Run 11369297

Paper: https://arxiv.org/abs/1706.05188

Formulas:

1-

E_{ex} = \frac{J_{eff}}{2} ϕ_{i j}^{2},

2-

ϕ_{i j} = acos (𝐦_{i} \cdot 𝐦_{j})

3-

A_{i j} (1 - 𝐦_{i} \cdot 𝐦_{j}) + B_{i j} [1 - {(𝐦_{i} \cdot 𝐦_{j})}^{2}] .

4-

ℋ = - \frac{1}{2} \sum_{i \neq j} J_{i j} 𝐒_{i} \cdot 𝐒_{j},

5-

μ_{s} = 1.44 μ_{B}

6-

J_{i j} = 5.6 × 10^{- 21} J

7-

T (ϕ) = - \frac{\partial E}{\partial ϕ} = - A \sin ϕ - 2 B \cos ϕ \sin ϕ .

8-

H_{exch} = A / M_{s} V

9-

M_{s} = 6 × 10^{5} {JT}^{- 1} m^{- 3}

10-

A (T) = A_{0} {(1 - \frac{T}{T_{C}})}^{α}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-th/9312025

Formulas:

1-

i = 1, \dots, m

2-

{\bar{D}}_{\pm} U_{i} = 0

3-

p = 1, \dots, n

4-

{\bar{D}}_{+} V_{p} = D_{-} V_{p} = 0

5-

K (U_{i}, {\bar{U}}_{i}, V_{p}, {\bar{V}}_{p})

6-

S = \frac{1}{2 π α^{'}} \int d^{2} x D_{+} D_{-} {\bar{D}}_{+} {\bar{D}}_{-} K (U_{i}, {\bar{U}}_{i}, V_{p}, {\bar{V}}_{p}) .

7-

\begin{matrix} S = & - \frac{1}{2 π α^{'}} \int d^{2} x [K_{u_{i} {\bar{u}}_{j}} \partial^{a} u_{i} \partial_{a} \bar{u}_{j} - K_{v_{p} {\bar{v}}_{q}} \partial^{a} v_{p} \partial_{a} \bar{v}_{q} \\ + ϵ_{a b} (K_{u_{i} {\bar{v}}_{p}} \partial_{a} u_{i} \partial_{b} \bar{v}_{p} + K_{v_{p} {\bar{u}}_{i}} \partial_{a} v_{p} \partial_{b} \bar{u}_{i})], \end{matrix}

8-

K_{u_{i} {\bar{u}}_{j}} = \frac{\partial^{2} K}{\partial U_{i} \partial {\bar{U}}_{j}}

9-

Φ (u_{i}, v_{p})

10-

δ c = c - \frac{3 D}{2}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9911504

Formulas:

1-

d r / v_{r}

2-

M_{b} (r)

3-

M_{b} (r) = \sum_{j} M_{b j} (r)

4-

M_{b j} (r)

5-

M_{b j} (r)

6-

= 0 for r < r_{\min} (E_{j}, J_{j})

7-

M_{b j} (r)

8-

= m_{b j} for r > r_{\max} (E_{j}, J_{j})

9-

r_{\min} < r < r_{\max}

10-

M_{b j} (r) = \frac{m_{b j}}{P_{j}} \int_{r_{\min}}^{r} \frac{d r}{v_{r}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0511112

Formulas:

1-

k T_{i n} \sim 1 - 2 keV

2-

k T_{i n} \sim 0.2 keV

3-

E_{c u t o f f} \sim 3 - 8 keV

4-

\sim 1 - 2.5 keV

5-

\sim 10^{39} erg s^{- 1}

6-

M_{B H} ≃ 10^{2} - 10^{4} M_{⊙}

7-

\sim 1 - 2 keV

8-

\sim 100 - 300 eV

9-

\sim 10^{3} - 10^{4} M_{⊙}

10-

\sim 0.1 L_{E d d}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2002.08698

Formulas:

1-

y = η (x, t)

2-

ω^{2} = g k + \frac{γ (μ)}{ρ} B^{2} k^{2} + \frac{σ}{ρ} k^{3},

3-

γ (μ) = {(μ - 1)}^{2} {(μ_{0} (μ + 1))}^{- 1}

4-

v_{m i n} = {(4 σ g / ρ)}^{1 / 4}

5-

v_{m i n}^{2}

6-

B_{c}^{2} = \frac{2 {(ρ g σ)}^{1 / 2}}{γ (μ)} .

