.m2.1.1.5.cmml"><</mo><msub id="p6.2.m2.1.1.6" xref="p6.2.m2.1.1.6.cmml"><mi id="p6.2.m2.1.1.6.2" xref="p6.2.m2.1.1.6.2.cmml">χ</mi><mi id="p6.2.m2.1.1.6.3" xref="p6.2.m2.1.1.6.3.cmml">c</mi></msub></mrow></math>, <math><mrow id="p7.6.m6.3.3" xref="p7.6.m6.3.3.cmml"><mrow id="p7.6.m6.3.3.3" xref="p7.6.m6.3.3.3.cmml"><msub id="p7.6.m6.3.3.3.2" xref="p7.6.m6.3.3.3.2.cmml"><mi id="p7.6.m6.3.3.3.2.2" xref="p7.6.m6.3.3.3.2.2.cmml">E</mi><mi id="p7.6.m6.3.3.3.2.3" xref="p7.6.m6.3.3.3.2.3.cmml">I</mi></msub><mo id="p7.6.m6.3.3.3.1" xref="p7.6.m6.3.3.3.1.cmml">⁢</mo><mrow id="p7.6.m6.3.3.3.3.2" xref="p7.6.m6.3.3.3.cmml"><mo stretchy="false" id="p7.6.m6.3.3.3.3.2.1" xref="p7.6.m6.3.3.3.cmml">(</mo><mi id="p7.6.m6.1.1" xref="p7.6.m6.1.1.cmml">α</mi><mo stretchy="false" id="p7.6.m6.3.3.3.3.2.2" xref="p7.6.m6.3.3.3.cmml">)</mo></mrow></mrow><mo id="p7.6.m6.3.3.2" xref="p7.6.m6.3.3.2.cmml">=</mo><mrow id="p7.6.m6.3.3.1" xref="p7.6.m6.3.3.1.cmml"><msub id="p7.6.m6.3.3.1.2" xref="p7.6.m6.3.3.1.2.cmml"><mo largeop="true" symmetric="true" id="p7.6.m6.3.3.1.2.2" xref="p7.6.m6.3.3.1.2.2.cmml">∑</mo><mi id="p7.6.m6.3.3.1.2.3" xref="p7.6.m6.3.3.1.2.3.cmml">m</mi></msub><mrow id="p7.6.m6.3.3.1.1" xref="p7.6.m6.3.3.1.1.cmml"><msub id="p7.6.m6.3.3.1.1.3" xref="p7.6.m6.3.3.1.1.3.cmml"><mi id="p7.6.m6.3.3.1.1.3.2" xref="p7.6.m6.3.3.1.1.3.2.cmml">e</mi><mi id="p7.6.m6.3.3.1.1.3.3" xref="p7.6.m6.3.3.1.1.3.3.cmml">I</mi></msub><mo id="p7.6.m6.3.3.1.1.2" xref="p7.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="p7.6.m6.3.3.1.1.4.2" xref="p7.6.m6.3.3.1.1.cmml"><mo stretchy="false" id="p7.6.m6.3.3.1.1.4.2.1" xref="p7.6.m6.3.3.1.1.cmml">(</mo><mi id="p7.6.m6.2.2" xref="p7.6.m6.2.2.cmml">m</mi><mo stretchy="false" id="p7.6.m6.3.3.1.1.4.2.2" xref="p7.6.m6.3.3.1.1.cmml">)</mo></mrow><mo id="p7.6.m6.3.3.1.1.2a" xref="p7.6.m6.3.3.1.1.2.cmml">⁢</mo><msup id="p7.6.m6.3.3.1.1.5" xref="p7.6.m6.3.3.1.1.5.cmml"><mi id="p7.6.m6.3.3.1.1.5.2" xref="p7.6.m6.3.3.1.1.5.2.cmml">e</mi><mrow id="p7.6.m6.3.3.1.1.5.3" xref="p7.6.m6.3.3.1.1.5.3.cmml"><mi id="p7.6.m6.3.3.1.1.5.3.2" xref="p7.6.m6.3.3.1.1.5.3.2.cmml">i</mi><mo id="p7.6.m6.3.3.1.1.5.3.1" xref="p7.6.m6.3.3.1.1.5.3.1.cmml">⁢</mo><mi id="p7.6.m6.3.3.1.1.5.3.3" xref="p7.6.m6.3.3.1.1.5.3.3.cmml">m</mi><mo id="p7.6.m6.3.3.1.1.5.3.1a" xref="p7.6.m6.3.3.1.1.5.3.1.cmml">⁢</mo><mi id="p7.6.m6.3.3.1.1.5.3.4" xref="p7.6.m6.3.3.1.1.5.3.4.cmml">α</mi></mrow></msup><mo id="p7.6.m6.3.3.1.1.2b" xref="p7.6.m6.3.3.1.1.2.cmml">⁢</mo><msub id="p7.6.m6.3.3.1.1.6" xref="p7.6.m6.3.3.1.1.6.cmml"><mi id="p7.6.m6.3.3.1.1.6.2" xref="p7.6.m6.3.3.1.1.6.2.cmml">J</mi><mi id="p7.6.m6.3.3.1.1.6.3" xref="p7.6.m6.3.3.1.1.6.3.cmml">m</mi></msub><mo id="p7.6.m6.3.3.1.1.2c" xref="p7.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="p7.6.m6.3.3.1.1.1.1" xref="p7.6.m6.3.3.1.1.1.1.1.cmml"><mo stretchy="false" id="p7.6.m6.3.3.1.1.1.1.2" xref="p7.6.m6.3.3.1.1.1.1.1.cmml">(</mo><mrow id="p7.6.m6.3.3.1.1.1.1.1" xref="p7.6.m6.3.3.1.1.1.1.1.cmml"><mi id="p7.6.m6.3.3.1.1.1.1.1.2" xref="p7.6.m6.3.3.1.1.1.1.1.2.cmml">n</mi><mo id="p7.6.m6.3.3.1.1.1.1.1.1" xref="p7.6.m6.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="p7.6.m6.3.3.1.1.1.1.1.3" xref="p7.6.m6.3.3.1.1.1.1.1.3.cmml">k</mi><mo id="p7.6.m6.3.3.1.1.1.1.1.1a" xref="p7.6.m6.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="p7.6.m6.3.3.1.1.1.1.1.4" xref="p7.6.m6.3.3.1.1.1.1.1.4.cmml">R</mi></mrow><mo stretchy="false" id="p7.6.m6.3.3.1.1.1.1.3" xref="p7.6.m6.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="p7.7.m7.1.1" xref="p7.7.m7.1.1.cmml"><mi id="p7.7.m7.1.1.2" xref="p7.7.m7.1.1.2.cmml">k</mi><mo id="p7.7.m7.1.1.1" xref="p7.7.m7.1.1.1.cmml">=</mo><mrow id="p7.7.m7.1.1.3" xref="p7.7.m7.1.1.3.cmml"><mrow id="p7.7.m7.1.1.3.2" xref="p7.7.m7.1.1.3.2.cmml"><mn id="p7.7.m7.1.1.3.2.2" xref="p7.7.m7.1.1.3.2.2.cmml">2</mn><mo id="p7.7.m7.1.1.3.2.1" xref="p7.7.m7.1.1.3.2.1.cmml">⁢</mo><mi id="p7.7.m7.1.1.3.2.3" xref="p7.7.m7.1.1.3.2.3.cmml">π</mi></mrow><mo id="p7.7.m7.1.1.3.1" xref="p7.7.m7.1.1.3.1.cmml">/</mo><mi id="p7.7.m7.1.1.3.3" xref="p7.7.m7.1.1.3.3.cmml">λ</mi></mrow></mrow></math>

