Run 11277703

Paper: https://arxiv.org/abs/2001.04876

Formulas:

1-

μ_{ν} / μ_{B} ≳ 10^{- 14} - 10^{- 15}

2-

{\bar{ν}}_{e} + p ⇌ n + e^{+}

3-

ν_{e} + n ⇌ p + e^{-}

4-

n_{{\bar{ν}}_{e}} / n_{ν_{e}} > 1

5-

n_{{\bar{ν}}_{e}} / n_{ν_{e}} < 1

6-

n_{{\bar{ν}}_{e}} / n_{ν_{e}} > 1

7-

\sim G_{F}^{- 1} n_{ν}^{- 1}

8-

\sim 𝒪 (1)

9-

ω = Δ m_{atm}^{2} / 2 E

10-

μ_{ν}^{i i, D} = \frac{3 e G_{F} m_{ν}}{8 \sqrt{2} π^{2}} ≃ 3.2 \times 10^{- 19} (\frac{m_{ν}}{1 e V}) μ_{B},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0305308

Formulas:

1-

R = R_{0} e^{β N}

2-

β = \sqrt{\frac{8 m}{ℏ^{2}} Φ_{A u}} a = 3.0

3-

Φ_{A u} = 5.3

4-

β = 2 (Im k) a

5-

\frac{d ϵ}{d k} = 0

6-

s p^{3} d^{5}

7-

e^{2} = 2 m = ℏ = 1

8-

E \hat{S} | ψ_{𝐤} ⟩ = [- \nabla^{2} + V_{eff} + {\hat{V}}_{N L}] | ψ_{𝐤} ⟩,

9-

{\hat{V}}_{N L} = \sum_{I m n} D_{m n}^{I} | β_{m}^{I} ⟩ ⟨ β_{n}^{I} |,

10-

\hat{S} = 1 + \sum_{I m n} q_{m n}^{I} | β_{m}^{I} ⟩ ⟨ β_{n}^{I} | .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-th/0104084

Formulas:

1-

E_{d} \sim Q_{d}^{(d / d + 1)}

2-

E \sim Q^{(3 / 4)}

3-

Q^{(3 / 4)}

4-

E_{d} \sim Q_{d}^{d / (d + 1)}

5-

ℒ = \int d^{d} x (\frac{1}{2} \partial_{μ} ϕ_{a} \partial^{μ} ϕ_{a} - V (| ϕ |))

6-

| ϕ | = \sqrt{ϕ_{1}^{2} + ϕ_{2}^{2}}

7-

S O (2)

8-

Q = \int d^{d} x ϵ_{a b} ϕ_{a} \partial_{t} ϕ_{b} .

9-

V (| ϕ |) = {\begin{matrix} \frac{1}{2} m^{2} {| ϕ |}^{2} & | ϕ | \to 0 \\ Λ & | ϕ | \to \infty \end{matrix}

10-

ϕ_{1} + i ϕ_{2} = e^{i ω t} φ (r)

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0103105

Formulas:

1-

α P_{s} + (1 - α) I / 4

2-

⟨ χ | M | χ ⟩

3-

\Pr (a_{i}, b_{j} | A, B, ρ) = \int 𝑑 λ ω^{ρ} (λ) \Pr (a_{i}, b_{j} | A, B, λ),

4-

ω^{ρ} (λ)

5-

\Pr (a_{i}, b_{j} | A, B, λ) = \Pr (a_{i} | A, λ) \Pr (b_{j} | B, λ) .

6-

{| ψ ⟩}_{A B}

7-

{| ψ ⟩}_{A D}

8-

(x_{1}, x_{2})

9-

x_{1}, x_{2} \in {0, 1, 2, 3}

10-

\begin{matrix} B_{0} = (\begin{matrix} 0 & 0 \\ 1 & 1 \\ 2 & 2 \\ 3 & 3 \end{matrix}) & B_{1} = (\begin{matrix} 0 & 3 \\ 1 & 0 \\ 2 & 1 \\ 3 & 2 \end{matrix}) \\ B_{2} = (\begin{matrix} 0 & 2 \\ 1 & 3 \\ 2 & 0 \\ 3 & 1 \end{matrix}) & B_{3} = (\begin{matrix} 0 & 1 \\ 1 & 2 \\ 2 & 3 \\ 3 & 0 \end{matrix}) . \end{matrix}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/0910.3140

Formulas:

1-

\sim T e V

2-

S = - T r (ρ \log ρ),

3-

R (E) \propto M_{D}^{- 1} {(E / M_{D})}^{1 / (D - 3)}

4-

τ_{i} \sim R (E) S (E)

5-

S (E) \propto R {(E)}^{D - 2}

6-

E \sim M_{s} = 1 / l_{s}

7-

R (E) \sim l_{s}

8-

b \sim M_{s}^{- 2} E

9-

b \overset{<}{\sim} R (E)

10-

b \sim R (E)

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/2001.04785

Formulas:

1-

\ddot{x} + a (t) x = 0

2-

a (t) = a (t + T)

3-

T = 2 π / ω

4-

τ = 2 π / ω_{H}

5-

w = \frac{N_{L} - N_{R}}{N_{L} + N_{R}}

6-

N_{L, (R)}

7-

ϕ = ϕ_{L} - ϕ_{R}

8-

ϕ_{L, (R)}

9-

\dot{w} (t) = - \frac{2 J}{ℏ} \sqrt{1 - w^{2} (t)} \sin ϕ (t) - \frac{η}{N} \dot{ϕ} (t)

10-

\dot{ϕ} (t) = \frac{2 J}{ℏ} [Λ_{0} w (t) + \frac{w (t)}{\sqrt{1 - w^{2} (t)}} \cos ϕ (t)]

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1506.04402

Formulas:

1-

H = \sum_{m} H_{m}

2-

H_{m} = p_{m}^{2} / (2 M) + (1 / 2) \sum_{n} x_{m} Φ_{m n} x_{n}

3-

{\dot{H}}_{m} = \sum_{n} [Φ_{m n} x_{m} p_{n} - Φ_{n m} x_{n} p_{m}] / (2 M)

4-

H = \sum_{k} (a_{k}^{+} a_{k} + 1 / 2) ℏ ω_{k}

5-

{\dot{H}}_{m} = \sum_{k, r} [e^{- i (k + r) R_{m}} T_{k, r} (t) + e^{i (k - r) R_{m}} W_{k, r} (t)] + h . c .

6-

a_{k (r)}

7-

a_{k (r)}^{+}

8-

H_{m + L} = H_{m}

9-

Λ = 2 π h / (L a)

10-

[- L / 2, L / 2]

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1304.4126

Formulas:

1-

1 W / {cm}^{2}

2-

21 kW / {cm}^{2}

3-

α_{surf} \approx 800 ppm / surface

4-

α = 264 ppm / cm

5-

11 k Ω cm

6-

\approx 30 k Ω cm

7-

≲ 5 ppm / cm

8-

1 W / {cm}^{2}

9-

21 kW / {cm}^{2}

10-

\approx 250 ppm / cm

Formulas (html):