7-

λ_{0} = 2 π {(\frac{σ}{g ρ})}^{1 / 2}, t_{0} = 2 π {(\frac{σ}{g^{3} ρ})}^{1 / 4} .

8-

ρ = 1324 {kg/m}^{3}, σ = 0.059 N/m, μ = 1.69 .

9-

2 \leq B / B_{c} \leq 6

10-

B_{m a x}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1211.5116

Formulas:

1-

P m \bar{3} m

2-

R \bar{3} c

3-

a^{-} a^{-} a^{-}

4-

𝝎_{i} = ω_{0} {(- 1)}^{n_{i x} + n_{i y} + n_{i z}} (1, 1, 1)

5-

𝝎_{i} = ω_{0} (1, 1, 1)

6-

𝐏^{I} = P_{0} (1, 1, 1)

7-

n_{i^{'} β} - n_{i β} = 2 n δ_{α β}

8-

𝝎_{i^{'}} = 𝝎_{i}

9-

𝐏^{II} = P_{0} (1, 1, 1)

10-

𝐏^{II} = P_{0} (- 1, - 1, - 1)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9302012

Formulas:

1-

ϑ \sim > {2.5}^{\circ}

2-

> 20 - 30 h^{- 1} Mpc

3-

𝒫 (k) \propto k T^{2} (k)

4-

L = 260 h^{- 1}

5-

b \equiv σ_{8}^{- 1} = 1.5

6-

3 \times 10^{- 2} h^{3}

7-

Φ (M) \sim dex [- 0.4 (α + 1) M] \exp [- dex 0.4 (M^{*} - M)]

8-

d N / d M = 0

9-

M > M^{*} + 3

10-

M < M^{*} - 2

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1706.02463

Formulas:

1-

H_{0} = 70.2 {km s}^{- 1} {Mpc}^{- 1}

2-

ℳ_{r}^{Petro} \leq - 18.78

3-

M_{vir} = (Δ_{v} / 2) H^{2} (z) r_{vir}^{3} / G

4-

v_{vir} = \sqrt{Δ_{v} / 2} H (z) r_{vir}

5-

[z_{\min} + 20 Δ z, z_{\max} - 20 Δ z]

6-

Δ z = \sqrt{2 / Δ_{v}} (1 + z_{group}) v_{vir} / c

7-

\log (M_{vir} / M_{⊙})

8-

ℳ_{r}^{Petro} \leq - 18.78

9-

d_{outer} (R, Δ z) = \sqrt{\frac{Δ_{v}}{2} {[\frac{c Δ z}{v_{vir} (1 + z_{group})}]}^{2} + {(\frac{R}{r_{vir}})}^{2}} .

10-

P_{M} (R, Δ z)

Formulas (html):