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1904.11087

Formulas:

1-

𝐗_{n \times n} = [𝐱_{0}, 𝐱_{1}, \dots, 𝐱_{n - 1}]

2-

𝐱_{0} = {(1, \dots, 1)}^{'}

3-

𝐱_{j} = {(\pm 1, \dots, \pm 1)}^{'}, j = 1, \dots, n - 1

4-

𝐘 = 𝐗 𝜷 + ϵ

5-

𝐘 = {(y_{1}, y_{2}, \dots, y_{n})}^{'}

6-

β_{1}, \dots, β_{n - 1}

7-

ϵ = {(ϵ_{1}, \dots, ϵ_{n})}^{'}

8-

ϵ_{i}, i = 1, \dots, n

9-

N (0, σ^{2})

10-

{\hat{β}}_{j}, j = 1, \dots, n - 1

Formulas (html):

Correct Categorie: stat

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1811.00202

Formulas:

1-

𝒳 \in ℝ^{W \times H \times K}

2-

𝒳_{k} \in ℝ^{W \times H}

3-

{[F_{1}, \dots, F_{K}]}^{T}

4-

{[F_{1}^{(SPoC)}, \dots, F_{K}^{(SPoC)}]}^{T}, F_{k}^{(SPoC)} = \frac{1}{| 𝒳_{k} |} \sum_{x \in 𝒳_{k}} x,

5-

{[F_{1}^{(MAC)}, \dots, F_{K}^{(MAC)}]}^{T}, F_{k}^{(MAC)} = \max_{x \in 𝒳_{k}} x,

6-

{[F_{1}^{(GeM)}, \dots, F_{K}^{(GeM)}]}^{T}, F_{k}^{(GeM)} = {(\frac{1}{| 𝒳_{k} |} \sum_{x \in 𝒳_{k}} x^{p_{k}})}^{\frac{1}{p_{k}}},

7-

{B_{2}, B_{3}, B_{4}, B_{5}}

8-

i \in {2, 3, 4, 5}

9-

𝒜_{4, 23} \otimes 𝒳_{5, 1}

10-

𝒜_{5, 1} \otimes 𝒳_{5, 2}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1709.03604

Formulas:

1-

Ψ_{tot} (𝐫, t) = Ψ_{h} (𝐫) + Ψ_{d} (𝐫) \times (1 + ζ (t) Ψ_{b} (𝐫)),

2-

Ψ_{b} = \cos (2 [φ - Ω_{b} t]) \times {\begin{matrix} - {(r_{b} / r)}^{3} & if r \geq r_{b}, \\ {(r / r_{b})}^{3} - 2 & if r \leq r_{b}, \end{matrix}

3-

ζ (t) = ε_{b} (\frac{3}{16} ξ {(t)}^{5} - \frac{5}{8} ξ {(t)}^{3} + \frac{15}{16} ξ (t) + \frac{1}{2}), ξ (t) \equiv 2 \frac{t}{h} - 1,

4-

ζ (t) = ε_{b}

5-

(2 - 4) \times 10^{6}

6-

SFE = Σ_{sfr} / Σ_{gas}

7-

(1 - 2) \times 10^{- 3}

8-

σ_{gas} = \sqrt{\frac{1}{3} (σ_{R}^{2} + σ_{ϕ}^{2} + σ_{z}^{2})}

9-

h_{gas} \propto σ_{gas}^{2}

10-

A_{2} = \sum_{k = 1}^{N} m_{k} \exp (2 i ϕ_{k}),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2010.15767

Formulas:

1-

ξ^{a} (x)

2-

L_{ξ} g_{d, m n} = 0

3-

x_{m} \to x_{m} + ξ^{m} (x)

4-

\partial_{a} J^{a} = 0

5-

J^{a} = \frac{\partial L_{m} (χ)}{\partial (\partial_{a} χ)} ξ^{b} \partial_{b} χ - ξ^{a} L_{m} (χ),

6-

Q = \int J^{0} \sqrt{| g_{4} |} \sqrt{| g_{d} |} d^{3} x d^{d} y .

7-

Q = Q (t)

8-

S = \frac{m_{D}^{D - 2}}{2} \int d^{D} Z \sqrt{| G |} [f (R) + L_{m}], f (R) = a R^{2} + R + c .

9-

L_{m} = \frac{1}{2} G^{M N} \partial_{M} χ \partial_{N} χ - V (χ), V (χ) = \frac{1}{2} m^{2} χ^{2} .

10-

d s^{2} = G_{M N} d Z^{M} d Z^{N} = g_{μ ν} (x) d x^{μ} d x^{ν} + g_{d, m n} (x, y) d y^{m} d y^{n},

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0607295

Formulas:

1-

5600 < T_{eff} < 5900

2-

5600 < T_{eff} < 5900

3-

u v b y

4-

5600 < T_{eff} < 5850

5-

5600 < T_{eff} < 5850

6-

5600 < T_{eff} < 5850

7-

5600 < T_{eff} < 5850

8-

5600 < T_{eff} < 5850

9-

- 0.2 < [Fe/H] < + 0.4

10-

5600 < T_{eff} < 5900

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0307607

Formulas:

1-

{(ω_{B} / ω_{D})}^{2} ≃ 3.1

2-

{(ω_{B} / ω_{D})}^{2} ≃ 4

3-

γ = \frac{m g_{1 D}}{ℏ^{2} n_{1 D}}

4-

l_{c} = ℏ / \sqrt{m n_{1 D} g_{1 D}}

5-

μ, k_{B} T ≪ ℏ ω_{r}

6-

a_{r} = \sqrt{ℏ / m ω_{r}}

7-

g_{1 D} = - \frac{2 ℏ^{2}}{m a_{1 D}}

8-

a_{1 D} ≃ - a_{r}^{2} / a

9-

μ = 2 ℏ ω_{r} n_{1 D} a

10-

n_{1 D} ≪ 1 / a

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1811.11625

Formulas:

1-

2 mm d^{- 1}

2-

7 mm d^{- 1}

3-

B = T_{p} - T_{e},

4-

2 × 10^{- 5} kg m^{- 2} s^{- 1}

5-

0.5 m s^{- 1}

6-

1 m s^{- 1}

7-

5 m s^{- 1}

8-

- 3 × 10^{- 6} s^{- 1}

9-

3 × 10^{- 6} s^{- 1}

10-

0.5 × 10^{- 6} s^{- 1}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1901.09208

Formulas:

1-

p (W_{i j}^{k}) = \frac{ϵ (n^{k} + n^{k - 1})}{n^{k} n^{k - 1}}

2-

n^{k}, n^{k - 1}

3-

𝐖^{k} \in 𝐑^{n^{k - 1} \times n^{k}}

4-

n^{k} n^{k - 1}

5-

n^{W} = ∣ 𝐖^{k} ∣ = ϵ (n^{k} + n^{k - 1})

6-

= σ (x_{t} \cdot 𝐖_{x i} + h_{t - 1} \cdot 𝐖_{h i} + b_{i})

7-

= σ (x_{t} \cdot 𝐖_{x f} + h_{t - 1} \cdot 𝐖_{h f} + b_{f})