<math><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mrow id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml"><mpadded width="+1.7pt" id="id2.2.m2.1.1.2.2" xref="id2.2.m2.1.1.2.2.cmml"><mn id="id2.2.m2.1.1.2.2a" xref="id2.2.m2.1.1.2.2.cmml">1</mn></mpadded><mo id="id2.2.m2.1.1.2.1" xref="id2.2.m2.1.1.2.1.cmml">⁢</mo><mi class="ltx_unit" mathvariant="normal" id="id2.2.m2.1.1.2.3" xref="id2.2.m2.1.1.2.3.cmml">W</mi></mrow><mo id="id2.2.m2.1.1.1" xref="id2.2.m2.1.1.1.cmml">/</mo><msup id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml"><mi class="ltx_unit" id="id2.2.m2.1.1.3.2" xref="id2.2.m2.1.1.3.2.cmml">cm</mi><mn id="id2.2.m2.1.1.3.3" xref="id2.2.m2.1.1.3.3.cmml">2</mn></msup></mrow></math>, <math><mrow id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml"><mrow id="id3.3.m3.1.1.2" xref="id3.3.m3.1.1.2.cmml"><mpadded width="+1.7pt" id="id3.3.m3.1.1.2.2" xref="id3.3.m3.1.1.2.2.cmml"><mn id="id3.3.m3.1.1.2.2a" xref="id3.3.m3.1.1.2.2.cmml">21</mn></mpadded><mo id="id3.3.m3.1.1.2.1" xref="id3.3.m3.1.1.2.1.cmml">⁢</mo><mi class="ltx_unit" id="id3.3.m3.1.1.2.3" xref="id3.3.m3.1.1.2.3.cmml">kW</mi></mrow><mo id="id3.3.m3.1.1.1" xref="id3.3.m3.1.1.1.cmml">/</mo><msup id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml"><mi class="ltx_unit" id="id3.3.m3.1.1.3.2" xref="id3.3.m3.1.1.3.2.cmml">cm</mi><mn id="id3.3.m3.1.1.3.3" xref="id3.3.m3.1.1.3.3.cmml">2</mn></msup></mrow></math>, <math><mrow id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml"><msub id="id5.5.m5.1.1.2" xref="id5.5.m5.1.1.2.cmml"><mi class="ltx_unit" id="id5.5.m5.1.1.2.2" xref="id5.5.m5.1.1.2.2.cmml">α</mi><mi class="ltx_unit" id="id5.5.m5.1.1.2.3" xref="id5.5.m5.1.1.2.3.cmml">surf</mi></msub><mo id="id5.5.m5.1.1.1" xref="id5.5.m5.1.1.1.cmml">≈</mo><mrow id="id5.5.m5.1.1.3" xref="id5.5.m5.1.1.3.cmml"><mrow id="id5.5.m5.1.1.3.2" xref="id5.5.m5.1.1.3.2.cmml"><mpadded width="+1.7pt" id="id5.5.m5.1.1.3.2.2" xref="id5.5.m5.1.1.3.2.2.cmml"><mn id="id5.5.m5.1.1.3.2.2a" xref="id5.5.m5.1.1.3.2.2.cmml">800</mn></mpadded><mo id="id5.5.m5.1.1.3.2.1" xref="id5.5.m5.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_unit" id="id5.5.m5.1.1.3.2.3" xref="id5.5.m5.1.1.3.2.3.cmml">ppm</mi></mrow><mo id="id5.5.m5.1.1.3.1" xref="id5.5.m5.1.1.3.1.cmml">/</mo><mi class="ltx_unit" id="id5.5.m5.1.1.3.3" xref="id5.5.m5.1.1.3.3.cmml">surface</mi></mrow></mrow></math>, <math><mrow id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi class="ltx_unit" id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">α</mi><mo id="S1.p2.2.m2.1.1.1" xref="S1.p2.2.m2.1.1.1.cmml">=</mo><mrow id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml"><mrow id="S1.p2.2.m2.1.1.3.2" xref="S1.p2.2.m2.1.1.3.2.cmml"><mpadded width="+1.7pt" id="S1.p2.2.m2.1.1.3.2.2" xref="S1.p2.2.m2.1.1.3.2.2.cmml"><mn id="S1.p2.2.m2.1.1.3.2.2a" xref="S1.p2.2.m2.1.1.3.2.2.cmml">264</mn></mpadded><mo id="S1.p2.2.m2.1.1.3.2.1" xref="S1.p2.2.m2.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_unit" id="S1.p2.2.m2.1.1.3.2.3" xref="S1.p2.2.m2.1.1.3.2.3.cmml">ppm</mi></mrow><mo id="S1.p2.2.m2.1.1.3.1" xref="S1.p2.2.m2.1.1.3.1.cmml">/</mo><mi class="ltx_unit" id="S1.p2.2.m2.1.1.3.3" xref="S1.p2.2.m2.1.1.3.3.cmml">cm</mi></mrow></mrow></math>, <math><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mpadded width="+1.7pt" id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml"><mn id="S1.p2.3.m3.1.1.2a" xref="S1.p2.3.m3.1.1.2.cmml">11</mn></mpadded><mo id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_unit" mathvariant="normal" id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">k</mi><mo id="S1.p2.3.m3.1.1.1a" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_unit" mathvariant="normal" id="S1.p2.3.m3.1.1.4" xref="S1.p2.3.m3.1.1.4.cmml">Ω</mi><mo id="S1.p2.3.m3.1.1.1b" xref="S1.p2.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_unit" id="S1.p2.3.m3.1.1.5" xref="S1.p2.3.m3.1.1.5.cmml">cm</mi></mrow></math>, <math><mrow id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml"><mi id="S1.p2.4.m4.1.1.2" xref="S1.p2.4.m4.1.1.2.cmml"/><mo id="S1.p2.4.m4.1.1.1" 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id="S1.p2.5.m5.1.1.3.2" xref="S1.p2.5.m5.1.1.3.2.cmml"><mpadded width="+1.7pt" id="S1.p2.5.m5.1.1.3.2.2" xref="S1.p2.5.m5.1.1.3.2.2.cmml"><mn id="S1.p2.5.m5.1.1.3.2.2a" xref="S1.p2.5.m5.1.1.3.2.2.cmml">5</mn></mpadded><mo id="S1.p2.5.m5.1.1.3.2.1" xref="S1.p2.5.m5.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_unit" id="S1.p2.5.m5.1.1.3.2.3" xref="S1.p2.5.m5.1.1.3.2.3.cmml">ppm</mi></mrow><mo id="S1.p2.5.m5.1.1.3.1" xref="S1.p2.5.m5.1.1.3.1.cmml">/</mo><mi class="ltx_unit" id="S1.p2.5.m5.1.1.3.3" xref="S1.p2.5.m5.1.1.3.3.cmml">cm</mi></mrow></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"><mpadded width="+1.7pt" id="S1.p3.1.m1.1.1.2.2" xref="S1.p3.1.m1.1.1.2.2.cmml"><mn id="S1.p3.1.m1.1.1.2.2a" xref="S1.p3.1.m1.1.1.2.2.cmml">1</mn></mpadded><mo id="S1.p3.1.m1.1.1.2.1" xref="S1.p3.1.m1.1.1.2.1.cmml">⁢</mo><mi class="ltx_unit" mathvariant="normal" id="S1.p3.1.m1.1.1.2.3" xref="S1.p3.1.m1.1.1.2.3.cmml">W</mi></mrow><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">/</mo><msup id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mi class="ltx_unit" id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml">cm</mi><mn id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml">2</mn></msup></mrow></math>, <math><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mrow id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml"><mpadded width="+1.7pt" id="S1.p3.2.m2.1.1.2.2" xref="S1.p3.2.m2.1.1.2.2.cmml"><mn id="S1.p3.2.m2.1.1.2.2a" xref="S1.p3.2.m2.1.1.2.2.cmml">21</mn></mpadded><mo id="S1.p3.2.m2.1.1.2.1" xref="S1.p3.2.m2.1.1.2.1.cmml">⁢</mo><mi class="ltx_unit" id="S1.p3.2.m2.1.1.2.3" xref="S1.p3.2.m2.1.1.2.3.cmml">kW</mi></mrow><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">/</mo><msup id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml"><mi class="ltx_unit" id="S1.p3.2.m2.1.1.3.2" xref="S1.p3.2.m2.1.1.3.2.cmml">cm</mi><mn id="S1.p3.2.m2.1.1.3.3" xref="S1.p3.2.m2.1.1.3.3.cmml">2</mn></msup></mrow></math>, <math><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml"/><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">≈</mo><mrow id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml"><mrow id="S1.p4.1.m1.1.1.3.2" xref="S1.p4.1.m1.1.1.3.2.cmml"><mpadded width="+1.7pt" id="S1.p4.1.m1.1.1.3.2.2" xref="S1.p4.1.m1.1.1.3.2.2.cmml"><mn id="S1.p4.1.m1.1.1.3.2.2a" xref="S1.p4.1.m1.1.1.3.2.2.cmml">250</mn></mpadded><mo id="S1.p4.1.m1.1.1.3.2.1" xref="S1.p4.1.m1.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_unit" id="S1.p4.1.m1.1.1.3.2.3" xref="S1.p4.1.m1.1.1.3.2.3.cmml">ppm</mi></mrow><mo id="S1.p4.1.m1.1.1.3.1" xref="S1.p4.1.m1.1.1.3.1.cmml">/</mo><mi class="ltx_unit" id="S1.p4.1.m1.1.1.3.3" xref="S1.p4.1.m1.1.1.3.3.cmml">cm</mi></mrow></mrow></math>

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0902.4870

Formulas:

1-

A u + A u

2-

\sqrt{s_{N N}} = 200 G e V

3-

G e V / f m^{3}

4-

A u + A u

5-

\sqrt{s_{N N}} = 200 G e V

6-

A u + A u

7-

\sqrt{s_{N N}} = 200 G e V

8-

2 G e V / c

9-

1 G e V / c

10-

p_{T} > 2.5 G e V / c

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-ex

Result: incorrect

Paper: https://arxiv.org/abs/1712.04069

Formulas:

1-

C_{2} H + {CH}_{4} ⟶ C_{2} H_{2} + {CH}_{3}

2-

C_{2} H_{2} + h ν ⟶ C_{2} H + H

3-

H + C_{2} H_{3} ⟶ C_{2} H_{2} + H_{2}

4-

{CH}_{3} + C_{3} H_{2} ⟶ C_{2} H_{2} + C_{2} H_{3}

5-

CH + C_{2} H_{2} ⟶ C_{3} H_{2} + H

6-

CH + {CH}_{4} ⟶ C_{2} H_{4} + H

7-

CN + C_{2} H_{4} ⟶ C_{2} H_{3} CN + H

8-

2 {CH}_{3} + M ⟶ C_{2} H_{6} + M

9-

N + {CH}_{3} ⟶ HCN + H_{2}

10-

C_{2} H + C_{2} H_{6} ⟶ C_{2} H_{2} + C_{2} H_{5}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1905.08333

Formulas:

1-

â € “ -

2-

Δ ν \propto \sqrt{\bar{ρ}}

3-

\frac{Δ ν}{Δ ν_{⊙}} ≃ \sqrt{\frac{M / M_{⊙}}{{(R / R_{⊙})}^{3}}} .

4-

ν_{\max} \propto ν_{ac} \propto g T_{eff}^{- 1 / 2}

5-

\frac{ν_{\max}}{ν_{\max, ⊙}} ≃ (\frac{M}{M_{⊙}}) {(\frac{R}{R_{⊙}})}^{- 2} {(\frac{T_{eff}}{T_{eff, ⊙}})}^{- 1 / 2} .

6-

P B S (ν) = \frac{P (ν) - B g (ν)}{B g (ν)},

7-

B g (ν)

8-

MCC = \frac{(\sum_{i = 1}^{n_{lags}} | A C F_{i} |) - 1}{n_{lags}}

9-

Δ ν \propto ν_{\max}^{β}

10-

s = \sqrt{\frac{\sum f x^{2} - \frac{{(\sum f x)}^{2}}{\sum f}}{\sum f - 1}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1406.5626

Formulas:

1-

ℋ = - J \sum_{⟨ i, j ⟩} s_{i} s_{j} - h \sum_{i} s_{i},

2-

s_{i} = \pm 1 / 2

3-

g (M, E)

4-

𝐬_{GS} = {s_{1}, \dots, s_{N}}

5-

𝐬_{GS} = \underset{𝐬}{\arg \min} ℋ (𝐬) .

6-

M_{j} = \sum_{i = 1}^{N} 𝐬_{GS}

7-

j = 1, \dots, W

8-

m = \sum_{j = 1}^{W} M_{j} / (W N)

9-

S / N = \ln W / N

10-

g (M, E)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/2002.00716

Formulas:

1-

E = \sum_{i = 1}^{N_{atoms}} \sum_{j > i}^{N_{atoms}} 4 ϵ [{(\frac{σ}{r_{i j}})}^{12} - {(\frac{σ}{r_{i j}})}^{6}]

2-

V (r) = {\begin{matrix} k \cdot {(r - r_{0})}^{2} & r < r_{0} - r_{z z} \lor r > r_{0} + r_{z z} \\ k \cdot {(r_{z z} - r_{0})}^{2} & r_{0} - r_{z z} \leq r \leq r_{0} + r_{z z} \end{matrix}

3-

r_{0} - r_{z z}

4-

r_{0} + r_{z z}

5-

V_{z z} = k \cdot {(r_{z z} - r_{0})}^{2}

6-

N_{grid} = N_{LJ} \cdot N_{mult}

7-

L = \sqrt[3]{N_{grid}} 2 r_{0}

8-

2 r_{z z}

9-

r_{0} = r_{z z}

10-

ϵ_{L J} / k_{b}

Formulas (html):