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xref="S2.SS1.p3.5.m5.1.2.3.2.2.3.cmml">2</mn></mrow></msqrt></mpadded><mo id="S2.SS1.p3.5.m5.1.2.3.1" xref="S2.SS1.p3.5.m5.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.5.m5.1.2.3.3" xref="S2.SS1.p3.5.m5.1.2.3.3.cmml">H</mi><mo id="S2.SS1.p3.5.m5.1.2.3.1a" xref="S2.SS1.p3.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.5.m5.1.2.3.4.2" xref="S2.SS1.p3.5.m5.1.2.3.cmml"><mo stretchy="false" id="S2.SS1.p3.5.m5.1.2.3.4.2.1" xref="S2.SS1.p3.5.m5.1.2.3.cmml">(</mo><mi id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml">z</mi><mo rspace="4.2pt" stretchy="false" id="S2.SS1.p3.5.m5.1.2.3.4.2.2" xref="S2.SS1.p3.5.m5.1.2.3.cmml">)</mo></mrow><mo id="S2.SS1.p3.5.m5.1.2.3.1b" xref="S2.SS1.p3.5.m5.1.2.3.1.cmml">⁢</mo><msub id="S2.SS1.p3.5.m5.1.2.3.5" xref="S2.SS1.p3.5.m5.1.2.3.5.cmml"><mi id="S2.SS1.p3.5.m5.1.2.3.5.2" xref="S2.SS1.p3.5.m5.1.2.3.5.2.cmml">r</mi><mi id="S2.SS1.p3.5.m5.1.2.3.5.3" xref="S2.SS1.p3.5.m5.1.2.3.5.3.cmml">vir</mi></msub></mrow></mrow></math>, <math><mrow id="S2.SS1.p4.5.m5.2.2.2" xref="S2.SS1.p4.5.m5.2.2.3.cmml"><mo stretchy="false" id="S2.SS1.p4.5.m5.2.2.2.3" xref="S2.SS1.p4.5.m5.2.2.3.cmml">[</mo><mrow id="S2.SS1.p4.5.m5.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.cmml"><msub id="S2.SS1.p4.5.m5.1.1.1.1.2" xref="S2.SS1.p4.5.m5.1.1.1.1.2.cmml"><mi id="S2.SS1.p4.5.m5.1.1.1.1.2.2" xref="S2.SS1.p4.5.m5.1.1.1.1.2.2.cmml">z</mi><mi id="S2.SS1.p4.5.m5.1.1.1.1.2.3" xref="S2.SS1.p4.5.m5.1.1.1.1.2.3.cmml">min</mi></msub><mo id="S2.SS1.p4.5.m5.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS1.p4.5.m5.1.1.1.1.3" xref="S2.SS1.p4.5.m5.1.1.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.SS1.p4.5.m5.1.1.1.1.3.2" xref="S2.SS1.p4.5.m5.1.1.1.1.3.2.cmml"><mn id="S2.SS1.p4.5.m5.1.1.1.1.3.2a" xref="S2.SS1.p4.5.m5.1.1.1.1.3.2.cmml">20</mn></mpadded><mo id="S2.SS1.p4.5.m5.1.1.1.1.3.1" xref="S2.SS1.p4.5.m5.1.1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.SS1.p4.5.m5.1.1.1.1.3.3" xref="S2.SS1.p4.5.m5.1.1.1.1.3.3.cmml">Δ</mi><mo id="S2.SS1.p4.5.m5.1.1.1.1.3.1a" xref="S2.SS1.p4.5.m5.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p4.5.m5.1.1.1.1.3.4" xref="S2.SS1.p4.5.m5.1.1.1.1.3.4.cmml">z</mi></mrow></mrow><mo id="S2.SS1.p4.5.m5.2.2.2.4" xref="S2.SS1.p4.5.m5.2.2.3.cmml">,</mo><mrow id="S2.SS1.p4.5.m5.2.2.2.2" xref="S2.SS1.p4.5.m5.2.2.2.2.cmml"><msub id="S2.SS1.p4.5.m5.2.2.2.2.2" xref="S2.SS1.p4.5.m5.2.2.2.2.2.cmml"><mi id="S2.SS1.p4.5.m5.2.2.2.2.2.2" xref="S2.SS1.p4.5.m5.2.2.2.2.2.2.cmml">z</mi><mi id="S2.SS1.p4.5.m5.2.2.2.2.2.3" xref="S2.SS1.p4.5.m5.2.2.2.2.2.3.cmml">max</mi></msub><mo id="S2.SS1.p4.5.m5.2.2.2.2.1" xref="S2.SS1.p4.5.m5.2.2.2.2.1.cmml">-</mo><mrow id="S2.SS1.p4.5.m5.2.2.2.2.3" xref="S2.SS1.p4.5.m5.2.2.2.2.3.cmml"><mpadded width="+1.7pt" id="S2.SS1.p4.5.m5.2.2.2.2.3.2" xref="S2.SS1.p4.5.m5.2.2.2.2.3.2.cmml"><mn id="S2.SS1.p4.5.m5.2.2.2.2.3.2a" xref="S2.SS1.p4.5.m5.2.2.2.2.3.2.cmml">20</mn></mpadded><mo id="S2.SS1.p4.5.m5.2.2.2.2.3.1" xref="S2.SS1.p4.5.m5.2.2.2.