Run 11339415

Paper: https://arxiv.org/abs/1901.01225

Formulas:

1-

P (m, n)

2-

P (m, n)

3-

P (m, n)

4-

P (m, n)

5-

P (3, 2)

6-

P (5, 2)

7-

P (2, 2)

8-

P (2, 2)

9-

P (3, 2)

10-

P (5, 2)

Formulas (html):

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1503.03375

Formulas:

1-

\frac{δ F}{F_{0}} \approx \frac{1}{\sqrt{2 π}} \frac{Δ}{σ} \frac{2 R Δ}{σ^{2}} e^{- R^{2} / 2 σ^{2}},

2-

M o d e = 3 \times M e d i a n - 2 \times M e a n .

3-

\overset{´}{m_{i, j, ref}} - \overset{´}{m_{i, j, k}}

4-

{\bar{Δ}}_{l} - {\bar{Δ}}_{r}

5-

\overset{´}{m_{i, j, ref}} - \overset{´}{m_{i, j, k}}

6-

{\bar{Δ}}_{l, r} = \frac{1}{N_{l, r}} \sum m_{i, j} - \overset{´}{m_{i, j}} + Z P_{l, r},

7-

Z P_{l, r}

8-

r - m_{i, j} = Z P_{l, r} - 0.07 X + 0.107 (r - i),

9-

{\bar{Δ}}_{l} - {\bar{Δ}}_{r}

10-

0 ″ .1 - 0 ″ .2

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0908.3893

Formulas:

1-

λ = 912 (1 + z) Å

2-

λ λ 2796, 2803

3-

W_{r}^{M g i λ 2852} >

4-

W_{r}^{M g ii λ 2796} / W_{r}^{F e ii λ 2600} < 2

5-

W_{r}^{F e ii λ 2600}

6-

W_{r}^{M g ii λ 2796}

7-

W_{r}^{M g i λ 2852}

8-

U_{AB} = U_{Vega} + 0.66

9-

R_{AB} = R_{Vega} + 0.18

10-

U_{FORS1} = U_{VIMOS} + 0.04 mag

Formulas (html):

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id="S2.T1.7.7.7.m1.1.1.2.2" xref="S2.T1.7.7.7.m1.1.1.2.2.cmml">W</mi><mi mathvariant="normal" id="S2.T1.7.7.7.m1.1.1.2.3" xref="S2.T1.7.7.7.m1.1.1.2.3.cmml">r</mi><mrow id="S2.T1.7.7.7.m1.1.1.3" xref="S2.T1.7.7.7.m1.1.1.3.cmml"><mi id="S2.T1.7.7.7.m1.1.1.3.2" xref="S2.T1.7.7.7.m1.1.1.3.2.cmml">M</mi><mo id="S2.T1.7.7.7.m1.1.1.3.1" xref="S2.T1.7.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.T1.7.7.7.m1.1.1.3.3" xref="S2.T1.7.7.7.m1.1.1.3.3.cmml">g</mi><mo id="S2.T1.7.7.7.m1.1.1.3.1a" xref="S2.T1.7.7.7.m1.1.1.3.1.cmml">⁢</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.T1.7.7.7.m1.1.1.3.4" xref="S2.T1.7.7.7.m1.1.1.3.4a.cmml">i</mtext><mo id="S2.T1.7.7.7.m1.1.1.3.1b" xref="S2.T1.7.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.T1.7.7.7.m1.1.1.3.5" xref="S2.T1.7.7.7.m1.1.1.3.5.cmml">λ</mi><mo id="S2.T1.7.7.7.m1.1.1.3.1c" xref="S2.T1.7.7.7.m1.1.1.3.1.cmml">⁢</mo><mn id="S2.T1.7.7.7.m1.1.1.3.6" xref="S2.T1.7.7.7.m1.1.1.3.6.cmml">2852</mn></mrow></msubsup></math>, <math><mrow id="S2.SS2.p3.5.m5.1.1" xref="S2.SS2.p3.5.m5.1.1.cmml"><msub id="S2.SS2.p3.5.m5.1.1.2" xref="S2.SS2.p3.5.m5.1.1.2.cmml"><mi id="S2.SS2.p3.5.m5.1.1.2.2" xref="S2.SS2.p3.5.m5.1.1.2.2.cmml">U</mi><mi id="S2.SS2.p3.5.m5.1.1.2.3" xref="S2.SS2.p3.5.m5.1.1.2.3.cmml">AB</mi></msub><mo id="S2.SS2.p3.5.m5.1.1.1" xref="S2.SS2.p3.5.m5.1.1.1.cmml">=</mo><mrow id="S2.SS2.p3.5.m5.1.1.3" xref="S2.SS2.p3.5.m5.1.1.3.cmml"><msub id="S2.SS2.p3.5.m5.1.1.3.2" xref="S2.SS2.p3.5.m5.1.1.3.2.cmml"><mi id="S2.SS2.p3.5.m5.1.1.3.2.2" xref="S2.SS2.p3.5.m5.1.1.3.2.2.cmml">U</mi><mi id="S2.SS2.p3.5.m5.1.1.3.2.3" xref="S2.SS2.p3.5.m5.1.1.3.2.3.cmml">Vega</mi></msub><mo id="S2.SS2.p3.5.m5.1.1.3.1" xref="S2.SS2.p3.5.m5.1.1.3.1.cmml">+</mo><mn id="S2.SS2.p3.5.m5.1.1.3.3" xref="S2.SS2.p3.5.m5.1.1.3.3.cmml">0.66</mn></mrow></mrow></math>, <math><mrow id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml"><msub id="S2.SS2.p3.6.m6.1.1.2" xref="S2.SS2.p3.6.m6.1.1.2.cmml"><mi id="S2.SS2.p3.6.m6.1.1.2.2" xref="S2.SS2.p3.6.m6.1.1.2.2.cmml">R</mi><mi id="S2.SS2.p3.6.m6.1.1.2.3" xref="S2.SS2.p3.6.m6.1.1.2.3.cmml">AB</mi></msub><mo id="S2.SS2.p3.6.m6.1.1.1" xref="S2.SS2.p3.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS2.p3.6.m6.1.1.3" xref="S2.SS2.p3.6.m6.1.1.3.cmml"><msub id="S2.SS2.p3.6.m6.1.1.3.2" xref="S2.SS2.p3.6.m6.1.1.3.2.cmml"><mi id="S2.SS2.p3.6.m6.1.1.3.2.2" xref="S2.SS2.p3.6.m6.1.1.3.2.2.cmml">R</mi><mi id="S2.SS2.p3.6.m6.1.1.3.2.3" xref="S2.SS2.p3.6.m6.1.1.3.2.3.cmml">Vega</mi></msub><mo id="S2.SS2.p3.6.m6.1.1.3.1" xref="S2.SS2.p3.6.m6.1.1.3.1.cmml">+</mo><mn id="S2.SS2.p3.6.m6.1.1.3.3" xref="S2.SS2.p3.6.m6.1.1.3.3.cmml">0.18</mn></mrow></mrow></math>, <math><mrow id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml"><msub id="S3.SS1.p2.7.m7.1.1.2" xref="S3.SS1.p2.7.m7.1.1.2.cmml"><mi id="S3.SS1.p2.7.m7.1.1.2.2" xref="S3.SS1.p2.7.m7.1.1.2.2.cmml">U</mi><mi id="S3.SS1.p2.7.m7.1.1.2.3" xref="S3.SS1.p2.7.m7.1.1.2.3.cmml">FORS1</mi></msub><mo id="S3.SS1.p2.7.m7.1.1.1" xref="S3.SS1.p2.7.m7.1.1.1.cmml">=</mo><mrow id="S3.SS1.p2.7.m7.1.1.3" xref="S3.SS1.p2.7.m7.1.1.3.cmml"><msub id="S3.SS1.p2.7.m7.1.1.3.2" xref="S3.SS1.p2.7.m7.1.1.3.2.cmml"><mi id="S3.SS1.p2.7.m7.1.1.3.2.2" xref="S3.SS1.p2.7.m7.1.1.3.2.2.cmml">U</mi><mi id="S3.SS1.p2.7.m7.1.1.3.2.3" xref="S3.SS1.p2.7.m7.1.1.3.2.3.cmml">VIMOS</mi></msub><mo id="S3.SS1.p2.7.m7.1.1.3.1" xref="S3.SS1.p2.7.m7.1.1.3.1.cmml">+</mo><mrow id="S3.SS1.p2.7.m7.1.1.3.3" xref="S3.SS1.p2.7.m7.1.1.3.3.cmml"><mpadded width="+3.3pt" id="S3.SS1.p2.7.m7.1.1.3.3.2" xref="S3.SS1.p2.7.m7.1.1.3.3.2.cmml"><mn id="S3.SS1.p2.7.m7.1.1.3.3.2a" xref="S3.SS1.p2.7.m7.1.1.3.3.2.cmml">0.04</mn></mpadded><mo id="S3.SS1.p2.7.m7.1.1.3.3.1" xref="S3.SS1.p2.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.7.m7.1.1.3.3.3" xref="S3.SS1.p2.7.m7.1.1.3.3.3.cmml">mag</mi></mrow></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/math/0406328