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xref="S2.E2.m1.2.2.2.2.2.1.3.3.1.cmml">⁢</mo><mi id="S2.E2.m1.2.2.2.2.2.1.3.3.3" xref="S2.E2.m1.2.2.2.2.2.1.3.3.3.cmml">z</mi></mrow></msub></mrow></mtd></mtr><mtr id="S2.E2.m1.4.4.4d" xref="S2.E2.m1.5.6.3.1.cmml"><mtd columnalign="left" id="S2.E2.m1.4.4.4e" xref="S2.E2.m1.5.6.3.1.cmml"><mrow id="S2.E2.m1.3.3.3.3.1.1" xref="S2.E2.m1.3.3.3.3.1.1.cmml"><mi id="S2.E2.m1.3.3.3.3.1.1.3" xref="S2.E2.m1.3.3.3.3.1.1.3.cmml">k</mi><mo id="S2.E2.m1.3.3.3.3.1.1.2" xref="S2.E2.m1.3.3.3.3.1.1.2.cmml">⋅</mo><msup id="S2.E2.m1.3.3.3.3.1.1.1" xref="S2.E2.m1.3.3.3.3.1.1.1.cmml"><mrow id="S2.E2.m1.3.3.3.3.1.1.1.1.1" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E2.m1.3.3.3.3.1.1.1.1.1.2" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.cmml"><msub id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.cmml"><mi id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.2" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.2.cmml">r</mi><mrow id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.cmml"><mi id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.2" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.2.cmml">z</mi><mo id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.1" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.3" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.2.3.3.cmml">z</mi></mrow></msub><mo id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.1" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.1.cmml">-</mo><msub id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3.cmml"><mi id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3.2" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3.2.cmml">r</mi><mn id="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3.3" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo stretchy="false" id="S2.E2.m1.3.3.3.3.1.1.1.1.1.3" xref="S2.E2.m1.3.3.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.E2.m1.3.3.3.3.1.1.1.3" xref="S2.E2.m1.3.3.3.3.1.1.1.3.cmml">2</mn></msup></mrow></mtd><mtd columnalign="left" id="S2.E2.m1.4.4.4f" xref="S2.E2.m1.5.6.3.1.cmml"><mrow id="S2.E2.m1.4.4.4.4.2.1" xref="S2.E2.m1.4.4.4.4.2.1.cmml"><mrow id="S2.E2.m1.4.4.4.4.2.1.2" xref="S2.E2.m1.4.4.4.4.2.1.2.cmml"><msub id="S2.E2.m1.4.4.4.4.2.1.2.2" xref="S2.E2.m1.4.4.4.4.2.1.2.2.cmml"><mpadded lspace="10pt" width="+10pt" id="S2.E2.m1.4.4.4.4.2.1.2.2.2" xref="S2.E2.m1.4.4.4.4.2.1.2.2.2.cmml"><mi id="S2.E2.m1.4.4.4.4.2.1.2.2.2a" xref="S2.E2.m1.4.4.4.4.2.1.2.2.2.cmml">r</mi></mpadded><mn id="S2.E2.m1.4.4.4.4.2.1.2.2.3" xref="S2.E2.m1.4.4.4.4.2.1.2.2.3.cmml">0</mn></msub><mo id="S2.E2.m1.4.4.4.4.2.1.2.1" xref="S2.E2.m1.4.4.4.4.2.1.2.1.cmml">-</mo><msub id="S2.E2.m1.4.4.4.4.2.1.2.3" xref="S2.E2.m1.4.4.4.4.2.1.2.3.cmml"><mi id="S2.E2.m1.4.4.4.4.2.1.2.3.2" xref="S2.E2.m1.4.4.4.4.2.1.2.3.2.cmml">r</mi><mrow id="S2.E2.m1.4.4.4.4.2.1.2.3.3" xref="S2.E2.m1.4.4.4.4.2.1.2.3.3.cmml"><mi id="S2.E2.m1.4.4.4.4.2.1.2.3.3.2" 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xref="S2.E2.m1.4.4.4.4.2.1.6.3.2.cmml">r</mi><mrow id="S2.E2.m1.4.4.4.4.2.1.6.3.3" xref="S2.E2.m1.4.4.4.4.2.1.6.3.3.cmml"><mi id="S2.E2.m1.4.4.4.4.2.1.6.3.3.2" xref="S2.E2.m1.4.4.4.4.2.1.6.3.3.2.cmml">z</mi><mo id="S2.E2.m1.4.4.4.4.2.1.6.3.3.1" xref="S2.E2.m1.4.4.4.4.2.1.6.3.3.1.cmml">⁢</mo><mi id="S2.E2.m1.4.4.4.4.2.1.6.3.3.3" xref="S2.E2.m1.4.4.4.4.2.1.6.3.3.3.cmml">z</mi></mrow></msub></mrow></mrow></mtd></mtr></mtable></mrow></mrow></math>, <math><mrow id="S2.SS3.SSS1.p8.4.m4.1.1" xref="S2.SS3.SSS1.p8.4.m4.1.1.cmml"><msub id="S2.SS3.SSS1.p8.4.m4.1.1.2" xref="S2.SS3.SSS1.p8.4.m4.1.1.2.cmml"><mi id="S2.SS3.SSS1.p8.4.m4.1.1.2.2" xref="S2.SS3.SSS1.p8.4.m4.1.1.2.2.cmml">r</mi><mn id="S2.SS3.SSS1.p8.4.m4.1.1.2.3" xref="S2.SS3.SSS1.p8.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS1.p8.4.m4.1.1.1" xref="S2.SS3.SSS1.p8.4.m4.1.1.1.cmml">-</mo><msub id="S2.SS3.SSS1.p8.4.m4.1.1.3" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.cmml"><mi id="S2.SS3.SSS1.p8.4.m4.1.1.3.2" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.2.cmml">r</mi><mrow id="S2.SS3.SSS1.p8.4.m4.1.1.3.3" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.3.cmml"><mi id="S2.SS3.SSS1.p8.4.m4.1.1.3.3.2" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.3.2.cmml">z</mi><mo id="S2.SS3.SSS1.p8.4.m4.1.1.3.3.1" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p8.4.m4.1.1.3.3.3" xref="S2.SS3.SSS1.p8.4.m4.1.1.3.3.3.cmml">z</mi></mrow></msub></mrow></math>, <math><mrow id="S2.SS3.SSS1.p8.5.m5.1.1" xref="S2.SS3.SSS1.p8.5.m5.1.1.cmml"><msub id="S2.SS3.SSS1.p8.5.m5.1.1.2" xref="S2.SS3.SSS1.p8.5.m5.1.1.2.cmml"><mi id="S2.SS3.SSS1.p8.5.m5.1.1.2.2" xref="S2.SS3.SSS1.p8.5.m5.1.1.2.2.cmml">r</mi><mn id="S2.SS3.SSS1.p8.5.m5.1.1.2.3" xref="S2.SS3.SSS1.p8.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS1.p8.5.m5.1.1.1" xref="S2.SS3.SSS1.p8.5.m5.1.1.1.cmml">+</mo><msub id="S2.SS3.SSS1.p8.5.m5.1.1.3" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.cmml"><mi id="S2.SS3.SSS1.p8.5.m5.1.1.3.2" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.2.cmml">r</mi><mrow id="S2.SS3.SSS1.p8.5.m5.1.1.3.3" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.3.cmml"><mi id="S2.SS3.SSS1.p8.5.m5.1.1.3.3.2" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.3.2.cmml">z</mi><mo id="S2.SS3.SSS1.p8.5.m5.1.1.3.3.1" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p8.5.m5.1.1.3.3.3" xref="S2.SS3.SSS1.p8.5.m5.1.1.3.3.3.cmml">z</mi></mrow></msub></mrow></math>, <math><mrow id="S2.SS3.SSS1.p8.6.m6.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.cmml"><msub id="S2.SS3.SSS1.p8.6.m6.1.1.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.3.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.2.cmml">V</mi><mrow id="S2.SS3.SSS1.p8.6.m6.1.1.3.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.3.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.3.3.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.3.2.cmml">z</mi><mo id="S2.SS3.SSS1.p8.6.m6.1.1.3.3.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p8.6.m6.1.1.3.3.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.3.3.3.cmml">z</mi></mrow></msub><mo id="S2.SS3.SSS1.p8.6.m6.1.1.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.2.cmml">=</mo><mrow id="S2.SS3.SSS1.p8.6.m6.1.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.1.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.3.cmml">k</mi><mo id="S2.SS3.SSS1.p8.6.m6.1.1.1.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.2.cmml">⋅</mo><msup id="S2.SS3.SSS1.p8.6.m6.1.1.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.cmml"><mrow id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.cmml"><msub id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.2.cmml">r</mi><mrow id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.2.cmml">z</mi><mo id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.2.3.3.cmml">z</mi></mrow></msub><mo id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.1" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.1.cmml">-</mo><msub id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3.2" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3.2.cmml">r</mi><mn id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo stretchy="false" id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.SS3.SSS1.p8.6.m6.1.1.1.1.3" xref="S2.SS3.SSS1.p8.6.m6.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow></math>, <math><mrow id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><msub id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.2.2" xref="S2.E3.m1.1.1.2.2.cmml">N</mi><mtext id="S2.E3.m1.1.1.2.3" xref="S2.E3.m1.1.1.2.3a.cmml">grid</mtext></msub><mo id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml"><msub id="S2.E3.m1.1.1.3.2" xref="S2.E3.m1.1.1.3.2.cmml"><mi id="S2.E3.m1.1.1.3.2.2" xref="S2.E3.m1.1.1.3.2.2.cmml">N</mi><mtext id="S2.E3.m1.1.1.3.2.3" xref="S2.E3.m1.1.1.3.2.3a.cmml">LJ</mtext></msub><mo id="S2.E3.m1.1.1.3.1" xref="S2.E3.m1.1.1.3.1.cmml">⋅</mo><msub id="S2.E3.m1.1.1.3.3" xref="S2.E3.m1.1.1.3.3.cmml"><mi id="S2.E3.m1.1.1.3.3.2" xref="S2.E3.m1.1.1.3.3.2.cmml">N</mi><mtext id="S2.E3.m1.1.1.3.3.3" xref="S2.E3.m1.1.1.3.3.3a.cmml">mult</mtext></msub></mrow></mrow></math>, <math><mrow id="S2.E4.m1.1.1" xref="S2.E4.m1.1.1.cmml"><mi id="S2.E4.m1.1.1.2" xref="S2.E4.m1.1.1.2.cmml">L</mi><mo id="S2.E4.m1.1.1.1" xref="S2.E4.m1.1.1.1.cmml">=</mo><mrow id="S2.E4.m1.1.1.3" xref="S2.E4.m1.1.1.3.cmml"><mroot id="S2.E4.m1.1.1.3.2" xref="S2.E4.m1.1.1.3.2.cmml"><msub id="S2.E4.m1.1.1.3.2.3" xref="S2.E4.m1.1.1.3.2.3.cmml"><mi id="S2.E4.m1.1.1.3.2.3.2" xref="S2.E4.m1.1.1.3.2.3.2.cmml">N</mi><mtext id="S2.E4.m1.1.1.3.2.3.3" xref="S2.E4.m1.1.1.3.2.3.3a.cmml">grid</mtext></msub><mn id="S2.E4.m1.1.1.3.2.2" xref="S2.E4.m1.1.1.3.2.2.cmml">3</mn></mroot><mo id="S2.E4.m1.1.1.3.1" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><mn id="S2.E4.m1.1.1.3.3" xref="S2.E4.m1.1.1.3.3.cmml">2</mn><mo id="S2.E4.m1.1.1.3.1a" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><msub 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">0</mn></msub></mrow></mrow></math>, <math><mrow id="S2.SS3.SSS2.p10.1.m1.1.1" xref="S2.SS3.SSS2.p10.1.m1.1.1.cmml"><mn id="S2.SS3.SSS2.p10.1.m1.1.1.2" xref="S2.SS3.SSS2.p10.1.m1.1.1.2.cmml">2</mn><mo id="S2.SS3.SSS2.p10.1.m1.1.1.1" xref="S2.SS3.SSS2.p10.1.m1.1.1.1.cmml">⁢</mo><msub id="S2.SS3.SSS2.p10.1.m1.1.1.3" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.cmml"><mi id="S2.SS3.SSS2.p10.1.m1.1.1.3.2" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.2.cmml">r</mi><mrow id="S2.SS3.SSS2.p10.1.m1.1.1.3.3" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.3.cmml"><mi id="S2.SS3.SSS2.p10.1.m1.1.1.3.3.2" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.3.2.cmml">z</mi><mo id="S2.SS3.SSS2.p10.1.m1.1.1.3.3.1" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS2.p10.1.m1.1.1.3.3.3" xref="S2.SS3.SSS2.p10.1.m1.1.1.3.3.3.cmml">z</mi></mrow></msub></mrow></math>, <math><mrow id="S2.SS3.SSS2.p10.2.m2.1.1" xref="S2.SS3.SSS2.p10.2.m2.1.1.cmml"><msub id="S2.SS3.SSS2.p10.2.m2.1.1.2" xref="S2.SS3.SSS2.p10.2.m2.1.1.2.cmml"><mi id="S2.SS3.SSS2.p10.2.m2.1.1.2.2" xref="S2.SS3.SSS2.p10.2.m2.1.1.2.2.cmml">r</mi><mn id="S2.SS3.SSS2.p10.2.m2.1.1.2.3" xref="S2.SS3.SSS2.p10.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS3.SSS2.p10.2.m2.1.1.1" xref="S2.SS3.SSS2.p10.2.m2.1.1.1.cmml">=</mo><msub id="S2.SS3.SSS2.p10.2.m2.1.1.3" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.cmml"><mi id="S2.SS3.SSS2.p10.2.m2.1.1.3.2" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.2.cmml">r</mi><mrow id="S2.SS3.SSS2.p10.2.m2.1.1.3.3" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.3.cmml"><mi id="S2.SS3.SSS2.p10.2.m2.1.1.3.3.2" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.3.2.cmml">z</mi><mo id="S2.SS3.SSS2.p10.2.m2.1.1.3.3.1" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS2.p10.2.m2.1.1.3.3.3" xref="S2.SS3.SSS2.p10.2.m2.1.1.3.3.3.cmml">z</mi></mrow></msub></mrow></math>, <math><mrow id="S3.T1.2.2.2.m1.1.1" xref="S3.T1.2.2.2.m1.1.1.cmml"><msub id="S3.T1.2.2.2.m1.1.1.2" xref="S3.T1.2.2.2.m1.1.1.2.cmml"><mi id="S3.T1.2.2.2.m1.1.1.2.2" xref="S3.T1.2.2.2.m1.1.1.2.2.cmml">ϵ</mi><mrow id="S3.T1.2.2.2.m1.1.1.2.3" xref="S3.T1.2.2.2.m1.1.1.2.3.cmml"><mi id="S3.T1.2.2.2.m1.1.1.2.3.2" xref="S3.T1.2.2.2.m1.1.1.2.3.2.cmml">L</mi><mo id="S3.T1.2.2.2.m1.1.1.2.3.1" xref="S3.T1.2.2.2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.T1.2.2.2.m1.1.1.2.3.3" xref="S3.T1.2.2.2.m1.1.1.2.3.3.cmml">J</mi></mrow></msub><mo id="S3.T1.2.2.2.m1.1.1.1" xref="S3.T1.2.2.2.m1.1.1.1.cmml">/</mo><msub id="S3.T1.2.2.2.m1.1.1.3" xref="S3.T1.2.2.2.m1.1.1.3.cmml"><mi id="S3.T1.2.2.2.m1.1.1.3.2" xref="S3.T1.2.2.2.m1.1.1.3.2.cmml">k</mi><mi id="S3.T1.2.2.2.m1.1.1.3.3" xref="S3.T1.2.2.2.m1.1.1.3.3.cmml">b</mi></msub></mrow></math>