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.SS1.p4.5.m5.2.2.2.2.3.3" xref="S2.SS1.p4.5.m5.2.2.2.2.3.3.cmml">Δ</mi><mo id="S2.SS1.p4.5.m5.2.2.2.2.3.1a" xref="S2.SS1.p4.5.m5.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p4.5.m5.2.2.2.2.3.4" xref="S2.SS1.p4.5.m5.2.2.2.2.3.4.cmml">z</mi></mrow></mrow><mo stretchy="false" id="S2.SS1.p4.5.m5.2.2.2.5" xref="S2.SS1.p4.5.m5.2.2.3.cmml">]</mo></mrow></math>, <math><mrow id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml"><mrow id="S2.SS1.p4.6.m6.1.1.3" xref="S2.SS1.p4.6.m6.1.1.3.cmml"><mi mathvariant="normal" id="S2.SS1.p4.6.m6.1.1.3.2" xref="S2.SS1.p4.6.m6.1.1.3.2.cmml">Δ</mi><mo id="S2.SS1.p4.6.m6.1.1.3.1" xref="S2.SS1.p4.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p4.6.m6.1.1.3.3" xref="S2.SS1.p4.6.m6.1.1.3.3.cmml">z</mi></mrow><mo id="S2.SS1.p4.6.m6.1.1.2" xref="S2.SS1.p4.6.m6.1.1.2.cmml">=</mo><mrow id="S2.SS1.p4.6.m6.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.cmml"><mrow id="S2.SS1.p4.6.m6.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.cmml"><mpadded width="+1.7pt" id="S2.SS1.p4.6.m6.1.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.1.3.cmml"><msqrt id="S2.SS1.p4.6.m6.1.1.1.1.3a" xref="S2.SS1.p4.6.m6.1.1.1.1.3.cmml"><mrow id="S2.SS1.p4.6.m6.1.1.1.1.3.2" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.cmml"><mn id="S2.SS1.p4.6.m6.1.1.1.1.3.2.2" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.2.cmml">2</mn><mo id="S2.SS1.p4.6.m6.1.1.1.1.3.2.1" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.1.cmml">/</mo><msub id="S2.SS1.p4.6.m6.1.1.1.1.3.2.3" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.3.cmml"><mi mathvariant="normal" id="S2.SS1.p4.6.m6.1.1.1.1.3.2.3.2" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.3.2.cmml">Δ</mi><mi mathvariant="normal" id="S2.SS1.p4.6.m6.1.1.1.1.3.2.3.3" xref="S2.SS1.p4.6.m6.1.1.1.1.3.2.3.3.cmml">v</mi></msub></mrow></msqrt></mpadded><mo id="S2.SS1.p4.6.m6.1.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p4.6.m6.1.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p4.6.m6.1.1.1.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.cmml"><mn id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml">+</mo><msub id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3.2" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3.2.cmml">z</mi><mi id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3.3" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3.3.cmml">group</mi></msub></mrow><mo rspace="4.2pt" stretchy="false" id="S2.SS1.p4.6.m6.1.1.1.