8-

= σ (x_{t} \cdot 𝐖_{x o} + h_{t - 1} \cdot 𝐖_{h o} + b_{o})

9-

= 𝑡𝑎𝑛ℎ (x_{t} \cdot 𝐖_{x g} + h_{t - 1} \cdot 𝐖_{h g} + b_{g})

10-

= f_{t} \otimes c_{t - 1} + i_{t} \otimes g_{t}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/0902.0425

Formulas:

1-

\sim 3 \times 10^{- 4}

2-

M_{t o t} ≲ 0.15

3-

F_{A} / F_{B} \sim 40

4-

u g r i z

5-

42' \times 28'

6-

L_{A} / L_{B} \sim 80

7-

L_{A} / L_{B} = 1.6

8-

(λ F_{λ}) / d

9-

= (- 16.5, - 13.2, - 11.0)

10-

μ_{α, *} = + 8.4

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0809.3655

Formulas:

1-

0 \leq ϕ - ϕ_{0} \leq 2 π

2-

I^{t} (μ, μ_{0}, ϕ - ϕ_{0}) = μ_{0} R (μ, μ_{0}, ϕ - ϕ_{0}) F_{0} .

3-

R (μ, μ_{0}, ϕ - ϕ_{0})

4-

I^{t} (μ, μ_{0}, ϕ - ϕ_{0})

5-

R (μ, μ_{0}, ϕ - ϕ_{0})

6-

μ = μ_{0} = 0

7-

μ = g (μ_{0})

8-

g (μ_{0}) / μ_{0}

9-

g (μ_{0})

10-

g (μ_{0})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0705.3254

Formulas:

1-

τ_{s} = 3.7 \times 10^{- 4} \frac{L}{\sqrt{T_{0}}} = 120 \frac{L_{9}}{\sqrt{T_{0, 7}}}

2-

τ_{s d} = \frac{L}{\sqrt{2 k_{B} T_{0} / m}} \approx 25 \frac{L_{9}}{\sqrt{T_{0, 7}}}

3-

τ_{c} = \frac{3 n_{c} k_{B} T_{0} L^{2}}{2 / 7 κ T_{0}^{7 / 2}} = \frac{10.5 n_{c} k_{B} L^{2}}{κ T_{0}^{5 / 2}} \approx 50 \frac{n_{c, 10} L_{9}^{2}}{T_{0, 7}^{5 / 2}}

4-

κ = 9 \times 10^{- 7}

5-

τ_{r} = \frac{3 k_{B} T_{M}}{n_{M} P (T)} = \frac{3 k_{B} T_{M}}{b T_{M}^{α} n_{M}} \approx 9 \times 10^{3} \frac{T_{M, 7}^{3 / 2}}{n_{M, 10}}

6-

P (T) = b T^{α}

7-

b = 1.5 \times 10^{- 19}

8-

α = - 1 / 2

9-

L = 2 \times 10^{9}

10-

t_{h e a t}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0808.1008

Formulas:

1-

\begin{matrix} H_{eff} = v (p_{x} - e A_{x}) σ_{x} \otimes τ_{z} + v (p_{y} - e A_{y}) σ_{y} \otimes τ_{0} \\ + m (x, y) σ_{z} \otimes τ_{0}, \end{matrix}

2-

m (x, y) \equiv 0

3-

𝒯 = (σ_{0} \otimes τ_{x}) 𝒞, 𝒯_{sl} = - i (σ_{y} \otimes τ_{0}) 𝒞, 𝒯_{v} = - i (σ_{0} \otimes τ_{y}) 𝒞 .

4-

𝒯_{v}^{2} = - I

5-

H_{tb} = \sum_{⟨ i, j ⟩} t_{i j} c_{i}^{†} c_{j} + \sum_{i} m_{i} c_{i}^{†} c_{i}

6-

m_{i} = m (x_{i}, y_{i})

7-

t_{i j} = - t \exp (i \frac{2 π}{Φ_{0}} \int_{{\vec{r}}_{i}}^{{\vec{r}}_{j}} \vec{A} \cdot 𝑑 \vec{r})

8-

Φ_{0} = h / e

9-

m (x, y) = ω^{2} {[δ (x, y) - W]}^{2} / 2

10-

δ (x, y)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0310563

Formulas:

1-

\sim 3 \times 10^{- 10} e r g s c m^{- 2} s^{- 1}

2-

\sim 3 \times 10^{- 11} e r g s c m^{- 2} s^{- 1}

3-

S = \sum_{i = 1}^{N} \frac{{(r_{i} - \bar{r})}^{2}}{\bar{r}} t_{i},

4-

\sim 3 \times 10^{- 11} e r g s c m^{- 2} s^{- 1}

5-

4.4 \times 10^{- 8} s / s

6-

ϕ (t) = ϕ (t_{0}) + ν_{0} (t - t_{0}) + \frac{1}{2} \dot{ν_{0}} {(t - t_{0})}^{2} + \frac{1}{6} \ddot{ν_{0}} {(t - t_{0})}^{3} + \dots

7-

≲ 2 \times 10^{- 10} e r g s c m^{- 2} s^{- 1}

8-

f_{p p} \equiv (F_{m a x} - F_{m i n}) / F_{m a x}

9-

19 \times 10^{22} c m^{- 2}

10-

r_{A} = 5.2 \times 10^{8} c m L_{x, 36}^{- 2 / 7} B_{12}^{4 / 7} M_{1.4}^{1 / 7} R_{6}^{10 / 7},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1906.11373

Formulas:

1-

\frac{\sum_{i = 1}^{n} {(x_{i} - μ)}^{2}}{n}

2-

\frac{\sum_{i = 1}^{n} {(y_{i} - μ)}^{2}}{n}

3-

\frac{\sum_{i = 1}^{n} {(s_{i} - μ)}^{2}}{n}

4-

\frac{\sum_{i = 1}^{n} {(d - μ)}^{2}}{n}

5-

d = \min_{j \in O} \sqrt{{(j_{x} - x_{i})}^{2} + {(j_{y} - y_{i})}^{2}}

6-

\frac{\sum_{i = 1}^{n} {(d - μ)}^{2}}{n}

7-

d = \min_{j \in D} \sqrt{{(j_{x} - x_{i})}^{2} + {(j_{y} - y_{i})}^{2}}

8-

\frac{\sum_{i = 1}^{n} d}{n}

9-

d = \min_{j \in O} \sqrt{{(j_{x} - x_{i})}^{2} + {(j_{y} - y_{i})}^{2}}

10-

\frac{\sum_{i = 1}^{n} d}{n}

Formulas (html):

Correct Categorie: stat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1012.0426

Formulas:

1-

R e = R^{+} = R u_{τ} / ν

2-

𝐯 (y, x, θ, t) = \sum_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} 𝐀_{k, n, ω} (y) e^{i (ω t - k x - n θ)} 𝑑 k 𝑑 ω,

3-

ℒ_{k, n, ω} (y)

4-

𝐯 (y, k, n, ω) = {(i ω - ℒ_{k, n, ω} (y))}^{- 1} 𝐟 (y, k, n, ω)

5-

{(i ω - ℒ_{k, n, ω} (y))}^{- 1}

6-

(k, n, ω) = (0, 0, 0)

7-

R e = 2 \times 10^{4}

8-

R e = 2 \times 10^{3}

9-

(λ_{x}^{+}, λ_{z}^{+}, U_{x}^{+}) = (1000, 100, 10)

10-

(λ_{x}^{+}, λ_{z}^{+}, U_{x}^{+}) = (1000, 100, 10)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0811.2022

Formulas:

1-

ψ_{\pm 1, 0} (x, t)

2-

m_{F} = \pm 1, 0

3-

i \partial_{t} ψ_{\pm 1}

4-

ℒ ψ_{\pm 1} + ν_{s} S ψ_{\pm 1} + ν_{a} (S - 2 {| ψ_{\mp 1} |}^{2}) ψ_{\pm 1} + ν_{a} ψ_{0}^{2} ψ_{\mp 1}^{*}

5-

i \partial_{t} ψ_{0}

6-

ℒ ψ_{0} + ν_{s} S ψ_{0} + ν_{a} (S - {| ψ_{0} |}^{2}) ψ_{0} + 2 ν_{a} ψ_{0}^{*} ψ_{1} ψ_{- 1} .