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1009.5239

Formulas:

1-

L = L_{k i n} + L_{i n t} + L_{m e s o n},

2-

L_{i n t} = - \sum_{i} \bar{ψ_{i}} [γ_{0} (g_{i ω} ω + g_{i ϕ} ϕ) + m_{i}^{*}] ψ_{i},

3-

L_{m e s o n} = - \frac{1}{2} (m_{ω}^{2} ω^{2} + m_{ϕ}^{2} ϕ^{2})

4-

- g_{4} (ω^{4} + \frac{ϕ^{4}}{4} + 3 ω^{2} ϕ^{2} + \frac{4 ω^{3} ϕ}{\sqrt{2}} + \frac{2 ω ϕ^{3}}{\sqrt{2}})

5-

+ \frac{1}{2} k_{0} (σ^{2} + ζ^{2}) - k_{1} {(σ^{2} + ζ^{2})}^{2}

6-

- k_{2} (\frac{σ^{4}}{2} + ζ^{4}) - k_{3} σ^{2} ζ

7-

+ m_{π}^{2} f_{π} σ

8-

+ (\sqrt{2} m_{k}^{2} f_{k} - \frac{1}{\sqrt{2}} m_{π}^{2} f_{π}) ζ,

9-

+ χ^{4} - χ_{0}^{4} + \ln \frac{χ^{4}}{χ_{0}^{4}} - k_{4} \frac{χ^{4}}{χ_{0}^{4}} \ln \frac{σ^{2} ζ}{σ_{0}^{2} ζ_{0}} .

10-

χ^{4} \ln χ

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/math/0301316

Formulas:

1-

Δ V (\vec{r}) = \frac{1}{2} ⟨ {(Δ r)}^{2} ⟩ \nabla^{2} V (\vec{r})

2-

Δ E = ⟨ Δ V (\vec{r}) ⟩

3-

\sim 1000 m c / s e c

4-

F = \frac{- π^{2}}{240} \frac{ℏ c A}{l^{4}}

5-

ψ (x) = {(\frac{m ω}{π ℏ})}^{1 / 4} e^{- (m ω / 2 ℏ) x^{2}}

6-

Δ x \sim {(ℏ / m ω)}^{1 / 2}

7-

ψ (ξ_{1}, ξ_{2}, \dots) = c o n s t . \exp [- (ξ_{1}^{2} + ξ_{2}^{2} + \dots)],

8-

B (x, y, z), ξ_{1}

9-

B = \frac{\sqrt{ℏ c}}{l^{2}}

10-

ı = \frac{N B A}{R Δ t}

Formulas (html):

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0704.2285

Formulas:

1-

δ P = n_{C} n_{A} [1 + ξ_{C A} (r)] δ V_{C} δ V_{A}

2-

L e f t P a n e l :

3-

R i g h t P a n e l

4-

\frac{b_{A}}{b_{G}} (s)

5-

ξ_{C A} (s) = {(\frac{s}{s_{0}})}^{- γ}

6-

s_{0} \sim 8.5 h_{70}^{- 1}

7-

{(\frac{δ ρ}{ρ})}^{2}

8-

ξ_{C A} (s) = b_{C} b_{A} ξ_{ρ} (s), ξ_{C G} (s) = b_{C} b_{G} ξ_{ρ} (s) .

9-

ξ_{ρ} (s)

10-

\frac{b_{A}}{b_{G}} (s)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/1011.0735

Formulas:

1-

≳ 10 h^{- 1}

2-

EW (H α)

3-

R_{Mg II} \propto M_{star}^{0.28} \times {sSFR}^{0.11}

4-

λ λ 2796, 2803

5-

N (H I) \approx 10^{18} - 10^{22}

6-

ρ \sim 10 h^{- 1}

7-

ρ ≲ 120 h^{- 1}

8-

W_{r} (2796)

9-

R_{Mg II} = 75 \times {(L_{B} / L_{B_{*}})}^{(0.35 \pm 0.03)} h^{- 1}

10-

B_{A B} - R_{A B}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1010.4510

Formulas:

1-

σ_{m i n} (ω, T) = σ_{m i n} (ω / T)

2-

S O (3)

3-

({\bar{ψ}}_{\vec{n}, α}, ψ_{\vec{n}, α}), α = 1, 2

4-

𝒮 [\bar{ψ}, ψ] = i \sum_{t, \vec{n}} ({\bar{ψ}}_{t, \vec{n}} \partial_{t} ψ_{t, \vec{n}} + {\bar{ψ}}_{t, \vec{n}} γ_{2} ψ_{t, \vec{n} + \vec{e}}),

5-

𝒮 [\bar{Ψ}, Ψ] = i \sum_{t, \vec{n}, i} ({\bar{Ψ}}_{t, \vec{n}} \partial_{t} Ψ_{t, \vec{n}} + {\bar{Ψ}}_{t, \vec{n}} γ_{2} Ψ_{t, \vec{n} + {\vec{e}}_{i}}^{'}),

6-

Ψ_{t, \vec{n}} = (\begin{matrix} ψ_{t, \vec{n}, 1} \\ ψ_{t, \vec{n}, 2} \end{matrix}), i = 1, 2, 3,

7-

γ_{0}, γ_{j}, j = 1, 2

8-

Ψ_{\vec{n} + {\vec{e}}_{1}}^{'}

9-

Ψ_{\vec{n} + {\vec{e}}_{1}} = e^{{\vec{e}}_{1} \cdot \vec{\partial}} Ψ_{\vec{n}},

10-

Ψ_{\vec{n} + {\vec{e}}_{2}}^{'}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1510.02417

Formulas:

1-

\frac{d 𝐫}{d t} = 𝐯, \frac{d 𝐯}{d t} = \frac{q}{m} (𝐄 + 𝐯 \times 𝐁),

2-

𝐄 = 𝐄 (𝐫, t)

3-

𝐁 = 𝐁 (𝐫, t)

4-

𝐄 = \frac{1}{ρ_{I}} (- 𝐉_{I} \times 𝐁 + μ_{0}^{- 1} (\nabla \times 𝐁) \times 𝐁 - \nabla p_{e}) + \frac{η}{μ_{0}} \nabla \times 𝐁,

5-

μ_{0} = 4 π \cdot 10^{- 7}

6-

\frac{\partial 𝐁}{\partial t} = - \nabla \times 𝐄 .