Formulas:

1-

a_{1}, \dots, a_{N}

2-

\frac{1}{(1 - a_{1} z) (1 - a_{2} z) \dots (1 - a_{N} z)} = \sum_{k = 1}^{N} \frac{\prod_{j = 1, j \neq k}^{N} {(1 - a_{j} / a_{k})}^{- 1}}{1 - a_{k} z} .

3-

h_{τ} (a_{1}, \dots, a_{N})

4-

a_{1}, \dots, a_{N}

5-

h_{0} (a_{1}, \dots, a_{N}) = 1

6-

h_{τ} (a_{1}, \dots, a_{N}) := \sum_{1 \leq i_{1} \leq i_{2} \leq \dots \leq i_{τ} \leq N} a_{i_{1}} a_{i_{2}} \dots a_{i_{τ}} = \sum_{k = 1}^{N} \prod_{j = 1, j \neq k}^{N} {(1 - a_{j} / a_{k})}^{- 1} a_{k}^{τ} .

7-

a_{k} = \frac{a - b q^{k + i - 1}}{c - z q^{k + i - 1}}

8-

N = n - i + 1

9-

h_{τ} (\frac{a - b q^{i}}{c - z q^{i}}, \frac{a - b q^{i + 1}}{c - z q^{i + 1}}, \dots, \frac{a - b q^{n}}{c - z q^{n}}) = \frac{c^{n - i + 1} {(z q^{i} / c)}_{n - i + 1}}{{(q)}_{n - i + 1} {(a z - b c)}^{n - i}}

10-

\cdot \sum_{k = i}^{n} {(- 1)}^{k - i} [\binom{n - i + 1}{n - k}] q^{(\binom{k - i + 1}{2}) - k (n - i)} \frac{(1 - q^{k - i + 1}) {(a - b q^{k})}^{τ + n - i}}{{(c - z q^{k})}^{τ + 1}},

Formulas (html):

Correct Categorie: math

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/1612.02550

Formulas:

1-

s_{12} = \sqrt{m_{d} / m_{s}}

2-

| V_{c d} | = | V_{u s} | | V_{t b} |

3-

| V_{c s} | = | V_{u d} | | V_{t b} |

4-

| V_{t s} | = | V_{u d} | | V_{c b} |

5-

s_{12} ≃ {(m_{d} / m_{s})}^{1 / 2}

6-

- ℒ_{Y} = \bar{Q_{L}} Y_{u} \tilde{H} U_{R} + \bar{Q_{L}} Y_{d} H D_{R} + h . c .,

7-

\tilde{H} \equiv i σ_{2} H^{*}

8-

M_{q}^{'} = Y_{q} ⟨ H ⟩

9-

⟨ H ⟩ \approx 174 GeV

10-

V_{u}^{†} M_{u}^{'} V_{u} = d i a g {m_{u}, m_{c}, m_{t}}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1511.00706

Formulas:

1-

θ_{o b s}

2-

E_{γ} \propto θ_{o b s}^{- s}

3-

θ_{o b s} / θ_{0}

4-

θ_{o b s} / θ_{0}

5-

θ = θ_{o b s} / θ_{0}

6-

P (θ, r)

7-

F (θ, r)

8-

F_{m i n}

9-

r_{m a x} (θ)

10-

F (θ, r_{m a x}) = F_{m i n}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9909456

Formulas:

1-

U B V R I

2-

q = p \cos 2 θ

3-

u = p \sin 2 θ

4-

d q = d u = d p

5-

28 \overset{\circ}{.} 65 (d p / p)

6-

p (λ) = p_{m a x} \exp [- 1.7 λ_{m a x} \ln^{2} (λ_{m a x} / λ)]

7-

U B V R I

8-

λ_{m a x}

9-

U B V R I

10-

U B V R I

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9907030

Formulas:

1-

\begin{matrix} d s^{2} = \frac{Δ_{r}}{ρ^{2}} {[d χ - \frac{a}{Ξ} \sin^{2} ϑ d ψ]}^{2} - \frac{ρ^{2}}{Δ_{r}} d r^{2} \\ - \frac{ρ^{2}}{Δ_{ϑ}} d ϑ^{2} - \frac{\sin^{2} ϑ Δ_{ϑ}}{ρ^{2}} {[a d χ - \frac{(r^{2} + a^{2})}{Ξ} d ψ]}^{2} \end{matrix}

2-

\begin{matrix} ρ^{2} & = r^{2} + a^{2} \cos^{2} ϑ \\ Δ_{r} & = - \frac{Λ}{3} r^{4} + (1 - \frac{a^{2} Λ}{3}) r^{2} - 2 M r + a^{2} \\ Δ_{ϑ} & = 1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ \\ Ξ & = 1 + \frac{a^{2} Λ}{3} \end{matrix}

3-

= β φ + γ t

4-

= {(1 + \frac{a^{2} Λ}{3})}^{2} t

5-

= \sqrt{1 + \frac{a^{2} Λ}{3}} φ + \frac{a Λ}{3} (1 + \frac{a^{2} Λ}{3}) t

6-

\begin{matrix} d s^{2} = \frac{(1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ) [(1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ) Δ_{r} - a^{2} \sin^{2} ϑ {(1 - \frac{Λ r^{2}}{3})}^{2}]}{r^{2} + a^{2} \cos^{2} ϑ} d t^{2} \\ - \frac{r^{2} + a^{2} \cos^{2} ϑ}{Δ_{r}} d r^{2} - \frac{r^{2} + a^{2} \cos^{2} ϑ}{1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ} d ϑ^{2} + \frac{4 M r a \sin^{2} ϑ (1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ)}{\sqrt{1 + \frac{a^{2} Λ}{3}} (r^{2} + a^{2} \cos^{2} ϑ)} d t d φ \\ - \frac{\sin^{2} ϑ [(1 + \frac{a^{2} Λ}{3} \cos^{2} ϑ) {(r^{2} + a^{2})}^{2} - a^{2} \sin^{2} ϑ Δ_{r}]}{(1 + \frac{a^{2} Λ}{3}) (r^{2} + a^{2} \cos^{2} ϑ)} d φ^{2} \end{matrix}

7-

1 + \frac{a^{2} Λ}{3} > 0

8-

(M^{2} Λ / 3, a^{2} / M^{2})

9-

Δ_{r} = Δ_{r}^{'} = 0

10-

x = M^{2} Λ / 3, y = a^{2} / M^{2}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1802.08456

Formulas:

1-

[10^{12.5}, 10^{14.2}] M_{⊙}

2-

z 1 = [0.01, 0.1]

3-

z 2 = [0.1, 0.2]

4-

z 3 = [0.2, 0.3]

5-

\log M_{star}^{limit} =

6-

[10^{12}, 10^{12.5}]