7-

S = {| ψ_{- 1} |}^{2} + {| ψ_{0} |}^{2} + {| ψ_{1} |}^{2}

8-

ℒ ψ_{j} = - \frac{1}{2} \partial_{x}^{2} ψ_{j} + V (x) ψ_{j},

9-

j = - 1, 0, 1

10-

V (x) = \frac{1}{2} Ω^{2} x^{2} + V_{0} {sech}^{2} (x / w) .

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/2002.07414

Formulas:

1-

10^{6} \leq 𝑅𝑎 \leq 6 \times 10^{7}

2-

𝐻𝑎 = B_{0} H \sqrt{\frac{σ}{ρ ν}}, 𝑃𝑚 = μ_{0} σ ν, 𝑃𝑟 = \frac{ν}{κ}, 𝑅𝑎 = \frac{g α Δ T H^{3}}{ν κ},

3-

{𝑅𝑎}_{c} \approx π^{2} {𝐻𝑎}^{2}

4-

{𝐻𝑎}_{c} \approx \sqrt{R a} / π

5-

D = 180 mm = 2 R

6-

Γ = D / H = 1

7-

σ = 3.2 \times 10^{6} S m^{- 1}

8-

ν = 3.2 \times 10^{- 7} m^{2} s^{- 1}

9-

κ = 1.1 \times 10^{- 5} m^{2} s^{- 1}

10-

η = {(μ_{0} σ)}^{- 1} = 0.25 m^{2} s^{- 1}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/hep-ex/0606034

Formulas:

1-

d_{μ} \sim 5 \times 10^{- 23} e cm

2-

d_{μ} \sim 5 \times 10^{- 25} e cm

3-

1.8 \times 10^{- 19} e cm

4-

< 10^{- 36} e cm

5-

d_{e} < 2.2 \times 10^{- 27} e cm

6-

d_{μ} ≃ (m_{μ} / m_{e}) d_{e} < 5 \times 10^{- 25} e cm

7-

10^{- 22} e cm

8-

10^{- 23} e cm

9-

a_{μ} = a_{μ}^{SM} + a_{μ}^{NP}

10-

a_{μ} = (g_{μ} - 2) / 2

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ex

Result: correct

Paper: https://arxiv.org/abs/1403.5377

Formulas:

1-

(c / 6) \ln n

2-

(p - 2) (q - 2) > 4

3-

r = 1 / \sqrt{- K}

4-

(c / 6) \ln n

5-

H {σ} = - J \sum_{⟨ i, j ⟩} σ_{i} σ_{j},

6-

Z = \sum_{{σ}} \exp [- \frac{1}{k_{B} T} H {σ}]

7-

ℒ_{1} = ℛ_{1} = 𝒲

8-

𝒞_{M + 1} = 𝒲 ℒ_{M} 𝒞_{M} ℛ_{M}

9-

6 (M + 1) M + 1

10-

𝒲 ℒ_{M} 𝒞_{M},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0505188

Formulas:

1-

E (S, S^{z}) = J S (S + 1) / 2 - g μ_{B} H S^{z}

2-

S^{z} = n - 1

3-

H_{n} = J n / (g μ_{B})

4-

H = H_{e x}

5-

H_{e x} = 2 J S / (g μ_{B})

6-

H < H_{e x}

7-

S^{z} = 2 S

8-

H > H_{e x} = J S / (g μ_{B})

9-

\begin{matrix} ℋ = J [S_{1}^{z} S_{2}^{z} + (1 - κ) 𝑺_{1}^{⟂} 𝑺_{2}^{⟂}] - \\ - \frac{B}{2} [{(S_{1}^{z})}^{2} + {(S_{2}^{z})}^{2}] - g μ_{B} H (S_{1}^{z} + S_{2}^{z}) . \end{matrix}

10-

𝒍 = (𝑺_{1} - 𝑺_{2}) / | 𝑺_{1} - 𝑺_{2} |

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1701.02492

Formulas:

1-

v ≃ {(G M / 2 R)}^{1 / 2}

2-

t_{r} = {(\frac{2}{3})}^{1 / 2} \frac{v^{3}}{4 π G^{2} m^{2} n Λ},

3-

Λ = \ln (0.4 N)

4-

M_{c} \propto {[1 - (t - t_{i}) / (κ t_{r i})]}^{ν_{1}}, R_{c} \propto {[1 - (t - t_{i}) / (κ t_{r i})]}^{ν_{2}} .

5-

ν_{1} = 2 α_{2} / (7 α_{2} - 3 α_{1})

6-

ν_{2} = 2 (2 α_{2} - α_{1}) / (7 α_{2} - 3 α_{1})

7-

κ = 2 / (7 α_{2} - 3 α_{1})

8-

α_{1} = 8.72 \times 10^{- 4}

9-

α_{2} = 1.24 \times 10^{- 3}

10-

t_{e} \equiv κ t_{r, i} \approx 330 t_{r, i}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1508.04830

Formulas:

1-

\log (τ) = 0

2-

\log (τ) = 0

3-

\log (τ) = 0

4-

\log (τ) = 0

5-

\log (τ) = 0

6-

\log (τ) = - 2.5

7-

\partial B / \partial z

8-

| Δ χ | > 90^{\circ}

9-

| Δ χ | > 90^{\circ}

10-

\log (τ) = - 2.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nlin/0005008

Formulas:

1-

E (k) = C ε^{2 / 3} k^{- 5 / 3}

2-

Ω = 1 / 2 {| \nabla \times 𝐯 |}^{2}

3-

E (k) = C^{'} η^{2 / 3} k^{- 3}

4-

𝜹 𝒗 (𝐱, 𝐫) = 𝐯 (𝐱 + 𝐫) - 𝐯 (𝐱)

5-

P (δ v (r))

6-

P (δ v (r))

7-

S_{p} (r) = ⟨ δ v^{p} (r) ⟩ = \int_{- \infty}^{\infty} δ v^{p} P (δ v (r)) d (δ v) .