7-

p (h, χ) = p_{0} e^{1 - y - \sec χ e^{- y}}, y = (h - h_{0}) / H, h \geq 0, 0 \leq χ < π / 2,

8-

(- 1.634, 2.516, 0.0)

9-

[- 186.6, 15] \times [- 100.8, 100.8] \times [- 100.8, 100.8] \cdot 10^{6}

10-

\approx [- 55.2, 4.44] \times {[- 29.8, 29.8]}^{2} R_{M}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1811.02206

Formulas:

1-

𝖾𝗑 F \subset 𝖾𝗑 K

2-

ℳ_{T} (X)

3-

𝖽𝗂𝗌𝗍 (μ, ν) = \sum_{n \geq 1} 2^{- n} | \int f_{n} 𝑑 μ - \int f_{n} 𝑑 ν |,

4-

{(f_{n})}_{n \geq 1}

5-

ℳ_{T}^{e} (X)

6-

ℳ_{T} (X)

7-

(X, Σ, μ)

8-

Φ (p) \approx \int Φ (e) 𝑑 ξ (e),

9-

π : K \to K^{'}

10-

Φ (p) \approx Φ^{'} (π (p))

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0802.1069

Formulas:

1-

9 \leq N_{200} < 200

2-

⟨ L_{X} ⟩ / 10^{42} h^{- 2} erg s^{- 1} = ({12.6}_{- 1.3}^{+ 1.4} (stat) \pm 1.6 (sys)) {(⟨ M_{200} ⟩ / 10^{14} h^{- 1} M_{☉})}^{1.65 \pm 0.13}

3-

P (L_{X} | M, z)

4-

{\bar{L}}_{X} \equiv ⟨ L_{X} | N_{200} ⟩

5-

{\bar{M}}_{200} \equiv ⟨ M_{200} | N_{200} ⟩

6-

200 ρ_{c} (z)

7-

H_{0} = 100 h km s^{- 1}

8-

Ω_{m} = 1 - Ω_{Λ} = 0.3

9-

\sim 10 % - 15 %

10-

⟨ L_{X} | N_{200} ⟩ = (42.1 \pm 1.7) {(N_{200} / 40)}^{1.88 \pm 0.06} \times 10^{42} erg s^{- 1} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1511.05524

Formulas:

1-

β = (β_{e})

2-

Σ := {- 1, + 1}^{X}

3-

P_{β} ((σ_{x})) = Z_{β}^{- 1} \prod_{e} \exp (β_{e} I_{σ} (e)),

4-

I_{σ} (e)

5-

{\hat{P}}_{β} ((n_{e})) = {\hat{Z}}_{β}^{- 1} \prod_{e} \frac{{(β_{e})}^{n_{e}}}{n_{e}!} .

6-

\exp (β_{e} I_{e} (σ))

7-

\sum_{n_{e}} {(σ_{x} σ_{y} β_{e})}^{n_{e}} / (n_{e})!

8-

{0, 1}^{E}

9-

w = (w_{e})

10-

{\tilde{P}}_{β} ((w_{e})) = {\tilde{Z}}_{β}^{- 1} \times 2^{k (w_{e})} \times \prod_{e} {(1 - \exp (- 2 β_{e}))}^{w_{e}} {(\exp (- 2 β_{e}))}^{1 - w_{e}}

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0409055

Formulas:

1-

| ψ^{+} ⟩ = \frac{| H ⟩ | V ⟩ + | V ⟩ | H ⟩}{\sqrt{2}}

2-

| ϕ^{-} ⟩ = \frac{| 45 ⟩ | 45 ⟩ - | 135 ⟩ | 135 ⟩}{\sqrt{2}}

3-

ρ = | V ⟩ ⟨ V |

4-

ρ = 1 / 2 [(1 + η_{1}) | V ⟩ ⟨ V | + (1 - η_{1}) | H ⟩ ⟨ H |]

5-

N_{2} = N_{0} \cdot α η_{2} [1 - η_{1} c o s (2 θ)]

6-

V = \frac{N_{2}^{V} - N_{2}^{H}}{N_{2}^{V} + N_{2}^{H}} = η_{1}

7-

S_{1} = S_{2} = S_{3} = 0

8-

S_{1} = η_{1} N

9-

S_{2} = S_{3} = 0

10-

P = \sqrt{\frac{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}{S_{0}^{2}}} .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1110.0453

Formulas:

1-

S_{m a x}

2-

S_{m i n}

3-

\log (ν L_{ν} (5 G H z) / L_{X})

4-

R_{X} = {\begin{matrix} - 0.67 \log L_{X} + 23.67 & 41.5 \leq \log L_{X} \leq 43 \\ - 5 & 43 < \log L_{X} \leq 44 \\ \log L_{X} - 49 & 44 < \log L_{X} \leq 45 \\ - 4 & \log L_{X} > 45 . \end{matrix}

5-

d N / d S

6-

d N / d S

7-

d N / d S

8-

I (ν) = \int_{S_{m i n}}^{S_{m a x}} \frac{d N}{d S} (ν) \cdot S 𝑑 S .

9-

T (ν) = I (ν) \frac{λ^{2}}{2 k},

10-

χ_{r e d}^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-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/0903.2680

Formulas:

1-

\frac{\partial}{\partial \ln r} \ln (\frac{κ^{2} Σ}{2 Ω B^{2}}) > 0

2-

R_{c o l}

3-

R_{i n} = R_{c o l} \times f^{2}

4-

R_{c o l}

5-

R_{c o l} > 3 r_{L S O}

6-

ν_{Q P O} \propto R_{c o l}^{- 3 / 2}

7-

1.3 r_{L S O} < R_{c o l} < 3 r_{L S O}

8-

< R_{c o l} <

9-

R_{c o l} <

10-

1.3 r_{L S O}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1506.05041

Formulas:

1-

N_{c} = N_{c}^{iso} = 2 N^{'} - 1

2-

𝒞 (ϕ, γ)

3-

𝒞_{1} (ϕ, γ)

4-

𝒞_{2} (ϕ, γ)

5-

𝒞_{1} (ϕ, γ)

6-

𝒞_{2} (ϕ, γ)

7-

ϕ = A {(γ - γ_{0})}^{2} + ϕ_{0}

8-

σ_{L} / σ_{S} = 1.4

9-

V / (N ϵ)

10-

10^{- 16} > V / (N ϵ) > 0

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1012.5724

Formulas:

1-

S_{x x} = V_{x x} / Δ T

2-

S_{x y} = - S_{y x} = V_{x y} / Δ T

3-

S_{x y} (B)

4-

V_{F G} = - 0.466

5-

V_{F G} = - 0.464

6-

S_{x y} = - α \frac{π^{2} k_{B}^{2} T}{3 | e | ϵ_{F}} (\frac{ω τ}{1 + ω^{2} τ^{2}})

7-

V_{F G} = - 0.466

8-

Δ S_{x y}

9-

Δ S_{x y}^{r m s}

10-

Δ S_{x y}^{r m s}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0005468

Formulas:

1-

E \sim n ω^{2} a^{4}

2-

K \sim {(n a^{2})}^{- 1}

3-

Z \sim n ω^{2} a^{2}

4-

n a^{2} \sim t^{- ξ / 2}

5-

a \sim t^{ξ / 4}

6-

Z \sim t^{- ξ / 2}

7-

K \sim t^{ξ / 2}

8-

E (k) = k^{30} / {(k + k_{0})}^{60}

9-

\bar{ω} = \sqrt{K Z}

10-

λ = \bar{ω} / \sqrt{E}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1502.05412

Formulas:

1-

L_{acc, p} \sim \frac{G M_{p} \dot{M_{p}}}{R_{p}} .

2-

{\dot{M}}_{p} \sim 10^{- 9}

3-

L_{acc, p} > 10^{- 4}

4-

R_{H} = a {(\frac{M_{p}}{3 M_{*}})}^{1 / 3} .

5-

M_{env} \approx M_{core} \approx 10

6-

\dot{M} = 10^{- 8}

7-

H / R \sim c_{s} / v_{K} ≲ 0.025

8-

R_{H} \sim 0.05 a

9-

R_{p} \approx R_{H} / 3 \approx 100

10-

\dot{M} M_{p} = 10^{- 5}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1710.09863

Formulas:

1-

𝐐 = (0, 0, L)

2-

(0.5, 0.5, 0)

3-

I 4 / m m m

4-

c_{214} = \frac{2}{3} c_{2126} = 12.91

5-

a^{'} \approx 2 \sqrt{2} a_{0}

6-

(H 0 L)

7-

𝐐 = (\frac{1}{2}, \frac{1}{2}, L)

8-

𝐐 = (0.45, 0.45, L)

9-

(\frac{1}{2}, \frac{1}{2}, L)

10-

𝐐 = (0.5, 0.5, L)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

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

Formulas:

1-

t_{form} \sim \frac{2 ω}{k_{⟂}^{2}}

2-

k_{⟂}^{2} \sim \frac{{⟨ q_{⟂}^{2} ⟩}_{med}}{λ} t_{form} \sim \sqrt{2 ω \hat{q}} .