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p4.6.m6.1.1.1.1.2a" xref="S2.SS1.p4.6.m6.1.1.1.1.2.cmml">⁢</mo><msub id="S2.SS1.p4.6.m6.1.1.1.1.4" xref="S2.SS1.p4.6.m6.1.1.1.1.4.cmml"><mi id="S2.SS1.p4.6.m6.1.1.1.1.4.2" xref="S2.SS1.p4.6.m6.1.1.1.1.4.2.cmml">v</mi><mi id="S2.SS1.p4.6.m6.1.1.1.1.4.3" xref="S2.SS1.p4.6.m6.1.1.1.1.4.3.cmml">vir</mi></msub></mrow><mo id="S2.SS1.p4.6.m6.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.2.cmml">/</mo><mi id="S2.SS1.p4.6.m6.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.3.cmml">c</mi></mrow></mrow></math>, <math><mrow id="S2.SS1.p4.9.m9.2.2.1" xref="S2.SS1.p4.9.m9.2.2.2.cmml"><mi id="S2.SS1.p4.9.m9.1.1" xref="S2.SS1.p4.9.m9.1.1.cmml">log</mi><mo id="S2.SS1.p4.9.m9.2.2.1a" xref="S2.SS1.p4.9.m9.2.2.2.cmml">⁡</mo><mrow id="S2.SS1.p4.9.m9.2.2.1.1" xref="S2.SS1.p4.9.m9.2.2.2.cmml"><mo stretchy="false" id="S2.SS1.p4.9.m9.2.2.1.1.2" xref="S2.SS1.p4.9.m9.2.2.2.cmml">(</mo><mrow id="S2.SS1.p4.9.m9.2.2.1.1.1" xref="S2.SS1.p4.9.m9.2.2.1.1.1.cmml"><msub id="S2.SS1.p4.9.m9.2.2.1.1.1.2" xref="S2.SS1.p4.9.m9.2.2.1.1.1.2.cmml"><mi id="S2.SS1.p4.9.m9.2.2.1.1.1.2.2" xref="S2.SS1.p4.9.m9.2.2.1.1.1.2.2.cmml">M</mi><mi id="S2.SS1.p4.9.m9.2.2.1.1.1.2.3" xref="S2.SS1.p4.9.m9.2.2.1.1.1.2.3.cmml">vir</mi></msub><mo id="S2.SS1.p4.9.m9.2.2.1.1.1.1" xref="S2.SS1.p4.9.m9.2.2.1.1.1.1.cmml">/</mo><msub id="S2.SS1.p4.9.m9.2.2.1.1.1.3" xref="S2.SS1.p4.9.m9.2.2.1.1.1.3.cmml"><mi mathvariant="normal" id="S2.SS1.p4.9.m9.2.2.1.1.1.3.2" xref="S2.SS1.p4.9.m9.2.2.1.1.1.3.2.cmml">M</mi><mo id="S2.SS1.p4.9.m9.2.2.1.1.1.3.3" xref="S2.SS1.p4.9.m9.2.2.1.1.1.3.3.cmml">⊙</mo></msub></mrow><mo stretchy="false" id="S2.SS1.p4.9.m9.2.2.1.1.3" xref="S2.SS1.p4.9.m9.2.2.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><msubsup id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.2.2.2" xref="S2.SS2.p3.1.m1.1.1.2.2.2.cmml">ℳ</mi><mi id="S2.SS2.p3.1.m1.1.1.2.3" xref="S2.SS2.p3.1.m1.1.1.2.3.cmml">r</mi><mi id="S2.SS2.p3.1.m1.1.1.2.2.3" xref="S2.SS2.p3.1.m1.1.1.2.2.3.cmml">Petro</mi></msubsup><mo id="S2.SS2.p3.1.m1.1.1.1" xref="S2.SS2.p3.1.m1.1.1.1.cmml">≤</mo><mrow id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml"><mo id="S2.SS2.p3.1.m1.1.1.3.1" xref="S2.SS2.p3.1.m1.1.1.3.1.cmml">-</mo><mn id="S2.SS2.p3.1.m1.1.1.3.2" xref="S2.SS2.p3.1.m1.1.1.3.2.cmml">18.78</mn></mrow></mrow></math>, <math><mrow id="S3.E1.m1.4.4.1" xref="S3.E1.m1.4.4.1.1.cmml"><mrow id="S3.E1.m1.4.4.1.1" xref="S3.E1.m1.4.4.1.1.cmml"><mrow id="S3.E1.m1.4.4.1.1.1" xref="S3.E1.m1.4.4.1.1.1.cmml"><msub id="S3.E1.m1.4.4.1.1.1.3" xref="S3.E1.m1.4.4.1.1.1.3.cmml"><mi id="S3.E1.m1.4.4.1.1.1.3.2" xref="S3.E1.m1.4.4.1.1.1.3.2.cmml">d</mi><mi id="S3.E1.m1.4.4.1.1.1.3.3" xref="S3.E1.m1.4.4.1.1.1.3.3.cmml">outer</mi></msub><mo id="S3.E1.m1.4.4.1.1.1.2" xref="S3.E1.