8-

G_{p} (r) = ⟨ {| δ v (r) |}^{p} ⟩

9-

S_{p} (r)

10-

G_{p} (r)

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0905.3702

Formulas:

1-

ψ_{j k} (\hat{γ}) = \sqrt{λ_{j k}} \sum_{ℓ} b (\frac{ℓ}{B^{j}}) \sum_{m = - ℓ}^{ℓ} {\bar{Y}}_{ℓ m} (\hat{γ}) Y_{ℓ m} (ξ_{j k})

2-

b (ℓ / B^{j})

3-

T (\hat{γ})

4-

ψ_{j k} (\hat{γ})

5-

β_{j k} = \sqrt{λ_{j k}} \sum_{ℓ} b (\frac{ℓ}{B^{j}}) \sum_{m = - ℓ}^{ℓ} a_{ℓ m} Y_{ℓ m} (ξ_{j k})

6-

S_{j_{1} j_{2} j_{3}}

7-

\frac{1}{{\tilde{N}}_{p}} \sum_{k} \frac{β_{j_{1} k} β_{j_{2} k} β_{j_{3} k}}{σ_{j_{1}} σ_{j_{2}} σ_{j_{3}}}

8-

\sum_{ℓ_{1} ℓ_{2} ℓ_{3}} b_{ℓ_{1}}^{(j_{1})} b_{ℓ_{2}}^{(j_{2})} b_{ℓ_{3}}^{(j_{3})} \sqrt{\frac{(2 ℓ_{1} + 1) (2 ℓ_{2} + 1) (2 ℓ_{3} + 1)}{4 π}}

9-

(\begin{matrix} ℓ_{1} & ℓ_{2} & ℓ_{3} \\ 0 & 0 & 0 \end{matrix}) {\hat{B}}_{ℓ_{1} ℓ_{2} ℓ_{3}}

10-

{\hat{B}}_{ℓ_{1} ℓ_{2} ℓ_{3}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0204526

Formulas:

1-

δ (2000) > 20^{\circ}

2-

\begin{matrix} b = B + c_{1, B} - 25 + c_{2, B} X + c_{3, B} (B - R) \\ r = R + c_{1, R} - 25 + c_{2, R} X + c_{3, R} (B - R) \end{matrix}

3-

μ (r) = μ_{0} + 1.0857 \frac{r}{h},

4-

μ_{0} = μ_{0}^{i} + 2.5 C \log (b / a),

5-

μ_{0, R} = 23

6-

- 2.5 \log {1 + \frac{b}{a} q (\frac{r_{\lim}}{h^{'}}) 10^{- 0.4 (m_{t} - m_{\lim})}},

7-

q (x) = (1 + x) e^{- x},

8-

m_{t} = μ_{0}^{'} - 5 \log h^{'} - 2.5 \log 2 π

9-

ϵ_{o u r} = 0

10-

ϵ_{U G C} = 0

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0101494

Formulas:

1-

^{3, 4, 5, 6}

2-

\approx 1.3 \times 10^{- 16}

3-

\approx 6.5 \times 10^{- 16}

4-

\overset{<}{\sim} 3 \times 10^{- 16}

5-

ℛ - K_{s} > 5.0

6-

\approx 1.9 \times 10^{- 17}

7-

\approx 17^{'} \times 19^{'}

8-

\approx 1.3 \times 10^{- 16}

9-

\approx 6.5 \times 10^{- 16}

10-

(1.6 \pm 0.4) \times 10^{20}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0207389

Formulas:

1-

E_{d e t} = 2.25

2-

E_{d e t} = 2.25

3-

E_{d e t} = 2.00

4-

E_{d e t} = 2.03

5-

E_{d e t}

6-

s i n g l e

7-

d I / d V

8-

d I / d V

9-

e x p (i π m z / L)

10-

n, L_{z}, m

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/9810315

Formulas:

1-

< A > = i τ_{2} \otimes D i a g (a, a, a, 0, 0)

2-

< A > = i τ_{2} \otimes D i a g (0, 0, 0, b, b)

3-

P A \bar{P}

4-

\bar{C} \bar{A} P + C \bar{A} \bar{P}

5-

S U {(4)}_{C} \times S U {(2)}_{L} \times S U {(2)}_{R}

6-

(𝟏𝟓, 𝟏, 𝟏) + (𝟔, 𝟐, 𝟐) + (𝟏, 𝟑, 𝟏) + (𝟏, 𝟏, 𝟑)

7-

(𝟒, 𝟐, 𝟏) + (\bar{𝟒}, 𝟏, 𝟐)

8-

(\bar{𝟒}, 𝟐, 𝟏) + (𝟒, 𝟏, 𝟐)

9-

P A \bar{P}

10-

(\bar{4}, 1, 2)

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9704008

Formulas:

1-

f_{s} = 1 - 2

2-

B^{1 / 4} \overset{>}{\approx} 200

3-

f_{s}^{i n i t} = 0

4-

f_{s} = (N_{s} - N_{\bar{s}}) / A

5-

Λ / (p - \bar{p})

6-

(Ω^{-} + {\bar{Ω}}^{-}) / (Ξ^{-} + {\bar{Ξ}}^{-})

7-

ρ \to π^{+} + π^{-}

8-

{(\bar{p} / p)}^{e q} \approx \frac{{\bar{p}}^{e q} + {\bar{p}}^{*}}{p^{e q} + p^{*}}

9-

P^{Q G P} = P^{H G}

10-

T^{Q G P} = T^{H G}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0608120

Formulas:

1-

E_{a} = ρ_{a b} j_{b}

2-

ρ_{a b} = (ρ_{0} + α^{'} {| 𝐁 |}^{2}) δ_{a b} + α B_{a} B_{b} - ρ_{H} ϵ_{a b c} B_{c},

3-

ρ = ρ^{s} + ρ^{a}

4-

E_{a}^{H} = 1 / 2 (ρ_{a b} - ρ_{b a}) j_{b}

5-

𝐄^{H} = ρ_{H} 𝐁 \times 𝐣 .

6-

S = ⟨ M_{x} M_{y} M_{z} ⟩ \neq 0

7-

a = x, y, z

8-

⟨ M_{a} M_{b} ⟩

9-

⟨ M_{a} M_{b} M_{c} ⟩

10-

S | ϵ_{a b c} | .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0606358

Formulas:

1-

τ_{vis} ≃ \frac{R_{⊙}^{2}}{ν_{⊙}} \sim 10^{12 - 13} years,

2-

\partial \ln Ω / \partial θ < 0

3-

\partial \ln Ω / \partial \ln r < 0

4-

(R, ϕ, Z)

5-

(r, θ, ϕ)

6-

\exp [i (𝒌 \cdot 𝒓 - ω t)]

7-

𝒌 = (k_{R}, 0, k_{Z}) = (k_{r}, k_{θ}, 0)

8-

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

9-

ρ \frac{\partial 𝒗}{\partial t} + (ρ 𝒗 \cdot \nabla) 𝒗 = - \nabla (P + \frac{B^{2}}{8 π}) - ρ \nabla Φ

10-

+ (\frac{𝑩}{4 π} \cdot \nabla) 𝑩 + μ (\nabla^{2} 𝒗 + \frac{1}{3} \nabla (\nabla \cdot 𝒗)),

Formulas (html):