3-

{⟨ q_{⟂}^{2} ⟩}_{med}

4-

\sin θ \equiv \frac{k_{⟂}}{ω} \sim {(\frac{2 \hat{q}}{ω^{3}})}^{1 / 4}

5-

\hat{ω} \sim {(2 \hat{q})}^{1 / 3}

6-

\hat{q} \sim 5 \dots 15

7-

\hat{ω} \sim 2 {(2 \hat{q})}^{1 / 3}

8-

k_{⟂}^{2} < \sqrt{2 ω \hat{q}}

9-

ω ≪ ω_{c} \equiv \frac{1}{2} \hat{q} L^{2}

10-

\frac{d I^{med}}{d ω d k_{⟂}^{2}} ≃ \frac{α_{s} C_{R}}{16 π} L \frac{1}{ω^{2}} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/physics/0512194

Formulas:

1-

a^{1} Δ, J = 2, M Ω = - 4

2-

A^{3} Π \leftarrow a^{1} Δ

3-

X^{3} Σ^{-}, v " = 0

4-

A^{3} Π \leftarrow a^{1} Δ

5-

c^{1} Π, v = 0 \leftarrow a^{1} Δ, v = 0

6-

A^{3} Π, v = 0 \leftarrow X^{3} Σ^{-}, v = 0

7-

A^{3} Π, v = 0 \leftarrow X^{3} Σ^{-}, v = 0

8-

c^{1} Π, v = 0 \leftarrow a^{1} Δ, v = 0

9-

T_{d i s s}

10-

T_{d i s s}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/2002.09042

Formulas:

1-

5 - 10 M_{\oplus}

2-

Δ M / M < 50 %

3-

α = 3 / 2, 5 / 2

4-

M_{i s o}

5-

M_{i s o} = 0.16 {(\frac{b Σ_{o}}{10})}^{α} {(\frac{a}{1 A U})}^{α (2 - α)} M_{\oplus}

6-

r_{H} = 6.9 \times 10^{- 3} {(\frac{b Σ_{o}}{10})}^{1 / 2} {(\frac{a}{1 A U})}^{(1 / 2) (4 - α)} A U

7-

Σ = Σ_{o} {(\frac{a}{1 A U})}^{- α}

8-

1.5 \times 10^{- 5} M_{\oplus}

9-

M_{e m b r y o} = 1.0 M_{\oplus}

10-

M_{d i s k} = 25 M_{\oplus}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0206299

Formulas:

1-

\frac{F_{M I R} (15 ″)}{F_{M I R} (135 ″)}

2-

\frac{F_{M I R} (15 ″)}{F_{M I R} (135 ″)}

3-

\frac{F_{M I R}}{F_{K}}

4-

\frac{F_{M I R}}{F_{K}}

5-

\frac{F_{M I R}}{F_{K}}

6-

\frac{F_{M I R}}{F_{K}}

7-

\frac{F_{M I R}}{F_{K}}

8-

S F R (M_{☉} y r^{- 1}) = 1.76 \times 10^{- 10} L_{F I R} (L_{☉})

9-

L_{F I R}

10-

τ_{R} = \frac{M_{g a s}}{S F R}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1705.08547

Formulas:

1-

η (0, y, t) = \frac{H}{2} \cos [\frac{k}{2} (\sin α_{w} + \sin β_{w}) y + σ t] \cos (\frac{π}{λ} y)

2-

λ = \frac{L}{\sin α_{w} - \sin β_{w}}

3-

θ_{m a x} = \tan^{- 1} (\frac{S / 2}{h})

4-

\frac{H}{S} = 4 (\frac{\sinh k h}{k h}) \frac{k h \sinh k h - \cosh k h + 1}{\sinh 2 k h + 2 k h}

5-

θ (0, y, t) = θ_{m a x} \cos [\frac{k}{2} (\sin α_{w} + \sin β_{w}) y + σ t] \cos (\frac{π}{λ} y)

6-

ξ_{b} = \frac{\tan ϕ}{\sqrt{H_{b} / L_{0}}}

7-

\bar{\vec{u} (x, y, z)} = \frac{1}{n_{w} T} \int_{t_{0}}^{t_{0} + n_{w} T} \vec{u} (x, y, z, t) d t

8-

t_{0} = 40 T

9-

(u, v, w)

10-

\bar{u (x, y, z)}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1703.08908

Formulas:

1-

π_{n + 1}^{Lip} (ℍ^{n}) \neq 0

2-

π_{n}^{Lip} (X)

3-

π_{k} (ℍ^{n}) {\begin{matrix} = 0 & if & 1 \leq k < n & [11], \\ \neq 0 & if & k = n & [2], \\ = 0 & if & n = 1, k \geq 2 & [12], \\ \neq 0 & if & n = 2 ℓ, k = 4 ℓ - 1 & [7] . \end{matrix}

4-

π_{n + 1}^{Lip} (ℍ^{n}) \neq 0

5-

π_{n} (ℍ^{n})

6-

π_{n} (ℍ^{n})

7-

π_{k}^{Lip} (ℍ^{n}) \neq 0

8-

f : 𝕊^{k} \to ℍ^{n}

9-

F : 𝔹^{k + 1} \to ℍ^{n}

10-

π_{k} (ℍ^{n}) \neq 0

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0107018

Formulas:

1-

G_{Δ t} (t) = \ln S (t) - \ln S (t - Δ t),

2-

g (t) = τ \frac{d}{d t} \ln S (t)

3-

G {(t)}_{Δ t} = \frac{1}{τ} \int_{t - Δ t}^{t} 𝑑 t^{'} g (t^{'}) .

4-

q (t) = τ \frac{d}{d t} Q (t)

5-

τ \partial_{t} g (t) = - r g (t) - κ g^{3} (t) + μ q (t) + ξ_{i} (t) .

6-

⟨ ξ (t) ξ (t^{'}) ⟩ = τ δ (t - t^{'})

7-

{⟨ {(G_{i} (Δ t))}^{2} ⟩}_{0} = \frac{2}{r^{2}} \frac{1}{τ} Δ t,

8-

D = \frac{2}{r^{2} τ}

9-

{⟨ G_{Δ t} ⟩}_{Q_{Δ t}}

10-

χ = lim_{Δ Q \to 0} \frac{{⟨ G_{Δ t} ⟩}_{Δ Q}}{Δ Q} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0808.2173

Formulas:

1-

I_{2} (m)

2-

X^{⟂} = ⋂_{x \in X} x^{⟂}

3-

Δ (Λ) ≅ Δ (Γ)

4-

Δ_{s} (Γ)

5-

Δ_{ℓ} (Γ)

6-

K (n, k)

7-

K (n - k, k)

8-

K (5, 2)

9-

K (6, 2)

10-

K (n, k)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0512511

Formulas:

1-

u v b y

2-

T_{eff} = 12500 \pm 250

3-

\log g = 4.3 \pm 0.1

4-

T_{eff} = 10500 \pm 500

5-

\log g = 4.1 \pm 0.2

6-

\log g = 4.2 \pm 0.2

7-

V γ_{C D}

8-

T_{C D} (periastron)

9-

V γ_{C D}

10-

a_{C D} \sin i_{C D}

Formulas (html):