7-

m h 1 = [10^{12}, 10^{12.5}] M_{⊙}

8-

m h 2 = [10^{12.5}, 10^{13}] M_{⊙}

9-

m h 3 = [10^{13}, 10^{13.5}] M_{⊙}

10-

m h 4 = [10^{13.5}, 10^{14}] M_{⊙}

Formulas (html):

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id="S3.F3.13.m6.2.2.4.4" xref="S3.F3.13.m6.2.2.4.4.cmml">3</mn></mrow><mo mathvariant="normal" id="S3.F3.13.m6.2.2.3" xref="S3.F3.13.m6.2.2.3.cmml">=</mo><mrow id="S3.F3.13.m6.2.2.2" xref="S3.F3.13.m6.2.2.2.cmml"><mrow id="S3.F3.13.m6.2.2.2.2.2" xref="S3.F3.13.m6.2.2.2.2.3.cmml"><mo mathvariant="normal" stretchy="false" id="S3.F3.13.m6.2.2.2.2.2.3" xref="S3.F3.13.m6.2.2.2.2.3.cmml">[</mo><msup id="S3.F3.13.m6.1.1.1.1.1.1" xref="S3.F3.13.m6.1.1.1.1.1.1.cmml"><mn mathvariant="normal" id="S3.F3.13.m6.1.1.1.1.1.1.2" xref="S3.F3.13.m6.1.1.1.1.1.1.2.cmml">10</mn><mn mathvariant="normal" id="S3.F3.13.m6.1.1.1.1.1.1.3" xref="S3.F3.13.m6.1.1.1.1.1.1.3.cmml">13</mn></msup><mo mathvariant="normal" id="S3.F3.13.m6.2.2.2.2.2.4" xref="S3.F3.13.m6.2.2.2.2.3.cmml">,</mo><msup id="S3.F3.13.m6.2.2.2.2.2.2" xref="S3.F3.13.m6.2.2.2.2.2.2.cmml"><mn mathvariant="normal" id="S3.F3.13.m6.2.2.2.2.2.2.2" xref="S3.F3.13.m6.2.2.2.2.2.2.2.cmml">10</mn><mn mathvariant="normal" id="S3.F3.13.m6.2.2.2.2.2.2.3" xref="S3.F3.13.m6.2.2.2.2.2.2.3.cmml">13.5</mn></msup><mo mathvariant="normal" stretchy="false" id="S3.F3.13.m6.2.2.2.2.2.5" xref="S3.F3.13.m6.2.2.2.2.3.cmml">]</mo></mrow><mo mathvariant="bold-sans-serif" id="S3.F3.13.m6.2.2.2.3" xref="S3.F3.13.m6.2.2.2.3.cmml">⁢</mo><msub id="S3.F3.13.m6.2.2.2.4" xref="S3.F3.13.m6.2.2.2.4.cmml"><mi id="S3.F3.13.m6.2.2.2.4.2" xref="S3.F3.13.m6.2.2.2.4.2.cmml">M</mi><mo mathvariant="normal" id="S3.F3.13.m6.2.2.2.4.3" xref="S3.F3.13.m6.2.2.2.4.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S3.F3.14.m7.2.2" xref="S3.F3.14.m7.2.2.cmml"><mrow id="S3.F3.14.m7.2.2.4" xref="S3.F3.14.m7.2.2.4.cmml"><mi id="S3.F3.14.m7.2.2.4.2" xref="S3.F3.14.m7.2.2.4.2.cmml">m</mi><mo mathvariant="bold-sans-serif" id="S3.F3.14.m7.2.2.4.1" xref="S3.F3.14.m7.2.2.4.1.cmml">⁢</mo><mi id="S3.F3.14.m7.2.2.4.3" xref="S3.F3.14.m7.2.2.4.3.cmml">h</mi><mo mathvariant="bold-sans-serif" id="S3.F3.14.m7.2.2.4.1b" xref="S3.F3.14.m7.2.2.4.1.cmml">⁢</mo><mn mathvariant="normal" id="S3.F3.14.m7.2.2.4.4" xref="S3.F3.14.m7.2.2.4.4.cmml">4</mn></mrow><mo mathvariant="normal" id="S3.F3.14.m7.2.2.3" xref="S3.F3.14.m7.2.2.3.cmml">=</mo><mrow id="S3.F3.14.m7.2.2.2" xref="S3.F3.14.m7.2.2.2.cmml"><mrow id="S3.F3.14.m7.2.2.2.2.2" xref="S3.F3.14.m7.2.2.2.2.3.cmml"><mo mathvariant="normal" stretchy="false" id="S3.F3.14.m7.2.2.2.2.2.3" xref="S3.F3.14.m7.2.2.2.2.3.cmml">[</mo><msup id="S3.F3.14.m7.1.1.1.1.1.1" xref="S3.F3.14.m7.1.1.1.1.1.1.cmml"><mn mathvariant="normal" id="S3.F3.14.m7.1.1.1.1.1.1.2" xref="S3.F3.14.m7.1.1.1.1.1.1.2.cmml">10</mn><mn mathvariant="normal" id="S3.F3.14.m7.1.1.1.1.1.1.3" xref="S3.F3.14.m7.1.1.1.1.1.1.3.cmml">13.5</mn></msup><mo mathvariant="normal" id="S3.F3.14.m7.2.2.2.2.2.4" xref="S3.F3.14.m7.2.2.2.2.3.cmml">,</mo><msup id="S3.F3.14.m7.2.2.2.2.2.2" xref="S3.F3.14.m7.2.2.2.2.2.2.cmml"><mn mathvariant="normal" id="S3.F3.14.m7.2.2.2.2.2.2.2" xref="S3.F3.14.m7.2.2.2.2.2.2.2.cmml">10</mn><mn mathvariant="normal" id="S3.F3.14.m7.2.2.2.2.2.2.3" xref="S3.F3.14.m7.2.2.2.2.2.2.3.cmml">14</mn></msup><mo mathvariant="normal" stretchy="false" id="S3.F3.14.m7.2.2.2.2.2.5" xref="S3.F3.14.m7.2.2.2.2.3.cmml">]</mo></mrow><mo mathvariant="bold-sans-serif" id="S3.F3.14.m7.2.2.2.3" xref="S3.F3.14.m7.2.2.2.3.cmml">⁢</mo><msub id="S3.F3.14.m7.2.2.2.4" xref="S3.F3.14.m7.2.2.2.4.cmml"><mi id="S3.F3.14.m7.2.2.2.4.2" xref="S3.F3.14.m7.2.2.2.4.2.cmml">M</mi><mo mathvariant="normal" id="S3.F3.14.m7.2.2.2.4.3" xref="S3.F3.14.m7.2.2.2.4.3.cmml">⊙</mo></msub></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1105.4898

Formulas:

1-

(\partial_{t} + 𝐯 \cdot \nabla) ϕ (𝐫, t) = M \nabla^{2} \frac{δ F}{δ ϕ (𝐫, t)} + θ (𝐫, t)

2-

v_{i} (𝐫, t)

3-

\int d^{2} r^{'} T_{i j} (𝐫 - 𝐫^{'}) [\frac{δ F}{δ ϕ (𝐫^{'}, t)} \nabla_{j}^{'} ϕ (𝐫^{'}, t) + ζ_{j} (𝐫^{'}, t)] .