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.4.4.1.1.1.1.1" xref="S3.E1.m1.4.4.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.E1.m1.4.4.1.1.1.1.1.2" xref="S3.E1.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S3.E1.m1.3.3" xref="S3.E1.m1.3.3.cmml">R</mi><mo id="S3.E1.m1.4.4.1.1.1.1.1.3" xref="S3.E1.m1.4.4.1.1.1.1.2.cmml">,</mo><mrow id="S3.E1.m1.4.4.1.1.1.1.1.1" xref="S3.E1.m1.4.4.1.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S3.E1.m1.4.4.1.1.1.1.1.1.2" xref="S3.E1.m1.4.4.1.1.1.1.1.1.2.cmml">Δ</mi><mo id="S3.E1.m1.4.4.1.1.1.1.1.1.1" xref="S3.E1.m1.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E1.m1.4.4.1.1.1.1.1.1.3" xref="S3.E1.m1.4.4.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo stretchy="false" id="S3.E1.m1.4.4.1.1.1.1.1.4" xref="S3.E1.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E1.m1.4.4.1.1.2" xref="S3.E1.m1.4.4.1.1.2.cmml">=</mo><mpadded width="+5pt" id="S3.E1.m1.2.2" xref="S3.E1.m1.2.2.cmml"><msqrt id="S3.E1.m1.2.2a" xref="S3.E1.m1.2.2.cmml"><mrow id="S3.E1.m1.2.2.2" xref="S3.E1.m1.2.2.2.cmml"><mrow id="S3.E1.m1.2.2.2.4" xref="S3.E1.m1.2.2.2.4.cmml"><mfrac id="S3.E1.m1.2.2.2.4.2" xref="S3.E1.m1.2.2.2.4.2.cmml"><msub id="S3.E1.m1.2.2.2.4.2.2" xref="S3.E1.m1.2.2.2.4.2.2.cmml"><mi mathvariant="normal" id="S3.E1.m1.2.2.2.4.2.2.2" xref="S3.E1.m1.2.2.2.4.2.2.2.cmml">Δ</mi><mi mathvariant="normal" id="S3.E1.m1.2.2.2.4.2.2.3" xref="S3.E1.m1.2.2.2.4.2.2.3.cmml">v</mi></msub><mn id="S3.E1.m1.2.2.2.4.2.3" xref="S3.E1.m1.2.2.2.4.2.3.cmml">2</mn></mfrac><mo id="S3.E1.m1.2.2.2.4.1" xref="S3.E1.m1.2.2.2.4.1.cmml">⁢</mo><msup id="S3.E1.m1.2.2.2.4.3" xref="S3.E1.m1.2.2.2.4.3.cmml"><mrow id="S3.E1.m1.2.2.2.4.3.2.2" xref="S3.E1.m1.2.2.2.4.3.2.1.cmml"><mo id="S3.E1.m1.2.2.2.4.3.2.2.1" xref="S3.E1.m1.2.2.2.4.3.2.1.1.cmml">[</mo><mfrac id="S3.E1.m1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.cmml"><mrow id="S3.E1.m1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.3.cmml"><mpadded width="+1.7pt" id="S3.E1.m1.1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.3.2.cmml"><mi id="S3.E1.m1.1.1.1.1.3.2a" xref="S3.E1.m1.1.1.1.1.3.2.cmml">c</mi></mpadded><mo id="S3.E1.m1.1.1.1.1.3.1" xref="S3.E1.m1.1.1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S3.E1.m1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.3.3.cmml">Δ</mi><mo id="S3.E1.m1.1.1.1.1.3.1a" xref="S3.E1.