<math><mrow id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">u</mi><mo id="S1.p2.2.m2.1.1.1" xref="S1.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml">v</mi><mo id="S1.p2.2.m2.1.1.1a" xref="S1.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p2.2.m2.1.1.4" xref="S1.p2.2.m2.1.1.4.cmml">b</mi><mo id="S1.p2.2.m2.1.1.1b" xref="S1.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p2.2.m2.1.1.5" xref="S1.p2.2.m2.1.1.5.cmml">y</mi></mrow></math>, <math><mrow id="S2.SS3.p3.7.m7.1.1" xref="S2.SS3.p3.7.m7.1.1.cmml"><msub id="S2.SS3.p3.7.m7.1.1.2" xref="S2.SS3.p3.7.m7.1.1.2.cmml"><mi id="S2.SS3.p3.7.m7.1.1.2.2" xref="S2.SS3.p3.7.m7.1.1.2.2.cmml">T</mi><mi id="S2.SS3.p3.7.m7.1.1.2.3" xref="S2.SS3.p3.7.m7.1.1.2.3.cmml">eff</mi></msub><mo id="S2.SS3.p3.7.m7.1.1.1" xref="S2.SS3.p3.7.m7.1.1.1.cmml">=</mo><mrow id="S2.SS3.p3.7.m7.1.1.3" xref="S2.SS3.p3.7.m7.1.1.3.cmml"><mn id="S2.SS3.p3.7.m7.1.1.3.2" xref="S2.SS3.p3.7.m7.1.1.3.2.cmml">12500</mn><mo id="S2.SS3.p3.7.m7.1.1.3.1" xref="S2.SS3.p3.7.m7.1.1.3.1.cmml">±</mo><mn id="S2.SS3.p3.7.m7.1.1.3.3" xref="S2.SS3.p3.7.m7.1.1.3.3.cmml">250</mn></mrow></mrow></math>, <math><mrow id="S2.SS3.p3.8.m8.1.1" xref="S2.SS3.p3.8.m8.1.1.cmml"><mrow id="S2.SS3.p3.8.m8.1.1.2" xref="S2.SS3.p3.8.m8.1.1.2.cmml"><mi id="S2.SS3.p3.8.m8.1.1.2.1" xref="S2.SS3.p3.8.m8.1.1.2.1.cmml">log</mi><mo id="S2.SS3.p3.8.m8.1.1.2a" xref="S2.SS3.p3.8.m8.1.1.2.cmml">⁡</mo><mi id="S2.SS3.p3.8.m8.1.1.2.2" xref="S2.SS3.p3.8.m8.1.1.2.2.cmml">g</mi></mrow><mo id="S2.SS3.p3.8.m8.1.1.1" xref="S2.SS3.p3.8.m8.1.1.1.cmml">=</mo><mrow id="S2.SS3.p3.8.m8.1.1.3" xref="S2.SS3.p3.8.m8.1.1.3.cmml"><mn id="S2.SS3.p3.8.m8.1.1.3.2" xref="S2.SS3.p3.8.m8.1.1.3.2.cmml">4.3</mn><mo id="S2.SS3.p3.8.m8.1.1.3.1" xref="S2.SS3.p3.8.m8.1.1.3.1.cmml">±</mo><mn id="S2.SS3.p3.8.m8.1.1.3.3" xref="S2.SS3.p3.8.m8.1.1.3.3.cmml">0.1</mn></mrow></mrow></math>, <math><mrow id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml"><msub id="S2.SS3.p4.1.m1.1.1.2" xref="S2.SS3.p4.1.m1.1.1.2.cmml"><mi id="S2.SS3.p4.1.m1.1.1.2.2" xref="S2.SS3.p4.1.m1.1.1.2.2.cmml">T</mi><mi id="S2.SS3.p4.1.m1.1.1.2.3" xref="S2.SS3.p4.1.m1.1.1.2.3.cmml">eff</mi></msub><mo id="S2.SS3.p4.1.m1.1.1.1" xref="S2.SS3.p4.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS3.p4.1.m1.1.1.3" xref="S2.SS3.p4.1.m1.1.1.3.cmml"><mn id="S2.SS3.p4.1.m1.1.1.3.2" xref="S2.SS3.p4.1.m1.1.1.3.2.cmml">10500</mn><mo id="S2.SS3.p4.1.m1.1.1.3.1" xref="S2.SS3.p4.1.m1.1.1.3.1.cmml">±</mo><mn id="S2.SS3.p4.1.m1.1.1.3.3" xref="S2.SS3.p4.1.m1.1.1.3.3.cmml">500</mn></mrow></mrow></math>, <math><mrow id="S2.SS3.p4.4.m4.1.1" xref="S2.SS3.p4.4.m4.1.1.cmml"><mrow id="S2.SS3.p4.4.m4.1.1.2" xref="S2.SS3.p4.4.m4.1.1.2.cmml"><mi id="S2.SS3.p4.4.m4.1.1.2.1" xref="S2.SS3.p4.4.m4.1.1.2.1.cmml">log</mi><mo id="S2.SS3.p4.4.m4.1.1.2a" xref="S2.SS3.p4.4.m4.1.1.2.cmml">⁡</mo><mi id="S2.SS3.p4.4.m4.1.1.2.2" xref="S2.SS3.p4.4.m4.1.1.2.2.cmml">g</mi></mrow><mo id="S2.SS3.p4.4.m4.1.1.1" xref="S2.SS3.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS3.p4.4.m4.1.1.3" xref="S2.SS3.p4.4.m4.1.1.3.cmml"><mn id="S2.SS3.p4.4.m4.1.1.3.2" xref="S2.SS3.p4.4.m4.1.1.3.2.cmml">4.1</mn><mo id="S2.SS3.p4.4.m4.1.1.3.1" xref="S2.SS3.p4.4.m4.1.1.3.1.cmml">±</mo><mn id="S2.SS3.p4.4.m4.1.1.3.3" xref="S2.SS3.p4.4.m4.1.1.3.3.cmml">0.2</mn></mrow></mrow></math>, <math><mrow id="S2.SS3.p4.6.m6.1.1" xref="S2.SS3.p4.6.m6.1.1.cmml"><mrow id="S2.SS3.p4.6.m6.1.1.2" xref="S2.SS3.p4.6.m6.1.1.2.cmml"><mi id="S2.SS3.p4.6.m6.1.1.2.1" xref="S2.SS3.p4.6.m6.1.1.2.1.cmml">log</mi><mo id="S2.SS3.p4.6.m6.1.1.2a" xref="S2.SS3.p4.6.m6.1.1.2.cmml">⁡</mo><mi id="S2.SS3.p4.6.m6.1.1.2.2" xref="S2.SS3.p4.6.m6.1.1.2.2.cmml">g</mi></mrow><mo id="S2.SS3.p4.6.m6.1.1.1" xref="S2.SS3.p4.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS3.p4.6.m6.1.1.3" xref="S2.SS3.p4.6.m6.1.1.3.cmml"><mn id="S2.SS3.p4.6.m6.1.1.3.2" xref="S2.SS3.p4.6.m6.1.1.3.2.cmml">4.2</mn><mo id="S2.SS3.p4.6.m6.1.1.3.1" xref="S2.SS3.p4.6.m6.1.1.3.1.cmml">±</mo><mn id="S2.SS3.p4.6.m6.1.1.3.3" xref="S2.SS3.p4.6.m6.1.1.3.3.cmml">0.2</mn></mrow></mrow></math>, <math><mrow id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml"><mpadded width="-1.7pt" id="S3.SS1.p2.4.m4.1.1.2" xref="S3.SS1.p2.4.m4.1.1.2.cmml"><mi id="S3.SS1.p2.4.m4.1.1.2a" xref="S3.SS1.p2.4.m4.1.1.2.cmml">V</mi></mpadded><mo id="S3.SS1.p2.4.m4.1.1.1" xref="S3.SS1.p2.4.m4.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p2.4.m4.1.1.3" xref="S3.SS1.p2.4.m4.1.1.3.cmml"><mi id="S3.SS1.p2.4.m4.1.1.3.2" xref="S3.SS1.p2.4.m4.1.1.3.2.cmml">γ</mi><mrow id="S3.SS1.p2.4.m4.1.1.3.3" xref="S3.SS1.p2.4.m4.1.1.3.3.cmml"><mpadded width="-1.7pt" id="S3.SS1.p2.4.m4.1.1.3.3.2" xref="S3.SS1.p2.4.m4.1.1.3.3.2.cmml"><mi id="S3.SS1.p2.4.m4.1.1.3.3.2a" xref="S3.SS1.p2.4.m4.1.1.3.3.2.cmml">C</mi></mpadded><mo id="S3.SS1.p2.4.m4.1.1.3.3.1" xref="S3.SS1.p2.4.m4.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.4.m4.1.1.3.3.3" xref="S3.SS1.p2.4.m4.1.1.3.3.3.cmml">D</mi></mrow></msub></mrow></math>, <math><mrow id="S3.T3.3.3.1.m1.1.2" xref="S3.T3.3.3.1.m1.1.2.cmml"><msub id="S3.T3.3.3.1.m1.1.2.2" xref="S3.T3.3.3.1.m1.1.2.2.cmml"><mi id="S3.T3.3.3.1.m1.1.2.2.2" xref="S3.T3.3.3.1.m1.1.2.2.2.cmml">T</mi><mrow id="S3.T3.3.3.1.m1.1.2.2.3" xref="S3.T3.3.3.1.m1.1.2.2.3.cmml"><mpadded width="-1.7pt" id="S3.T3.3.3.1.m1.1.2.2.3.2" xref="S3.T3.3.3.1.m1.1.2.2.3.2.cmml"><mi id="S3.T3.3.3.1.m1.1.2.2.3.2a" xref="S3.T3.3.3.1.m1.1.2.2.3.2.cmml">C</mi></mpadded><mo id="S3.T3.3.3.1.m1.1.2.2.3.1" xref="S3.T3.3.3.1.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.T3.3.3.1.m1.1.2.2.3.3" xref="S3.T3.3.3.1.m1.1.2.2.3.3.cmml">D</mi></mrow></msub><mo id="S3.T3.3.3.1.m1.1.2.1" xref="S3.T3.3.3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.T3.3.3.1.m1.1.2.3.2" xref="S3.T3.3.3.1.m1.1.2.cmml"><mo stretchy="false" id="S3.T3.3.3.1.m1.1.2.3.2.1" xref="S3.T3.3.3.1.m1.1.2.cmml">(</mo><mi id="S3.T3.3.3.1.m1.1.1" xref="S3.T3.3.3.1.m1.1.1.cmml">periastron</mi><mo stretchy="false" id="S3.T3.3.3.1.m1.1.2.3.2.2" xref="S3.T3.3.3.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S3.T3.5.5.1.m1.1.1" xref="S3.T3.5.5.1.m1.1.1.cmml"><mpadded width="-1.7pt" id="S3.T3.5.5.1.m1.1.1.2" xref="S3.T3.5.5.1.m1.1.1.2.cmml"><mi id="S3.T3.5.5.1.m1.1.1.2a" xref="S3.T3.5.5.1.m1.1.1.2.cmml">V</mi></mpadded><mo id="S3.T3.5.5.1.m1.1.1.1" xref="S3.T3.5.5.1.m1.1.1.1.cmml">⁢</mo><msub id="S3.T3.5.5.1.m1.1.1.3" xref="S3.T3.5.5.1.m1.1.1.3.cmml"><mi id="S3.T3.5.5.1.m1.1.1.3.2" xref="S3.T3.5.5.1.m1.1.1.3.2.cmml">γ</mi><mrow id="S3.T3.5.5.1.m1.1.1.3.3" xref="S3.T3.5.5.1.m1.1.1.3.3.cmml"><mpadded width="-1.7pt" id="S3.T3.5.5.1.m1.1.1.3.3.2" xref="S3.T3.5.5.1.m1.1.1.3.3.2.cmml"><mi id="S3.T3.5.5.1.m1.1.1.3.3.2a" xref="S3.T3.5.5.1.m1.1.1.3.3.2.cmml">C</mi></mpadded><mo id="S3.T3.5.5.1.m1.1.1.3.3.1" xref="S3.T3.5.5.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.T3.5.5.1.m1.1.1.3.3.3" xref="S3.T3.5.5.1.m1.1.1.3.3.3.cmml">D</mi></mrow></msub></mrow></math>, <math><mrow id="S3.T3.18.18.1.m1.1.1" xref="S3.T3.18.18.1.m1.1.1.cmml"><msub id="S3.T3.18.18.1.m1.1.1.2" xref="S3.T3.18.18.1.m1.1.1.2.cmml"><mi id="S3.T3.18.18.1.m1.1.1.2.2" xref="S3.T3.18.18.1.m1.1.1.2.2.cmml">a</mi><mrow id="S3.T3.18.18.1.m1.1.1.2.3" xref="S3.T3.18.18.1.m1.1.1.2.3.cmml"><mpadded width="-1.7pt" id="S3.T3.18.18.1.m1.1.1.2.3.2" xref="S3.T3.18.18.1.m1.1.1.2.3.2.cmml"><mi id="S3.T3.18.18.1.m1.1.1.2.3.2a" xref="S3.T3.18.18.1.m1.1.1.2.3.2.cmml">C</mi></mpadded><mo id="S3.T3.18.18.1.m1.1.1.2.3.1" xref="S3.T3.18.18.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.T3.18.18.1.m1.1.1.2.3.3" xref="S3.T3.18.18.1.m1.1.1.2.3.3.cmml">D</mi></mrow></msub><mo id="S3.T3.18.18.1.m1.1.1.1" xref="S3.T3.18.18.1.m1.1.1.1.cmml">⁢</mo><mrow id="S3.T3.18.18.1.m1.1.1.3" xref="S3.T3.18.18.1.m1.1.1.3.cmml"><mi id="S3.T3.18.18.1.m1.1.1.3.1" xref="S3.T3.18.18.1.m1.1.1.3.1.cmml">sin</mi><mo id="S3.T3.18.18.1.m1.1.1.3a" xref="S3.T3.18.18.1.m1.1.1.3.cmml">⁡</mo><msub id="S3.T3.18.18.1.m1.1.1.3.2" xref="S3.T3.18.18.1.m1.1.1.3.2.cmml"><mi id="S3.T3.18.18.1.m1.1.1.3.2.2" xref="S3.T3.18.18.1.m1.1.1.3.2.2.cmml">i</mi><mrow id="S3.T3.18.18.1.m1.1.1.3.2.3" xref="S3.T3.18.18.1.m1.1.1.3.2.3.cmml"><mpadded width="-1.7pt" id="S3.T3.18.18.1.m1.1.1.3.2.3.2" xref="S3.T3.18.18.1.m1.1.1.3.2.3.2.cmml"><mi id="S3.T3.18.18.1.m1.1.1.3.2.3.2a" xref="S3.T3.18.18.1.m1.1.1.3.2.3.2.cmml">C</mi></mpadded><mo id="S3.T3.18.18.1.m1.1.1.3.2.3.1" xref="S3.T3.18.18.1.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.T3.18.18.1.m1.1.1.3.2.3.3" xref="S3.T3.18.18.1.m1.1.1.3.2.3.3.cmml">D</mi></mrow></msub></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1302.1732

Formulas:

1-

\dot{x} = v (x, t), x \in U \subset ℝ^{2}, t \in [t_{0}, t_{1}],

2-

v (x, t)

3-

x (t_{1}; t_{0}, x_{0})

4-

F_{t_{0}}^{t_{1}} : 𝐱_{0} ⟼ 𝐱 (t_{1}; t_{0}, x_{0}),

5-

C_{t_{0}}^{t_{1}} (x_{0}) = {[D F_{t_{0}}^{t_{1}} (x_{0})]}^{⊤} D F_{t_{0}}^{t_{1}} (x_{0}),

6-

D F_{t_{0}}^{t_{1}}

7-

0 < λ_{1} \leq λ_{2}

8-

{ξ_{1}, ξ_{2}}

9-

C_{t_{0}}^{t_{1}} (x_{0}) ξ_{i} (x_{0}) = λ_{i} (x_{0}) ξ_{i} (x_{0}), | ξ_{i} (x_{0}) | = 1, i \in {1, 2},

10-

ξ_{2} (x_{0}) = Ω ξ_{1} (x_{0}), Ω = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) .