4-

T_{i j} (𝐪) = \int d^{2} r T_{i j} (𝐫) e^{- i 𝐪 \cdot 𝐫} = \frac{1}{η_{m} (q^{2} + q / L_{s d})} (δ_{i j} - \frac{q_{i} q_{j}}{q^{2}})

5-

ϕ_{𝐪} = \int_{0}^{ℒ} \int_{0}^{ℒ} d^{2} r ϕ (𝐫) e^{- i 𝐪 \cdot 𝐫}

6-

\partial_{t} ϕ_{𝐪} (t) + {𝐯 \cdot \nabla ϕ (𝐫, t)}_{𝐪} = - M q^{2} {\frac{δ F}{δ ϕ (𝐫, t)}}_{𝐪} + θ_{𝐪}

7-

v_{𝐪, i} (t) = T_{i j} (𝐪) {\frac{δ F}{δ ϕ (𝐫, t)} \nabla_{j} ϕ (𝐫, t) + ζ_{j}}_{𝐪}

8-

⟨ θ_{𝐪} (t) θ_{𝐪^{'}}^{*} (t^{'}) ⟩ = 2 k_{B} T M q^{2} ℒ^{2} δ_{𝐪, 𝐪^{'}} δ (t - t^{'})

9-

⟨ ζ_{𝐪, i} (t) ζ_{𝐪^{'}, j}^{*} (t^{'}) ⟩ = 2 k_{B} T ℒ^{2} η_{m} (q^{2} + q / L_{s d}) δ_{i j} δ_{𝐪, 𝐪^{'}} δ (t - t^{'}) .

10-

{f (𝐫)}_{𝐪}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0802.3717

Formulas:

1-

ε_{F} τ / ℏ ≫ 1

2-

ε_{F} τ / ℏ ≫ 1

3-

\frac{d \hat{ρ}}{d t} + \frac{i}{ℏ} [\hat{H}, \hat{ρ}] = 0 .

4-

| u_{𝒌 s} ⟩

5-

ρ_{s s^{'}} (𝒌, 𝒌^{'}) \equiv ρ (𝒌, 𝒌^{'})

6-

H_{v} \equiv H_{v} (𝒌)

7-

H_{v} = \frac{ℏ^{2}}{2 m} [γ_{1} k^{2} + \bar{γ} (\frac{5}{2} k^{2} - 2 {(𝒌 \cdot 𝑺)}^{2})],

8-

H_{Mn} (𝒓) = \sum_{I} [𝒰 (𝒓 - 𝑹_{I}) 𝟙 + 𝒱 (𝕣 - ℝ_{𝕀}) 𝕤_{𝕀} \cdot 𝕊],

9-

𝒰 (𝒓 - 𝑹_{I}) = U δ (𝒓 - 𝑹_{I})

10-

𝒱 (𝒓 - 𝑹_{I}) = (J_{p d} / V) δ (𝒓 - 𝑹_{I})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1510.03627

Formulas:

1-

\frac{d^{3} N}{p_{T} d p_{T} d ϕ d y} \propto 1 + 2 \sum_{n} v_{n} \cos (n ϕ) .

2-

R_{AA} (p_{T}) = \frac{1}{N_{coll}} \frac{d N_{A A} / d p_{T}}{d N_{p p} / d p_{T}} .

3-

m_{c} \sim 1.3 - 1.5

4-

m_{b} \sim 4.5 - 5.1

5-

τ_{f} = 1 / Q_{HQ} = 1 / \sqrt{Q_{0}^{2} + m_{HQ}^{2}}

6-

τ_{QGP} \sim 0.3 - 1.0

7-

\frac{d}{d t} f_{HQ} (t, \vec{x}, \vec{p}) =

8-

𝒞 [f_{HQ}]

9-

with 𝒞 [f_{H Q}] =

10-

\int d \vec{k} [\underset{gain term}{\underset{⏟}{w (\vec{p} + \vec{k}, \vec{k}) f_{HQ} (\vec{p} + \vec{k})}} - \underset{loss term}{\underset{⏟}{w (\vec{p}, \vec{k}) f_{HQ} (\vec{p})}}],

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: hep-ex

Result: incorrect

Paper: https://arxiv.org/abs/1311.2167

Formulas:

1-

λ_{1} < λ_{2} < λ_{3}

2-

Δ x_{l} = 0.00125

3-

Δ x_{r} = 0.01

4-

Δ x_{i} = x_{i + \frac{1}{2}} - x_{i - \frac{1}{2}}

5-

{\hat{e}}_{t} + {(p u)}_{ξ}

6-

\hat{e} = \frac{1}{2} u^{2} + e

7-

\frac{d}{d t} (*) = \frac{\partial}{\partial t} (*) + u \nabla (*)

8-

d ξ = ρ d x

9-

C = {(γ p ρ)}^{\frac{1}{2}}

10-

x_{j + \frac{1}{2}}^{n + 1}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0806.2318

Formulas:

1-

< 0.001 - 0.003 M_{⊙}

2-

8 \overset{''}{.} 1

3-

1.55 " \times 1.0 "

4-

0 \overset{''}{.} 53

5-

≲ 300 A U

6-

M_{dust} = \frac{S_{ν} D^{2}}{B_{ν} (T_{D}) κ_{ν}}

7-

κ_{ν} \propto λ^{- β}

8-

M_{K} = 3.8 \dots 6.9

9-

T_{eff} = 2700 \dots 3200

10-

M = 0.03 \dots 0.16 M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1106.2867

Formulas:

1-

H_{N} (T) - H_{c} = A T^{ϕ},

2-

H_{N} (T)

3-

H {(T)}_{N}

4-

H {(T)}_{N}

5-

T_{\min} \leq T \leq T_{\max}

6-

H_{N} (T)

7-

(0, 0, 1)

8-

H - H_{c} < 1

9-

H {(T)}_{N}

10-

H_{N} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1605.08150

Formulas:

1-

N o n l i n e a r F u n c t i o n * G a u s s i a n

2-

\begin{matrix} E [{| {\hat{x}}_{T} (ω) |}^{2}] = E [\frac{1}{T} \int_{T}^{0} x^{*} (t) e^{i ω t} 𝑑 t \int_{T}^{0} x (t^{'}) e^{- i ω t^{'}} 𝑑 t^{'}] \\ = \frac{1}{T} \int_{T}^{0} \int_{T}^{0} E [x^{*} (t) x (t^{'}) e^{i ω (t - t^{'})} d t d t^{'}] \end{matrix}

3-

x_{i} = A s_{i} + μ + η_{i}

4-

η_i \sim N (0, σ^{2} \sum i)

5-

\hat{x} = W x

6-

μ_{M L} = < x_{i} >

7-

x_{i} = A (s_{i} + W_{μ M L}) + \tilde{η_{i}}

8-

\tilde{η_{i}} = η_{i} + {(I - A W)}_{μ M L}

9-

{\hat{A}}_{M L} = U_{q} {(△_{q} - σ^{2} I_{q})}^{1 / 2}

10-

{\hat{S}}_{M L} = {({\hat{A}}^{t} \hat{A})}^{- 1} {\hat{A}}^{t} x

Formulas (html):

Correct Categorie: cs

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0705.3411

Formulas:

1-

d_{i} \equiv c / ω_{p i}

2-

\frac{\partial 𝐕_{i}}{\partial t} - \frac{e}{m_{i}} 𝐄 - 𝐕_{i} \times (𝛀_{i} + \frac{c}{m_{i} c} 𝐁) = - \nabla (\frac{𝐕^{2}}{2} + P_{i})

3-

\frac{\partial}{\partial t} (𝐕_{i} + \frac{e}{m_{i} c} 𝐀) - 𝐕_{i} \times (𝛀_{i} + \frac{e}{m_{i} c} 𝐁) = - \nabla (\frac{𝐕_{i}^{2}}{2} + P_{i})

4-

𝐄 \equiv - \frac{1}{c} \frac{\partial 𝐀}{\partial t}, 𝛀_{i} \equiv \nabla \times 𝐯_{i},

5-

\nabla P_{e, i} \equiv \frac{1}{n_{e, i}} \nabla p_{e, i} .

6-

\frac{\partial}{\partial t} (𝛀_{i} + \frac{e}{m_{i} c} 𝐁) - \nabla \times [𝐕_{i} \times (𝛀_{i} + \frac{e}{m_{i} c} 𝐁)] = 𝟎

7-

\frac{\partial 𝐕_{e}}{\partial t} + \frac{e}{m_{e}} 𝐄 - 𝐕_{e} \times (\nabla \times 𝐕_{e} - \frac{e}{m_{e} c} 𝐁) = - \nabla (\frac{𝐕_{e}^{2}}{2} + P_{e}) .