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E1.m1.1.1.1.1.3.4" xref="S3.E1.m1.1.1.1.1.3.4.cmml">z</mi></mrow><mrow id="S3.E1.m1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.cmml"><msub id="S3.E1.m1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.3.2.cmml">v</mi><mi id="S3.E1.m1.1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.3.3.cmml">vir</mi></msub><mo id="S3.E1.m1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.E1.m1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.cmml"><mn id="S3.E1.m1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.E1.m1.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><msub id="S3.E1.m1.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.3.2.cmml">z</mi><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.3.3.cmml">group</mi></msub></mrow><mo stretchy="false" id="S3.E1.m1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S3.E1.m1.2.2.2.4.3.2.2.2" xref="S3.E1.m1.2.2.2.4.3.2.1.1.cmml">]</mo></mrow><mn id="S3.E1.m1.2.2.2.4.3.3" xref="S3.E1.m1.2.2.2.4.3.3.cmml">2</mn></msup></mrow><mo id="S3.E1.m1.2.2.2.3" xref="S3.E1.m1.2.2.2.3.cmml">+</mo><msup id="S3.E1.m1.2.2.2.5" xref="S3.E1.m1.2.2.2.5.cmml"><mrow id="S3.E1.m1.2.2.2.5.2.2" xref="S3.E1.m1.2.2.2.2.cmml"><mo id="S3.E1.m1.2.2.2.5.2.2.1" xref="S3.E1.m1.2.2.2.2.cmml">(</mo><mfrac id="S3.E1.m1.2.2.2.2" xref="S3.E1.m1.2.2.2.2.cmml"><mi id="S3.E1.m1.2.2.2.2.2" xref="S3.E1.m1.2.2.2.2.2.cmml">R</mi><msub id="S3.E1.m1.2.2.2.2.3" xref="S3.E1.m1.2.2.2.2.3.cmml"><mi id="S3.E1.m1.2.2.2.2.3.2" xref="S3.E1.m1.2.2.2.2.3.2.cmml">r</mi><mi id="S3.E1.m1.2.2.2.2.3.3" xref="S3.E1.m1.2.2.2.2.3.3.cmml">vir</mi></msub></mfrac><mo id="S3.E1.m1.2.2.2.5.2.2.2" xref="S3.E1.m1.2.2.2.2.cmml">)</mo></mrow><mn id="S3.E1.m1.2.2.2.5.3" xref="S3.E1.m1.2.2.2.5.3.cmml">2</mn></msup></mrow></msqrt></mpadded></mrow><mo id="S3.E1.m1.4.4.1.2" xref="S3.E1.m1.4.4.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S3.p1.3.m1.2.2" xref="S3.p1.3.m1.2.2.cmml"><msub id="S3.p1.3.m1.2.2.3" xref="S3.p1.3.m1.2.2.3.cmml"><mi id="S3.p1.3.m1.2.2.3.2" xref="S3.p1.3.m1.2.2.3.2.cmml">P</mi><mi id="S3.p1.3.m1.2.2.3.3" xref="S3.p1.3.m1.2.2.3.3.cmml">M</mi></msub><mo id="S3.p1.3.m1.2.2.2" xref="S3.p1.3.m1.2.2.2.cmml">⁢</mo><mrow id="S3.p1.3.m1.2.2.1.1" xref="S3.p1.3.m1.2.2.1.2.cmml"><mo stretchy="false" id="S3.p1.3.m1.2.2.1.1.2" xref="S3.p1.3.m1.2.2.1.2.cmml">(</mo><mi id="S3.p1.3.m1.1.1" xref="S3.p1.3.m1.1.1.cmml">R</mi><mo id="S3.p1.3.m1.2.2.1.1.3" xref="S3.p1.3.m1.2.2.1.2.cmml">,</mo><mrow id="S3.p1.3.m1.2.2.1.1.1" xref="S3.p1.3.m1.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S3.p1.3.m1.2.2.1.1.1.2" xref="S3.p1.3.m1.2.2.1.1.1.2.cmml">Δ</mi><mo id="S3.p1.3.m1.2.2.1.1.1.1" xref="S3.p1.3.m1.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.p1.3.m1.2.2.1.1.1.3" xref="S3.p1.3.m1.2.2.1.1.1.3.cmml">z</mi></mrow><mo stretchy="false" id="S3.p1.3.m1.2.2.1.1.4" xref="S3.p1.3.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0807.4217