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0603630

Formulas:

1-

(\arg r^{+} - \arg r^{-}) / 2

2-

r^{\pm} = r_{1} + r_{2}^{\pm}

3-

r_{1} = (n_{0} - n_{1}) / (n_{0} + n_{1})

4-

r_{2}^{\pm} = | t_{01} t_{10} | e^{i (ϕ_{0} + ϕ)} (n_{1} - n_{2}^{\pm}) / (n_{1} + n_{2}^{\pm})

5-

ϕ = 4 π n_{1} d / λ

6-

ϕ_{0} = \arg t_{01} t_{10}

7-

| r_{1} | >> | r_{2}^{\pm} |

8-

θ (Δ t)

9-

θ (Δ t)

10-

θ (Δ t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/2007.06755

Formulas:

1-

M_{k} = B^{- 1} * τ * M_{p}

2-

v^{'} = \sum_{k = 1}^{K} (w_{k} * v * M_{k})

3-

N * K = 272, 272

4-

L_{v} = \frac{1}{N} \sqrt{\sum_{n = 1}^{N} {(v_{p, n} - v_{g, n})}^{2}}

5-

L_{m} = \frac{1}{K} \sum_{k = 1}^{K} (| R_{k} | + | T_{k} | + | S_{k} - 1 |)

6-

L_{x} = \frac{1}{K} \sum_{k = 1}^{K} {\begin{matrix} | x_{k} - x_{m a x} |, & if x_{k} > x_{m a x} \\ | x_{k} - x_{m i n} |, & if x_{k} < x_{m i n} \\ 0, & if x_{m i n} <= x_{k} <= x_{m a x} \end{matrix}

7-

L_{p} = \frac{1}{N} \sqrt{\sum_{n = 1}^{N} {(\nabla^{2} (v_{p, n}) - \nabla^{2} (v_{g, n}))}^{2}}

8-

L_{r} = L_{v} + λ_{m} L_{m} + λ_{x} (L_{R} + L_{T} + L_{S}) + λ_{p} L_{p}

9-

L_{C} = \frac{1}{m} \sum_{i = 1}^{m} (C (J_{i}) - C (D (Z_{i})))

10-

L_{G} = \sum_{i = 1}^{m} (| J_{i} - D (Z_{i}) | + λ_{0} | D (Z_{i}) | - λ_{1} C (D (Z_{i})))

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1302.0478

Formulas:

1-

H = \sum_{n = 0}^{N} (\frac{p_{n}^{2}}{2 m_{n}} + \frac{1}{2} K {(x_{n + 1} - x_{n})}^{2}),

2-

x_{n} \cos β_{n} = \frac{1}{2} (x_{n + 1} + x_{n - 1}),

3-

\cos β_{n} = (1 - ω^{2} m_{n} / 2 K) .

4-

x_{n} = \int_{k_{1}}^{k_{2}} f (k) \exp (i k n) 𝑑 k .

5-

{(- i n)}^{p} x_{n} = \int_{k_{1}}^{k_{2}} \frac{d^{p} f (k)}{d k^{p}} \exp (i k n) 𝑑 k .

6-

\cos β (n / N)

7-

\cos β (i \frac{1}{N} \frac{d}{d k}) f = f \cos (k),

8-

m_{n} = M_{0} + \frac{M_{1} - M_{0}}{N} n,

9-

x_{n} = H_{n + Δ n}^{(1)} (z), H_{n + Δ n}^{(2)} (z)

10-

Δ n = z \cos β (0), z = N / (\cos β (1) - \cos β (0))

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cs/0010017

Formulas:

1-

t = 13^{\circ} C

2-

j_{n}^{'} (x) = 0

3-

d_{l, m, n} = \frac{2}{c \sqrt{{(l / X)}^{2} + {(m / Y)}^{2} + {(n / Z)}^{2}}}

4-

(l, m, n)

5-

(2 l, 2 m, 2 n)

6-

(3 l, 3 m, 3 n)

7-

j_{n}^{'} (x) = 0, n = 0, 1, \dots

8-

f_{n s} = \frac{c}{2 π a} z_{n s}

9-

j_{n} (x) = \sqrt{\frac{π}{2 x}} J_{n + \frac{1}{2}} (x) .

10-

j_{n}^{'} (x)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1712.00459

Formulas:

1-

10^{14} < ν_{peak} < 10^{15}

2-

χ_{ν}^{2} / d o f

3-

F (t) = F_{c} + F_{0} {(e^{\frac{t_{0} - t}{τ_{r}}} + e^{\frac{t - t_{0}}{τ_{d}}})}^{- 1},

4-

τ_{r} \times \ln 2

5-

τ_{d} \times \ln 2

6-

t_{p} = t_{0} + \frac{τ_{r} τ_{d}}{τ_{r} + τ_{d}} \ln (\frac{τ_{d}}{τ_{r}}),

7-

ξ = \frac{τ_{d} - τ_{r}}{τ_{d} + τ_{r}},

8-

χ_{ν}^{2} = χ^{2} / d o f

9-

σ_{net}^{2} = σ_{obs}^{2} + k^{2} σ_{bkg}^{2}

10-

0 \leq k^{2} < 1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1207.3599

Formulas:

1-

T^{F r a m e}

2-

T^{F r a m e}

3-

T_{n, n + 1}^{G B} = \frac{F}{100} \times \frac{1}{2} [T S_{n} + T S_{n + 1}]

4-

T_{1}^{G B} = \frac{F \times T S_{1}}{100}

5-

T_{n}^{G B} = \frac{F \times T S_{n}}{100}

6-

D = M i n (T S_{1} \dots \dots T S_{n}) \times \frac{F}{100}

7-

△ T = E x p e c t e d A r r i v a l T i m e - C u r r e n t A r r i v a l T i m e

8-

D V = {\begin{matrix} 0 & if ∣ △ T ∣ < D \\ △ T & if ∣ △ T ∣ > D \end{matrix}

9-

E_{T o t a l} = \sum_{k = 1}^{N} E_{S l e e p_{k}} + \sum_{k = 1}^{N} E_{A c t i v e_{k}}

10-

T_{S l e e p} = T_{F r a m e} - T_{A c t i v e}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1101.5417

Formulas:

1-

| ψ ⟩ = \int \frac{d ω}{2 π} \tilde{f} (ω) | ω, - ω ⟩ \otimes \frac{| {\underline{u}}_{A}, {\underline{u}}_{B} ⟩ + e^{i α} | {\underline{u}}_{A}^{'}, {\underline{u}}_{B}^{'} ⟩}{\sqrt{2}},

2-

\tilde{f} (ω)

3-

\int d ω {| \tilde{f} (ω) |}^{2} = 2 π T^{- 1}

4-

{\underline{u}}_{A, B}^{'}

5-

{\underline{u}}_{A, B} \cdot {\underline{u}}_{A, B}^{'} = 0

6-

\int \frac{d ω}{2 π} \tilde{f} (ω) | ω, - ω ⟩ = \iint d t_{A} d t_{B} f (t_{A} - t_{B}) | t_{A}, t_{B} ⟩

7-

\tilde{f} (ω)

8-

{| f (τ) |}^{2}

9-

| ψ_{p} ⟩ = [| {\underline{u}}_{A}, {\underline{u}}_{B} ⟩ + e^{i α} | {\underline{u}}_{A}^{'}, {\underline{u}}_{B}^{'} ⟩] / \sqrt{2}

10-

| f (t_{A} - t_{B}) ⟩

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1203.0110

Formulas:

1-

E_{x} = E_{0} s i n [ω (t - z / c) + ϕ (t - z / c)]

2-

E_{y} = - E_{0} c o s [ω (t - z / c) + ϕ (t - z / c)]

3-

B_{x} = (E_{0} / c) c o s [ω (t - z / c) + ϕ (t - z / c)]

4-

B_{y} = (E_{0} / c) s i n [ω (t - z / c) + ϕ (t - z / c)]

5-

ϕ (t - z / c)

6-

ϕ (t - z / c) = π s i n [α ω (t - z / c)]

7-

\frac{d 𝐩}{d t}

8-

q [𝐄 + 𝐯 \times 𝐁]

9-

\frac{𝐩 / m_{0}}{\sqrt{1 + {| 𝐩 |}^{2} / {(m_{0} c)}^{2}}}

10-

\frac{d 𝐫}{d t}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1311.4471

Formulas:

1-

{(K C l)}_{32}

2-

C_{i, j} = \exp (- r_{i, j}^{2} / (2 σ_{i j}^{2}))

3-

T \in [0.20, 0.31]

4-

Q (t) = \int 𝑑 \vec{r} ρ (\vec{r}, t_{0}) ρ (\vec{r}, t + t_{0}) \sim \sum_{i = 1}^{N} w (| {\vec{r}}_{i} (t_{0}) - {\vec{r}}_{i} (t + t_{0}) |),

5-

ρ (\vec{r}, t_{0})

6-

w (r) = 1

7-

χ_{4} (t) = \frac{1}{N} [⟨ Q^{2} (t) ⟩ - {⟨ Q (t) ⟩}^{2}]

8-

χ_{4} (t)

9-

χ_{4} (t)

10-

B L J_{55}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

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