8-

n_{e} \approx n_{i} = n

9-

𝐉 = n e (𝐕_{i} - 𝐕_{e})

10-

\frac{\partial}{\partial t} [(𝐕_{i} - \frac{𝐉}{n e}) - \frac{e}{m_{e} c} 𝐀] - (𝐕_{i} - \frac{𝐉}{n e}) \times [\nabla \times (𝐕_{i} - \frac{𝐉}{n e}) - \frac{e}{m_{e} c} 𝐁] = - \nabla (\frac{𝐕_{e}^{2}}{2} + P_{e})

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

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

Formulas:

1-

B_{s} \bar{B_{s}}

2-

f_{B_{s}} \sqrt{M_{B_{s}}} / (f_{B} \sqrt{M_{B}})

3-

n_{f} = 2 + 1

4-

f_{B_{s}} \sqrt{M_{B_{s}}} / (f_{B} \sqrt{M_{B}})

5-

m_{l} \geq m_{s} / 2

6-

m_{l} \geq m_{s} / 2

7-

f_{B} = 0.216 (9) (19) (4) (6) GeV .

8-

f_{B_{s}} / f_{B} = 1.20 (3) (1),

9-

f_{B_{s}} / f_{B} = 1.13 (3) (\binom{+ 13}{- 2})

10-

f_{B_{s}} = 0.260 (7) (26) (9) GeV .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1311.1511

Formulas:

1-

S U {(2)}_{L}

2-

Z \to \bar{ν} ν

3-

ℒ_{5} = \frac{2 C_{1}^{(5)}}{Λ} {| H |}^{2} \bar{χ} χ + \frac{2 C_{2}^{(5)}}{Λ} {| H |}^{2} \bar{χ} γ_{5} χ .

4-

H = \frac{1}{\sqrt{2}} (\begin{matrix} 0 \\ v + h (x) \end{matrix}) .

5-

ℒ_{5} = \frac{C_{1}^{(5)}}{Λ} h^{2} \bar{χ} χ + \frac{C_{2}^{(5)}}{Λ} h^{2} \bar{χ} γ_{5} χ .

6-

ℒ_{6} = \frac{C_{1}^{(6)}}{Λ^{2}} H^{†} {\overset{\leftrightarrow}{D}}_{μ} H \bar{χ} γ^{μ} χ + \frac{C_{2}^{(6)}}{Λ^{2}} H^{†} {\overset{\leftrightarrow}{D}}_{μ} H \bar{χ} γ^{μ} γ_{5} χ,

7-

D_{μ} = \partial_{μ} - i (g / 2) σ^{a} W_{μ}^{a} - i (g^{'} / 2) B_{μ}

8-

ℒ_{6} = \frac{i C_{1}^{(6)} m_{Z}}{Λ^{2}} h Z_{μ} \bar{χ} γ^{μ} χ + \frac{i C_{2}^{(6)} m_{Z}}{Λ^{2}} h Z_{μ} \bar{χ} γ^{μ} γ_{5} χ,

9-

Z_{μ} = {(g^{2} + g^{', 2})}^{- 1 / 2} (g W_{μ}^{3} - g^{'} B_{μ})

10-

2 m_{Z} = v \sqrt{g^{2} + g^{', 2}}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1105.5431

Formulas:

1-

{(A - Z + ϵ Z)}^{2}

2-

ϵ = 1 - 4 \sin^{2} θ_{W} \sim 0.08

3-

f_{n} / f_{p} ≃ - 0.7

4-

f_{n} / f_{p} ≃ - 0.7

5-

U {(1)}_{ϕ}

6-

ℒ = \partial_{μ} ϕ^{*} \partial^{μ} ϕ - (M_{ϕ}^{2} - d_{H} \frac{v_{E W}^{2}}{2}) ϕ^{*} ϕ - d_{H} H^{†} H ϕ^{*} ϕ + \frac{d_{B}}{Λ^{2}} e J_{μ} \partial_{ν} F^{μ ν},

7-

H = (π^{+}, \frac{1}{\sqrt{2}} (v_{E W} + h + i π^{0}))

8-

J_{μ} = i ϕ^{*} \overset{\leftrightarrow}{\partial_{μ}} ϕ

9-

v_{E W} ≃ 246

10-

σ_{A} = \frac{μ_{A}^{2}}{4 π} {| Z f_{p} + (A - Z) f_{n} |}^{2},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0103008

Formulas:

1-

ℋ = g_{e} μ_{B} B_{0} S_{z} + g_{n} μ_{n} B_{0} I_{z} - A \vec{S} \vec{I},

2-

S = I = 1 / 2

3-

A / 2 π ℏ \approx 3.5

4-

f = f_{e 0} = g_{e} μ_{B} B_{0} / 2 π ℏ + A / 4 π ℏ .

5-

f = f_{e 1} = g_{e} μ_{B} B_{0} / 2 π ℏ - A / 4 π ℏ .

6-

A / 2 π ℏ

7-

Δ f_{e} = (g_{e} μ_{B} / 2 π ℏ) a \frac{\partial B_{0}}{\partial x} .

8-

Δ f_{n} = (g_{n} μ_{n} / 2 π ℏ) a \frac{\partial B_{0}}{\partial x} .

9-

\partial B_{0} / \partial x = 10^{5}

10-

Δ f_{n} \approx 6.75

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0207523

Formulas:

1-

L_{X} \sim 10^{43} - 10^{45} e r g / s

2-

T \sim 10^{8} K

3-

n_{e} \sim 10^{- 3} c m^{- 3}

4-

0.3 - 0.4 μ G

5-

Ω_{Λ} = 1 - Ω_{m} = 0.7

6-

70 km / s / Mpc

7-

\frac{d n (M, t)}{d M} = \sqrt{\frac{2}{π}} \frac{ϱ}{M^{2}} \frac{δ_{c} (t)}{σ (M)} | \frac{d l n σ (M)}{d l n M} | e x p [- \frac{δ_{c}^{2} (t)}{2 σ^{2} (M)}] .

8-

R (M, M^{'}, t) d M^{'} =

9-

\sqrt{\frac{2}{π}} | \frac{d δ_{c} (t)}{d t} | \frac{1}{σ^{2} (M^{'})} | \frac{d σ (M^{'})}{d M^{'}} | {(1 - \frac{σ^{2} (M^{'})}{σ^{2} (M)})}^{- 3 / 2}

10-

\exp [- \frac{δ_{c}^{2} (t)}{2} (\frac{1}{σ^{2} (M^{'})} - \frac{1}{σ^{2} (M)})] {dM}^{'},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1012.4812

Formulas:

1-

U_{s w i m}

2-

U_{s w i m}

3-

(R e ≪ 1)

4-

\nabla p = μ \nabla^{2} 𝐮, \nabla \cdot 𝐮 = 0

5-

U_{s w i m}

6-

k = \frac{2 π}{λ}

7-

r_{s} = c + b \sin (k z - ω t), z_{s} = z + a \cos (k z - ω t + ϕ)

8-

U_{s w i m}

9-

{u_{r} |}_{r = c} = - b ω \cos (k z - ω t), {u_{z} |}_{r = c} = a ω \sin (k z - ω t + ϕ)

10-

{u_{r} |}_{r = R} = 0, {u_{z} |}_{r = R} = U_{s w i m}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1409.3311

Formulas:

1-

u_{j}^{t} \in {0, 1}

2-

j = 1, 2, \dots, N

3-

k = 1, \dots, K

4-

1 < ϵ_{1} < ϵ_{2} < \dots < ϵ_{K} < N

5-

1 < ι_{1} < ι_{2} < \dots < ι_{K} = N

6-

k = 1, 2, \dots, K

7-

f (k, l)

8-

f (j, j + 1) = \min (u_{j}^{t}, 1 - u_{j + 1}^{t})

9-

j \neq ϵ_{k}, ι_{k}

10-

k = 1, 2, \dots, K

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1908.01846

Formulas:

1-

A [[t]]

2-

H^{*} (A, A)

3-

A [[t]]