<math><mrow id="S1.E1.m1.1.1.1" xref="S1.E1.m1.1.1.1.1.cmml"><mrow id="S1.E1.m1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.cmml"><msub id="S1.E1.m1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.2.cmml"><mi id="S1.E1.m1.1.1.1.1.2.2" xref="S1.E1.m1.1.1.1.1.2.2.cmml">τ</mi><mi id="S1.E1.m1.1.1.1.1.2.3" xref="S1.E1.m1.1.1.1.1.2.3.cmml">vis</mi></msub><mo id="S1.E1.m1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.3.cmml">≃</mo><mfrac id="S1.E1.m1.1.1.1.1.4" xref="S1.E1.m1.1.1.1.1.4.cmml"><msubsup id="S1.E1.m1.1.1.1.1.4.2" xref="S1.E1.m1.1.1.1.1.4.2.cmml"><mi id="S1.E1.m1.1.1.1.1.4.2.2.2" xref="S1.E1.m1.1.1.1.1.4.2.2.2.cmml">R</mi><mo id="S1.E1.m1.1.1.1.1.4.2.2.3" xref="S1.E1.m1.1.1.1.1.4.2.2.3.cmml">⊙</mo><mn id="S1.E1.m1.1.1.1.1.4.2.3" xref="S1.E1.m1.1.1.1.1.4.2.3.cmml">2</mn></msubsup><msub id="S1.E1.m1.1.1.1.1.4.3" xref="S1.E1.m1.1.1.1.1.4.3.cmml"><mi id="S1.E1.m1.1.1.1.1.4.3.2" xref="S1.E1.m1.1.1.1.1.4.3.2.cmml">ν</mi><mo id="S1.E1.m1.1.1.1.1.4.3.3" xref="S1.E1.m1.1.1.1.1.4.3.3.cmml">⊙</mo></msub></mfrac><mo id="S1.E1.m1.1.1.1.1.5" xref="S1.E1.m1.1.1.1.1.5.cmml">∼</mo><mrow id="S1.E1.m1.1.1.1.1.6" xref="S1.E1.m1.1.1.1.1.6.cmml"><mpadded width="+3.3pt" id="S1.E1.m1.1.1.1.1.6.2" xref="S1.E1.m1.1.1.1.1.6.2.cmml"><msup id="S1.E1.m1.1.1.1.1.6.2a" xref="S1.E1.m1.1.1.1.1.6.2.cmml"><mn id="S1.E1.m1.1.1.1.1.6.2.2" xref="S1.E1.m1.1.1.1.1.6.2.2.cmml">10</mn><mrow id="S1.E1.m1.1.1.1.1.6.2.3" xref="S1.E1.m1.1.1.1.1.6.2.3.cmml"><mn id="S1.E1.m1.1.1.1.1.6.2.3.2" xref="S1.E1.m1.1.1.1.1.6.2.3.2.cmml">12</mn><mo id="S1.E1.m1.1.1.1.1.6.2.3.1" xref="S1.E1.m1.1.1.1.1.6.2.3.1.cmml">-</mo><mn id="S1.E1.m1.1.1.1.1.6.2.3.3" xref="S1.E1.m1.1.1.1.1.6.2.3.3.cmml">13</mn></mrow></msup></mpadded><mo id="S1.E1.m1.1.1.1.1.6.1" xref="S1.E1.m1.1.1.1.1.6.1.cmml">⁢</mo><mi id="S1.E1.m1.1.1.1.1.6.3" xref="S1.E1.m1.1.1.1.1.6.3.cmml">years</mi></mrow></mrow><mo id="S1.E1.m1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.cmml">,</mo></mrow></math>, <math><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mrow id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml"><mrow id="footnote1.m1.1.1.2.2" xref="footnote1.m1.1.1.2.2.cmml"><mrow id="footnote1.m1.1.1.2.2.1" xref="footnote1.m1.1.1.2.2.1.cmml"><mo id="footnote1.m1.1.1.2.2.1.1" xref="footnote1.m1.1.1.2.2.1.1.cmml">∂</mo><mo id="footnote1.m1.1.1.2.2.1b" xref="footnote1.m1.1.1.2.2.1.cmml">⁡</mo><mi id="footnote1.m1.1.1.2.2.1.2" xref="footnote1.m1.1.1.2.2.1.2.cmml">ln</mi></mrow><mo id="footnote1.m1.1.1.2.2b" xref="footnote1.m1.1.1.2.2.cmml">⁡</mo><mi mathvariant="normal" id="footnote1.m1.1.1.2.2.2" xref="footnote1.m1.1.1.2.2.2.cmml">Ω</mi></mrow><mo id="footnote1.m1.1.1.2.1" xref="footnote1.m1.1.1.2.1.cmml">/</mo><mrow id="footnote1.m1.1.1.2.3" xref="footnote1.m1.1.1.2.3.cmml"><mo id="footnote1.m1.1.1.2.3.1" xref="footnote1.m1.1.1.2.3.1.cmml">∂</mo><mo id="footnote1.m1.1.1.2.3b" xref="footnote1.m1.1.1.2.3.cmml">⁡</mo><mi id="footnote1.m1.1.1.2.3.2" xref="footnote1.m1.1.1.2.3.2.cmml">θ</mi></mrow></mrow><mo id="footnote1.m1.1.1.1" xref="footnote1.m1.1.1.1.cmml"><</mo><mn id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml">0</mn></mrow></math>, <math><mrow id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml"><mrow id="footnote1.m2.1.1.2" xref="footnote1.m2.1.1.2.cmml"><mrow id="footnote1.m2.1.1.2.2" xref="footnote1.m2.1.1.2.2.cmml"><mrow id="footnote1.m2.1.1.2.2.1" xref="footnote1.m2.1.1.2.2.1.cmml"><mo id="footnote1.m2.1.1.2.2.1.1" xref="footnote1.m2.1.1.2.2.1.1.cmml">∂</mo><mo id="footnote1.m2.1.1.2.2.1b" xref="footnote1.m2.1.1.2.2.1.cmml">⁡</mo><mi id="footnote1.m2.1.1.2.2.1.2" xref="footnote1.m2.1.1.2.2.1.2.cmml">ln</mi></mrow><mo id="footnote1.m2.1.1.2.2b" xref="footnote1.m2.1.1.2.2.cmml">⁡</mo><mi mathvariant="normal" id="footnote1.m2.1.1.2.2.2" xref="footnote1.m2.1.1.2.2.2.cmml">Ω</mi></mrow><mo id="footnote1.m2.1.1.2.1" xref="footnote1.m2.1.1.2.1.cmml">/</mo><mrow id="footnote1.m2.1.1.2.3" xref="footnote1.m2.1.1.2.3.cmml"><mrow id="footnote1.m2.1.1.2.3.1" xref="footnote1.m2.1.1.2.3.1.cmml"><mo id="footnote1.m2.1.1.2.3.1.1" xref="footnote1.m2.1.1.2.3.1.1.cmml">∂</mo><mo id="footnote1.m2.1.1.2.3.1b" xref="footnote1.m2.1.1.2.3.1.cmml">⁡</mo><mi id="footnote1.m2.1.1.2.3.1.2" xref="footnote1.m2.1.1.2.3.1.2.cmml">ln</mi></mrow><mo id="footnote1.m2.1.1.2.3b" xref="footnote1.m2.1.1.2.3.cmml">⁡</mo><mi id="footnote1.m2.1.1.2.3.2" xref="footnote1.m2.1.1.2.3.2.cmml">r</mi></mrow></mrow><mo id="footnote1.m2.1.1.1" xref="footnote1.m2.1.1.1.cmml"><</mo><mn id="footnote1.m2.1.1.3" xref="footnote1.m2.1.1.3.cmml">0</mn></mrow></math>, <math><mrow id="S2.SS1.p1.1.m1.3.4.2" xref="S2.SS1.p1.1.m1.3.4.1.cmml"><mo stretchy="false" id="S2.SS1.p1.1.m1.3.4.2.1" xref="S2.SS1.p1.1.m1.3.4.1.cmml">(</mo><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">R</mi><mo id="S2.SS1.p1.1.m1.3.4.2.2" xref="S2.SS1.p1.1.m1.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.2.2" xref="S2.SS1.p1.1.m1.2.2.cmml">ϕ</mi><mo id="S2.SS1.p1.1.m1.3.4.2.3" xref="S2.SS1.p1.1.m1.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.3.3" xref="S2.SS1.p1.1.m1.3.3.cmml">Z</mi><mo stretchy="false" id="S2.SS1.p1.1.m1.3.4.2.4" xref="S2.SS1.p1.1.m1.3.4.1.cmml">)</mo></mrow></math>, <math><mrow id="S2.SS1.p1.2.m2.3.4.2" xref="S2.SS1.p1.2.m2.3.4.1.cmml"><mo stretchy="false" id="S2.SS1.p1.2.m2.3.4.2.1" xref="S2.SS1.p1.2.m2.3.4.1.cmml">(</mo><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">r</mi><mo id="S2.SS1.p1.2.m2.3.4.2.2" xref="S2.SS1.p1.2.m2.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.2.m2.2.2" xref="S2.SS1.p1.2.m2.2.2.cmml">θ</mi><mo id="S2.SS1.p1.2.m2.3.4.2.3" xref="S2.SS1.p1.2.m2.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.2.m2.3.3" xref="S2.SS1.p1.2.m2.3.3.cmml">ϕ</mi><mo stretchy="false" id="S2.SS1.p1.2.m2.3.4.2.4" xref="S2.SS1.p1.2.m2.3.4.1.cmml">)</mo></mrow></math>, <math><mrow id="S2.SS1.p1.3.m3.3.3.1" xref="S2.SS1.p1.3.m3.3.3.2.cmml"><mi id="S2.SS1.p1.3.m3.2.2" xref="S2.SS1.p1.3.m3.2.2.cmml">exp</mi><mo id="S2.SS1.p1.3.m3.3.3.1a" xref="S2.SS1.p1.3.m3.3.3.2.cmml">⁡</mo><mrow id="S2.SS1.p1.3.m3.3.3.1.1" xref="S2.SS1.p1.3.m3.3.3.2.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m3.3.3.1.1.2" xref="S2.SS1.p1.3.m3.3.3.2.cmml">[</mo><mrow id="S2.SS1.p1.3.m3.3.3.1.1.1" xref="S2.SS1.p1.3.m3.3.3.1.1.1.cmml"><mi id="S2.SS1.p1.3.m3.3.3.1.1.1.3" xref="S2.SS1.p1.3.m3.3.3.1.1.1.3.cmml">i</mi><mo id="S2.SS1.p1.3.m3.3.3.1.1.1.2" xref="S2.SS1.p1.3.m3.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.3.m3.3.3.1.1.1.1.1" xref="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.2" xref="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.1" xref="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.1.cmml"><mrow id="S2.SS1.p1.3.m3.1.1.1" xref="S2.SS1.p1.3.m3.1.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.1.3" xref="S2.SS1.p1.3.m3.1.1.1.3.cmml">𝒌</mi><mo mathvariant="bold" id="S2.SS1.p1.3.m3.1.1.1.2" xref="S2.SS1.p1.3.m3.1.1.1.2.cmml">⋅</mo><mi id="S2.SS1.p1.3.m3.1.1.1.4" xref="S2.SS1.p1.3.m3.1.1.1.4.cmml">𝒓</mi></mrow><mo id="S2.SS1.p1.3.m3.3.3.1.1.1.1.1.1.1" 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xref="S2.SS1.p1.4.m4.7.7.4.3.cmml"><mo stretchy="false" id="S2.SS1.p1.4.m4.7.7.4.2.3" xref="S2.SS1.p1.4.m4.7.7.4.3.cmml">(</mo><msub id="S2.SS1.p1.4.m4.6.6.3.1.1" xref="S2.SS1.p1.4.m4.6.6.3.1.1.cmml"><mi id="S2.SS1.p1.4.m4.6.6.3.1.1.2" xref="S2.SS1.p1.4.m4.6.6.3.1.1.2.cmml">k</mi><mi id="S2.SS1.p1.4.m4.6.6.3.1.1.3" xref="S2.SS1.p1.4.m4.6.6.3.1.1.3.cmml">r</mi></msub><mo id="S2.SS1.p1.4.m4.7.7.4.2.4" xref="S2.SS1.p1.4.m4.7.7.4.3.cmml">,</mo><msub id="S2.SS1.p1.4.m4.7.7.4.2.2" xref="S2.SS1.p1.4.m4.7.7.4.2.2.cmml"><mi id="S2.SS1.p1.4.m4.7.7.4.2.2.2" xref="S2.SS1.p1.4.m4.7.7.4.2.2.2.cmml">k</mi><mi id="S2.SS1.p1.4.m4.7.7.4.2.2.3" xref="S2.SS1.p1.4.m4.7.7.4.2.2.3.cmml">θ</mi></msub><mo id="S2.SS1.p1.4.m4.7.7.4.2.5" xref="S2.SS1.p1.4.m4.7.7.4.3.cmml">,</mo><mn id="S2.SS1.p1.4.m4.3.3" xref="S2.SS1.p1.4.m4.3.3.cmml">0</mn><mo stretchy="false" id="S2.SS1.p1.4.m4.7.7.4.2.6" xref="S2.SS1.p1.4.m4.7.7.4.3.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E2.m3.4.4.1" 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Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1702.04838

Formulas:

1-

x \bar{\lor} y = (1 - x) (1 - y)

2-

\begin{matrix} U_{1} & = (1 - I_{1}) (1 - I_{2}), \\ U_{2} & = (1 - I_{3}) (1 - I_{4}); \end{matrix}

3-

f : {[0, 1]}^{2} \times [- 2, 2] \to [- 2, 2]

4-

g : {[0, 1]}^{2} \times [- 1, 1] \to [- 1, 1]

5-

\begin{matrix} ξ_{n + 1} & = f (I_{1}, I_{2}, ξ_{n}), \\ η_{n + 1} & = g (I_{1}, I_{2}, η_{n}); \end{matrix}

6-

\begin{matrix} f (0, 0, 0) = 0 & f (1, 0, 1) = f (0, 1, 1) = 1 & f (1, 1, ⋆) = ⋆, \\ g (0, 0, - 1) = - 1 & g (1, 0, 0) = g (0, 1, 0) = 0 & g (1, 1, ⋆) = ⋆ . \end{matrix}

7-

y_{f} : [- 2, 2] \to [- 2, 2]

8-

y_{g} : [- 1, 1] \to [- 1, 1]

9-

\begin{matrix} f (I_{1}, I_{2}, x_{1}) & := | I_{1} - I_{2} | + I_{1} I_{2} y_{f} (x_{1}), \\ g (I_{3}, I_{4}, x_{2}) & := | I_{3} - I_{4} | - 1 + I_{3} I_{4} y_{g} (x_{2}); \end{matrix}

10-

(I_{1}, I_{2}) = (1, 1)

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/q-bio/0610033

Formulas:

1-

V_{n + 1}^{(i)} = s V_{n}^{(i)} + 1

2-

V_{n + 1}^{(i)} = s V_{n}^{(i)}

3-

V_{n + 1}^{(i)}

4-

(s^{n + 1} - 1) / (s - 1)

5-

V_{n + 1}^{(i)}

6-

0 \leq V^{(i)} < 1 / (1 - s)

7-

w_{n + 1}^{(i)} = (1 - ϵ) w_{n}^{(i)} + ϵ V_{n + 1}^{(i)} / [(s^{n + 1} - 1) / (s - 1)]

8-

w_{n + 1}^{(i)}

9-

V_{0}^{(i)} = 0

10-

w_{0}^{(i)} = U [0, 1)

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1311.4939

Formulas:

1-

𝒰 (𝒜) = {u \in 𝒜 : u u^{*} = u^{*} u = I}

2-

(𝒜, ℍ, D)

3-

(𝒜, ℍ, D)

4-

V = \sum_{j} a_{j} [D, b_{j}]

5-

a_{j}, b_{j} \in 𝒜

6-

[a, b] = a b - b a .