4-

m_{t} : A [[t]] \otimes A [[t]] ⟶ A [[t]]

5-

m_{t} (a \otimes b) = a b + c_{1} (a \otimes b) t + c_{2} (a \otimes b) t^{2} + \dots

6-

c_{i} : A \otimes A ⟶ A

7-

m o d t^{2}

8-

m o d t^{n + 1}

9-

m o d t^{n + 2}

10-

H^{3} (A, A)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1702.00997

Formulas:

1-

\log τ = - 2

2-

\log τ = - 2

3-

\log τ = - 2

4-

\log τ = - 2

5-

\log τ = - 2

6-

\log τ = - 2

7-

\log τ = - 2

8-

\log τ = - 2

9-

\log τ = - 2

10-

l o g τ = - 2

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1805.01260

Formulas:

1-

^{2, 8, 10, 11, *}

2-

\dot{x} (t) = 𝐀 x (t) + 𝐁 u (t),

3-

𝒦 = u_{1}, \dots, u_{p}

4-

u (t) = β \times I \times l o g (ω) \times (Δ t)

5-

p = 3.31 \times 10^{- 5}

6-

F (2, 45) = 17.4

7-

p = 2.51 \times 10^{- 6}

8-

F (2, 45) = 25.6

9-

p = 3.77 \times 10^{- 8}

10-

p = 8.76 \times 10^{- 6}

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/1801.04669

Formulas:

1-

H_{lf} (n)

2-

H_{lf} (n)

3-

H_{lf} (n)

4-

H_{pf} (n)

5-

H (n, [a, b])

6-

i = 1, \dots, n

7-

p (s_{i})

8-

x_{i} = x - ε / 2

9-

x_{j} = x + ε / 2

10-

x_{i} = x_{j} = x

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0509083

Formulas:

1-

| F = 1, m_{F} = \pm 1 ⟩

2-

| F = 1, m_{F} = 0 ⟩

3-

𝜻_{\frac{π}{2}} = {(ζ_{+}, ζ_{0}, ζ_{-})}^{⊤} = {(- 1, - i \sqrt{2}, 1)}^{⊤} / 2

4-

𝝍 (𝒓, t) = (\begin{matrix} ψ_{+} (𝒓, t) \\ ψ_{0} (𝒓, t) \\ ψ_{-} (𝒓, t) \end{matrix}) = \sqrt{n (𝒓)} (\begin{matrix} ζ_{+} (t) \\ ζ_{0} (t) \\ ζ_{-} (t) \end{matrix}),

5-

\int d^{3} r n (𝒓) = N

6-

{|| 𝜻 ||}^{2} = 1

7-

i \partial_{t} 𝜻

8-

(H_{Z} + H_{mf}) 𝜻,

9-

H_{LZ} + H_{QZ} = - p F_{z} - q (𝟙 - F_{z}^{2}),

10-

g_{2} ⟨ n ⟩ \sum_{α = x, y, z} F_{α} 𝜻 𝜻^{†} F_{α},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0908.2768

Formulas:

1-

16^{3} \times 32 \times 16

2-

S_{0} [A]

3-

d A e x p (- S_{0}) δ (c - Q [A]) \to d μ_{γ} = d A e x p (- S_{0}) e x p (- γ V Q [A])

4-

{⟨ Q [A] ⟩}_{γ} = c

5-

⟨ c \bar{c} ⟩

6-

u (k^{2})

7-

{⟨ D_{μ} c^{a} {(A_{ν} \times c)}^{b} ⟩}_{k} = (δ_{μ ν} - \frac{k_{μ} k_{ν}}{k^{2}}) δ^{a b} u (k^{2}),

8-

u (0) = - 1

9-

F (k) = \frac{1}{1 + u (k^{2}) + o (k^{2})}

10-

u (0) \sim - 1

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1006.0008

Formulas:

1-

u (𝐱) - α^{2} Δ u (𝐱) = 0

2-

O (N \log N)

3-

u (𝐱) - α^{2} Δ u (𝐱) = f (𝐱),

4-

u (𝐱) - α^{2} Δ u (𝐱) = 0,

5-

O (N \log N)

6-

O (N^{3 / 2})

7-

Γ_{1}, \dots, Γ_{M}

8-

G (𝐲 - 𝐱) = G (y_{1} - x_{1}, y_{2} - x_{2})

9-

1 - α^{2} Δ

10-

G (𝐱) = \frac{1}{2 π α^{2}} K_{0} (\frac{| 𝐱 |}{α}),

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9511062

Formulas:

1-

\sim \sqrt{k T / ϵ_{F}}

2-

δ 𝐩 = \sqrt{2 m Δ (T)}

3-

Δ (T) / v_{F}

4-

α \equiv \frac{δ R}{R} = {\begin{matrix} {(\frac{k T}{ϵ_{F}})}^{2} \frac{l_{0}}{L} & k T ≪ Δ (T) \\ 1.45 \frac{l_{0}}{L} \frac{Δ_{0}}{k T_{c}} {(\frac{Δ_{0}}{ϵ_{F}})}^{2} {(\frac{T_{c} - T}{T_{c}})}^{\frac{3}{2}} & Δ (T) ≪ k T_{c} \\ \sim (\frac{l_{0}}{L}) {(\frac{Δ_{0}}{ϵ_{F}})}^{2} & Δ (T) \sim k T_{c} \end{matrix}

5-

\frac{δ R}{R} = (V (L) / I - R) / R

6-

Δ (T) = 1.8 Δ_{0} {(\frac{T_{c} - T}{T_{c}})}^{1 / 2}

7-

l < λ_{T} < l_{0}

8-

λ_{T} = ℏ v_{F} / k T

9-

T_{𝐩} = 2 ρ^{2} \frac{1 + \cos ϕ}{1 + 2 ρ^{2} \cos ϕ + ρ^{4}}

10-

ϕ = 2 m ϵ l / ℏ (p_{F}^{2} - p^{2}) \sim l / λ_{T}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/math/0112294

Formulas:

1-

| ∥ f (x) - f (y) ∥ - ∥ x - y ∥ | \leq ε

2-

f (0) = 0

3-

f - f (0)

4-

f - f (0)

5-

U + f (0)

6-

f (0) = 0

7-

∥ f (x) - U x ∥ \leq 10 ε

8-

{∥ f (x) - U x ∥ : x \in 𝐑}

9-

U : 𝐑 \to 𝐑^{2}

10-

f (x) = (x, \sqrt{2 ε | x |})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0903.4075

Formulas:

1-

ϕ = π σ^{3} N / 6 V \approx 0.74

2-

Γ = d σ / d t

3-

β P σ^{3}

4-

1 / (β P σ^{3})

5-

β P σ^{3}

6-

1 / (β P σ^{3})

7-

β = 1 / k_{B} T

8-

β P / ρ \propto 1 / (ϕ_{J} - ϕ)

9-

β P σ^{3} = a \frac{ϕ_{J}^{1 / 3} ϕ^{2 / 3}}{{(ϕ_{J} / ϕ)}^{1 / 3} - 1},

10-

ϕ_{r c p} = 0.683

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0202330

Formulas:

1-

- (1 + 1 / d)

2-

1 - s (𝐫) - F

3-

a g d s_{m}^{(d - 1) / d}

4-

1 / s (𝐫)

5-

F = a g d \sum_{m} s_{m}^{(d - 1) / d} = a g d \sum_{m} s_{m}^{(d - 1) / d} \int_{m} \frac{d^{d} r}{s (𝐫)} = a g d \int s {(𝐫)}^{- 1 / d} d^{d} r .

6-

Y = 1 - \int p (𝐫) s (𝐫) d^{d} r - a g d \int s {(𝐫)}^{- 1 / d} d^{d} r,

7-

δ Y / δ s (𝐫) = 0

8-

s (𝐫) = {[\frac{a g}{p (𝐫)}]}^{d / (d + 1)},

9-

Y_{opt} = 1 - (d + 1) {(a g)}^{d / (d + 1)} \int p {(𝐫)}^{1 / (d + 1)} d^{d} r,

10-

n = \int \frac{d^{d} r}{s (𝐫)} = {(a g)}^{- d / (d + 1)} \int p {(𝐫)}^{d / (d + 1)} d^{d} r .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9707033

Formulas:

1-

\int d^{3} x [\frac{1}{4} {(\partial_{i} A_{j} - \partial_{j} A_{i})}^{2} + {| (\partial_{i} + i e_{3} A_{i}) ϕ |}^{2} + m_{3}^{2} ϕ^{*} ϕ + λ_{3} {(ϕ^{*} ϕ)}^{2}] .

2-

(e_{3}, m_{3}, λ_{3})

3-

y \equiv m_{3}^{2} (e_{3}^{2}) / e_{3}^{4}, x \equiv λ_{3} / e_{3}^{2}

4-

y_{c} (x)

5-

β_{G} \sum_{𝐱, i < j} {1 - \cos [α_{i j} (𝐱)]} - β_{H} \sum_{𝐱, i} Re ϕ^{*} (𝐱) U_{i} (𝐱) ϕ (𝐱 + \hat{i})

6-

+ \sum_{𝐱} ϕ^{*} (𝐱) ϕ (𝐱) + β_{R} \sum_{𝐱} {[ϕ^{*} (𝐱) ϕ (𝐱)]}^{2},

7-

β_{G} = 1 / e_{3}^{2} a

8-

α_{i j} (𝐱) = α_{i} (𝐱) + α_{j} (𝐱 + \hat{i}) - α_{i} (𝐱 + \hat{j}) - α_{j} (𝐱)

9-

m_{γ} / e_{3}^{2} = 0.003

10-

ϕ^{*} (𝐱) ϕ (𝐱), O_{i} (𝐱) = Im ϕ^{*} (𝐱) U_{i} (𝐱) ϕ (𝐱 + \hat{i}),

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/0704.3714

Formulas:

1-

F ≃ F_{n} + a (T) {| ϕ |}^{2} + \frac{λ}{4} {| ϕ |}^{4} .

2-

a (T) ≃ \dot{a} (T_{C}) (T - T_{C})

3-

ℒ = {(D_{μ} ϕ)}^{†} (D^{μ} ϕ) + m^{2} ϕ^{†} ϕ - \frac{λ}{4} {(ϕ^{†} ϕ)}^{2} - \frac{1}{4} F_{μ ν} F^{μ ν},

4-

F_{μ ν} \equiv \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

5-

D_{μ} \equiv \partial_{μ} + 2 i e A_{μ}

6-

ϕ \equiv \frac{1}{\sqrt{2}} (η_{0} + η) e^{i θ / η_{0}},

7-

F_{I} = F_{n} + \frac{1}{2} a_{I} (T) η_{0}^{2} + \frac{λ}{16} η_{0}^{4}

8-

a_{I} = - m^{2} + \frac{λ + 4 e^{2}}{16} T^{2} .

9-

T_{1} = 2 \sqrt{\frac{4 m^{2}}{λ + 4 e^{2}}} .

10-

η_{0}^{2} (T) = - \frac{4 a_{I} (T)}{λ} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/1009.5707

Formulas:

1-

{\tilde{W}}_{Λ^{'} Λ}^{[Γ]} (P, k, Δ, N; η) = \frac{1}{2} \int \frac{d^{4} z}{{(2 π)}^{4}} e^{i k \cdot z} ⟨ p^{'}, Λ^{'} | \bar{ψ} (- \frac{1}{2} z) Γ 𝒲 ψ (\frac{1}{2} z) | p, Λ ⟩ .

2-

P = (p^{'} + p) / 2

3-

Δ = p^{'} - p

4-

{𝟙, γ_{5}, γ^{μ}, γ^{μ} γ_{5}, i σ^{μ ν}}

5-

𝒲 \equiv 𝒲 (- \frac{1}{2} z, \frac{1}{2} z | n)

6-

- \frac{1}{2} z + \infty \cdot n

7-

\frac{1}{2} z + \infty \cdot n

8-

N = \frac{M^{2} n}{P \cdot n}

9-

η = sign (n^{0})

10-

\begin{matrix} W_{Λ^{'} Λ}^{[Γ]} (P, x, {\vec{k}}_{⟂}, Δ, N; η) & = \int d k^{-} {\tilde{W}}_{Λ^{'} Λ}^{[Γ]} (P, k, Δ, N; η) \\ = {\frac{1}{2} \int \frac{d z^{-} d^{2} z_{⟂}}{{(2 π)}^{3}} e^{i k \cdot z} ⟨ p^{'}, Λ^{'} | \bar{ψ} (- \frac{1}{2} z) Γ 𝒲 ψ (\frac{1}{2} z) | p, Λ ⟩ |}_{z^{+} = 0}, \end{matrix}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1808.03157

Formulas:

1-

(1 + o_{k} (1)) 2^{- k} N

2-

r (B_{n}^{(k)}) = 2^{k} n + o_{k} (n)

3-

{\sqrt{2}}^{t} \leq r (K_{t}) \leq 4^{t}

4-

r (K_{t})

5-

2^{k} n + o_{k} (n) \leq r (B_{n}^{(k)}) \leq 4^{k} n,

6-

G (n, 1 / 2)

7-

r (B_{n}^{(k)})

8-

r (B_{n}^{(k)}) \leq 2^{k} (n + k - 2) + 2 .

9-

r (B_{n}^{(k)}) = 2^{k} n + o_{k} (n) .

10-

r (K_{t - k}, K_{t})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/2004.00699

Formulas:

1-

\frac{d θ_{j}}{d t} = 1 - \cos θ_{j} + (1 + \cos θ_{j}) (η_{j} + κ I_{j}); j = 1, \dots N; θ_{j} \in S^{1},

2-

I_{j} (t) = \frac{2 π}{N} \sum_{k = 1}^{N} K_{j k} P_{n} (θ_{k} (t))

3-

P_{n} (θ_{k}) = a_{n} {(1 - \cos θ_{k})}^{n}

4-

a_{n} = 2^{n} {(n!)}^{2} / (2 n)!

5-

\int_{0}^{2 π} P_{n} (θ) 𝑑 θ = 2 π

6-

g (η) = \frac{γ / π}{{(η - η_{0})}^{2} + γ^{2}},

7-

K_{j k} = K (2 π | j - k | / N)

8-

K (x) = 0.1 + 0.3 \cos x + B \sin x

9-

z (x, t)

10-

\frac{\partial z}{\partial t} = \frac{(i η_{0} - γ) {(1 + z)}^{2} - i {(1 - z)}^{2}}{2} + κ \frac{i {(1 + z)}^{2}}{2} I

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0209047

Formulas:

1-

Δ A = \frac{1 - R - T}{1 - R} - \frac{1 - R_{0} - T_{0}}{1 - R_{0}}

2-

\frac{d n}{d t} = - C_{2} n^{2},

3-

n (t) = \frac{1}{C_{2} t + 1 / n (0)}, t \geq 0

4-

τ^{- 1} (t) = α n^{β} (t) + τ_{i}^{- 1} .

5-

ϵ_{ω} = ϵ_{\infty} [1 + ϵ_{p h} (ω) - \frac{ω_{p} {(t)}^{2}}{ω (ω - i / τ (t))}] .

6-

{[4 π n (t) e^{2} / m_{e} ϵ_{\infty}]}^{1 / 2}

7-

ϵ_{p h} (ω) = \frac{ω_{L}^{2} - ω_{T}^{2}}{ω_{T}^{2} - ω^{2} + i Γ ω}

8-

r_{ω} (t) = \frac{ϵ_{ω} \cos (π / 4) - \sqrt{ϵ_{ω} - \sin^{2} (π / 4)}}{ϵ_{ω} \cos (π / 4) + \sqrt{ϵ_{ω} - \sin^{2} (π / 4)}} .