7-

u \in 𝒰 (𝒜)

8-

V ⟼ G_{u} (V) = u V u^{*} + u [D, u^{*}] .

9-

| ψ ⟩ ⟨ ψ |

10-

| ψ ⟩ ⟨ ψ | = | ψ ⟩ ⟨ φ | i [D, b] | φ ⟩ ⟨ ψ |

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1906.10137

Formulas:

1-

{p_{1} \dots p_{n}}

2-

{k_{1}^{(t)} \dots k_{t}^{(t)}}

3-

k_{1}^{(1)} = p_{1} + \dots + p_{n}

4-

{k_{1}^{(n)} \dots k_{n}^{(n)}} = {p_{1} \dots p_{n}}

5-

{k_{1}^{(t)} \dots k_{t}^{(t)}} \to {k_{1}^{(t + 1)} \dots k_{t + 1}^{(t + 1)}}

6-

{p_{1} \dots p_{n}}

7-

P_{𝒥} (jet)

8-

= [\prod_{t = 1}^{n - 1} P^{(t)} (k_{1}^{(t + 1)} \dots k_{t + 1}^{(t + 1)} | k_{1}^{(t)} \dots k_{t}^{(t)})]

9-

\times P^{(n)} (end | k_{1}^{(n)} \dots k_{n}^{(n)})

10-

{k_{1}^{(t)} \dots k_{t}^{(t)}}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/2009.09830

Formulas:

1-

ϵ \frac{d u_{1 i}}{d t}

2-

= u_{1 i} - {(u_{1 i})}^{3} / 3 - v_{1 i} + \frac{σ}{2} \sum_{j = i - 1}^{i + 1} (u_{1 j} - u_{1 i}) + μ_{i} σ_{12} (u_{2 i} - u_{1 i}) + \sqrt{2 D_{1}} ζ_{1 i} (t),

3-

\frac{d v_{1 i}}{d t}

4-

= u_{1 i} + a,

5-

ϵ \frac{d u_{2 i}}{d t}

6-

= u_{2 i} - {(u_{2 i})}^{3} / 3 - v_{2 i} + \frac{σ}{2} \sum_{j = i - 1}^{i + 1} (u_{2 j} - u_{2 i}) + μ_{i} σ_{12} (u_{1 i} - u_{2 i}) + \sqrt{2 D_{2}} ζ_{2 i} (t),

7-

\frac{d v_{2 i}}{d t}

8-

= u_{2 i} + a .

9-

i = 1 \dots N

10-

σ = σ_{12} = 0

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0308477

Formulas:

1-

6 m s^{- 1}

2-

20.6 m s^{- 1}

3-

m \sin i \approx 2 M_{Jup}

4-

m \sin i = 0.56 M_{Jup}

5-

10 m s^{- 1}

6-

70 m s^{- 1}

7-

3 - 4 m s^{- 1}

8-

m \sin i = 0.0126 - 0.019 M_{Jup}

9-

m \sin i = 4 - 6 M_{\oplus}

10-

5.7 m s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1106.3243

Formulas:

1-

\sinh ρ = - \frac{1 - q^{2} τ^{2} + q^{2} r^{2}}{2 q τ}, \tan θ = \frac{2 q r}{1 + q^{2} τ^{2} - q^{2} r^{2}}

2-

τ = \sqrt{t^{2} - z^{2}}

3-

T = T_{0} (1 + δ)

4-

\frac{\partial^{2} δ}{\partial ρ^{2}} - \frac{1}{3 \cosh^{2} ρ} (\frac{\partial^{2} δ}{\partial θ^{2}} + \frac{1}{\tan θ} \frac{\partial δ}{\partial θ} + \frac{1}{\sin^{2} θ} \frac{\partial^{2} δ}{\partial ϕ^{2}}) + \frac{4}{3} \tanh ρ \frac{\partial δ}{\partial ρ} = 0

5-

Y_{l m} (θ, ϕ)

6-

δ_{l} (f o)

7-

4 π η / s = 0, 1, 1.68, 2

8-

\frac{4 π η}{s} \approx 2

9-

r = 4.1 f m

10-

η / s = 0, 0.134

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/0311037

Formulas:

1-

({35.68}^{\circ} N, {139.54}^{\circ} E)

2-

5 \times 10^{- 21} / \sqrt{Hz}

3-

3 \times 10^{- 21} / \sqrt{Hz}

4-

S_{n} (f)

5-

h (t) = 𝒜 [h_{c} (t - t_{c}) \cos ϕ_{0} + h_{s} (t - t_{c}) \sin ϕ_{0}],

6-

h_{c} (t)

7-

h_{s} (t)

8-

ℳ (\equiv M η^{3 / 5})

9-

(M = m_{1} + m_{2})

10-

η (\equiv m_{1} m_{2} / M^{2})

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1104.2334

Formulas:

1-

ρ (r, θ, φ) = ⟨ ρ ⟩ (r) + δ ρ (r, θ, φ)

2-

Φ (r, θ, φ) = ⟨ Φ ⟩ (r) + δ Φ (r, θ, φ) .

3-

⟨ Φ ⟩ (r) = \frac{1}{4 π} \int_{0}^{2 π} d φ \int_{0}^{π} d θ \sin θ Φ (r, θ, φ),

4-

\frac{d ⟨ Φ ⟩ (r)}{d r} = \frac{G M (r)}{r^{2}},

5-

M (R) = 4 π \int_{0}^{R} d r r^{2} ⟨ ρ ⟩ (r) .

6-

W (R) = \frac{1}{2} \int_{0}^{R} d r r^{2} \int_{0}^{2 π} d φ \int_{0}^{π} d θ \sin θ ρ (r, θ, φ) Φ (r, θ, φ),

7-

W (R) = 2 π \int_{0}^{R} d r r^{2} [⟨ ρ ⟩ (r) ⟨ Φ ⟩ (r) + ⟨ δ ρ δ Φ ⟩]

8-

= - 4 π \int_{0}^{R} d r r^{2} ⟨ ρ ⟩ (r) \frac{G M (r)}{r} + 2 π \int_{0}^{R} d r r^{2} ⟨ δ ρ δ Φ ⟩

9-

\equiv 𝒲 (R) + δ 𝒲 (R),

10-

𝒲 (R) = - 4 π \int_{0}^{R} d r r^{2} ⟨ ρ ⟩ (r) \frac{G M (r)}{r} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Run 11314749 (Agent202)