9-

r_{ω} r_{ω}^{*}

10-

10^{- 15} - 10^{- 11}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0604180

Formulas:

1-

H (x (t))

2-

x^{μ} (t)

3-

P_{m} (x (t))

4-

H (t) = \sum_{m} ϵ_{m} (t) P_{m} (t)

5-

P_{m} (t) = \sum_{k} | m_{k} (t) ⟩ ⟨ m_{k} (t) |

6-

D (t, t_{0})

7-

R (t, t_{0}) P_{m} (t_{0}) = P_{m} (t) R (t, t_{0}), D (t, t_{0}) = - i R {(t, t_{0})}^{†} \frac{\partial}{\partial t} R (t, t_{0}) .

8-

P_{m} (t) = P_{m} (t_{0})

9-

U (t, t_{0}) \sim \sum_{m} P_{m} (t_{0}) e^{- i \int_{t_{0}}^{t} ϵ_{m} (s) 𝑑 s} U_{ad}^{m} P_{m} (t_{0}), U_{ad}^{m} = 𝐏 \exp {- \oint_{C} A^{m} (x)},

10-

A^{m} (x) = \sum_{μ} A_{μ}^{m} d x^{μ}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9605171

Formulas:

1-

Φ_{0} \equiv h / 2 e

2-

V_{0} \equiv ℏ ⟨ \dot{φ} ⟩ / 2 e

3-

I_{l} (t) = J_{i} \sin (φ_{i}) + \frac{ℏ}{2 e R_{i}} {\dot{φ}}_{i} + \frac{ℏ C_{i}}{2 e} {\ddot{φ}}_{i},

4-

I_{l} (t)

5-

C_{1} = C_{2} \equiv 2 C_{l}

6-

R_{1} = R_{2} \equiv R_{l} / 2

7-

J_{1} = J_{2} \equiv J_{l}

8-

V = V_{1} + V_{2}

9-

V_{i} = (ℏ / 2 e) {\dot{φ}}_{i}

10-

φ_{1} (t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1612.00013

Formulas:

1-

0.4 μ m - 1.0 μ m

2-

0.35 - 2.5 μ m

3-

> 1 μ m

4-

d n / d m \propto m^{- α}

5-

0.08 < M / M_{⊙} < 0.5

6-

0.5 < M / M_{⊙} < 1.0

7-

m_{c} = 0.08 M_{⊙}

8-

0.08 < m_{c} / M_{⊙} < 0.40

9-

0.2 - 1.0 M_{⊙}

10-

0.08 - 0.2 M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9609196

Formulas:

1-

G_{I}^{- 1} = G_{s (A)}^{- 1} + G_{s (B)}^{- 1} - I_{B} H^{i} I_{A} - I_{A} H^{i} I_{B},

2-

G_{s (A)}

3-

G_{s (B)}

4-

- ℐ_{A} H^{i} ℐ_{B}

5-

- ℐ_{B} H^{i} ℐ_{A}

6-

{\bar{E}}_{α, α^{'}} (x) = x E_{α, α^{'}}^{(1)} + (1 - x) E_{α, α^{'}}^{(2)}, α, α^{'} = s, p^{3}, s^{*}

7-

E_{α, α^{'}}^{(1, 2)}

8-

{\bar{E}}_{s, ν} (x, b_{ν}) = {\bar{E}}_{s, ν} (x) + x (1 - x) b_{ν}, ν = a, c

9-

{\bar{E}}_{s, ν} (x)

10-

B_{I h}, B_{I l}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/1009.3555

Formulas:

1-

⟨ a^{†} b ⟩ \neq 0, ⟨ a ⟩ = 0

2-

⟨ a_{i} b_{i}^{†} a_{j}^{†} b_{j} ⟩

3-

⟨ a^{†} b ⟩ \neq 0, ⟨ a ⟩ = 0

4-

⟨ a_{i} a_{j}^{†} ⟩

5-

⟨ a_{i} b_{i}^{†} a_{j}^{†} b_{j} ⟩

6-

⟨ a^{†} b ⟩

7-

⟨ a_{i} b_{i}^{†} a_{j}^{†} b_{j} ⟩

8-

C_{pair} (r)

9-

C_{a} (r)

10-

- \sum_{⟨ i, j ⟩} t_{a} a_{i}^{†} a_{j} - \sum_{⟨ i, j ⟩} t_{b} b_{i}^{†} b_{j} + \sum_{i} U n_{a i} n_{b i},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1101.1490

Formulas:

1-

𝒜 𝒞 (n) = 1

2-

𝒜 𝒞 (n) = 3

3-

𝒜 𝒞 (n) = 3

4-

𝒜 𝒞 (n) \in {3, 4, 5, 6, 7}

5-

{n \in ℕ | 𝒜 𝒞 (n) = 3}

6-

x^{3} - p x^{2} - x + 1

7-

𝒜 = {A, B}

8-

w = w_{0} w_{1} w_{2} \dots w_{n - 1}

9-

{| w |}_{A}, {| w |}_{B}

10-

w = w_{0} w_{1} \dots w_{n - 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1604.00844

Formulas:

1-

\frac{\partial^{2} 𝐏}{\partial t^{2}} + δ_{e} ω_{e} \frac{\partial 𝐏}{\partial t} + ω_{e}^{2} {(1 + \frac{{| 𝐏 |}^{2}}{P_{s}^{2}})}^{- 3 / 2} 𝐏 = ϵ_{0} (ϵ_{s} - 1) ω_{e}^{2} 𝐄

2-

| 𝐏 | / P_{s}

3-

\exp (- i ω t)

4-

ϵ (ω) = 1 + \frac{ϵ_{s} - 1}{1 - i δ_{e} (\frac{ω}{ω_{e}}) - {(\frac{ω}{ω_{e}})}^{2}},

5-

Re [ϵ (ω_{0})] = 0

6-

Re [ϵ (ω_{0})] = 0

7-

ω_{0} = \frac{ω_{e}}{\sqrt{2}} {(ϵ_{s} + 1 - δ_{e}^{2}) + {[{(ϵ_{s} + 1 - δ_{e}^{2})}^{2} - 4 ϵ_{s}]}^{1 / 2}}^{1 / 2} .

8-

L = 1.25 λ_{e}

9-

λ_{e} = 2 π c / ω_{e}

10-

ω_{0} = 1.095 ω_{e}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0704.0146

Formulas:

1-

Γ = \oint d 𝐫 \cdot 𝐯 (𝐫)

2-

ψ (𝐫, t)

3-

n (𝐫, t)

4-

S (𝐫, t)

5-

ψ (𝐫) = \sqrt{n (𝐫, t)} \exp [i S (𝐫, t)]

6-

\int_{C} \nabla S \cdot d 𝐥 = 2 π q,

7-

𝐯 (𝐫, t) = (ℏ / m) \nabla S (𝐫, t)

8-

Γ = \int_{C} 𝐯 \cdot 𝑑 𝐥 = q (\frac{h}{m}) .

9-

𝐯 (r, θ) = q ℏ / (m r) \hat{𝜽}

10-

V = \frac{1}{2} m (ω_{r}^{2} r^{2} + ω_{z}^{2} z^{2})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0109059

Formulas:

1-

ϕ = R \exp (i S)

2-

(\partial^{μ} + i e A^{μ}) (\partial_{μ} + i e A_{μ}) ϕ = - m_{0}^{2} ϕ

3-

\partial^{μ} (R^{2} (\partial_{μ} S + e A^{μ})) = 0

4-

(\partial_{μ} S + e A_{μ}) (\partial^{μ} S + e A^{μ}) - \frac{□ R}{R} = m_{0}^{2}

5-

\sqrt{m_{0}^{2} + \frac{□ R}{R}}

6-

v_{d B}^{k} = \frac{S^{k} + e A^{k}}{- (S^{0} + e A^{0})}

7-

k = 1, 2, 3

8-

S^{μ} = \partial^{μ} S

9-

R^{2} \partial_{0} S,

10-

Φ (x, t) = ϕ_{1} (x) e^{- i ω_{1} t} + ϕ_{2} (x) e^{- i ω_{2} t}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: hep-th

Result: incorrect

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