Formulas:

1-

o s p (32 | 1)

2-

o s p (32 | 1)

3-

o s p (32 | 1) \times o s p (32 | 1)

4-

𝐙_{a b c d e}

5-

\begin{matrix} {𝐐_{α}, 𝐐_{β}} & = & {(C Γ^{a})}_{α β} 𝐏_{a} + {(C Γ^{a b})}_{α β} 𝐙_{a b} + {(C Γ^{a b c d e})}_{α β} 𝐙_{a b c d e} \\ \equiv & 𝐏_{α β}, \end{matrix}

6-

𝐙_{a b c d e}

7-

o s p (32 | 1)

8-

o s p (32 | 1)

9-

𝒜 = \frac{1}{2} ω^{a b} 𝐉_{a b} + e^{a} 𝐏_{a} + \frac{1}{\sqrt{2}} ψ^{α} 𝐐_{α} + b_{[2]}^{a b} 𝐙_{a b} + b_{[5]}^{a b c d e} 𝐙_{a b c d e},

10-

ℱ = d 𝒜 + 𝒜 \land 𝒜

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/physics/0508027

Formulas:

1-

P (k, k^{'})

2-

(2 - δ_{k, k^{'}}) P (k, k^{'})

3-

P (k, k^{'})

4-

P (k, k^{'}) = \frac{k k^{'} P (k) P (k^{'})}{{⟨ k ⟩}^{2}} .

5-

P (k, k^{'})

6-

{\bar{k}}_{n n} (k) = ⟨ k ⟩ \frac{\sum_{k^{'}} k^{'} P (k, k^{'})}{k P (k)} .

7-

P (k, k^{'})

8-

r = ⟨ k ⟩ \frac{\sum_{k} k^{2} {\bar{k}}_{n n} (k) P (k) - {⟨ k^{2} ⟩}^{2}}{⟨ k ⟩ ⟨ k^{3} ⟩ - {⟨ k^{2} ⟩}^{2}} .

9-

{\bar{k}}_{n n} (k) = ⟨ k^{2} ⟩ / ⟨ k ⟩

10-

{\bar{k}}_{n n} (k)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0406084

Formulas:

1-

θ = 1 / 2 δ

2-

N (γ) = K γ^{- n}

3-

α_{r} = 0.7 - 0.8

4-

θ_{\max} \sim 1 / δ

5-

P_{j} = π R^{2} Γ^{2} (U_{B}^{'} + U_{e}^{'} + U_{p}^{'}) c

6-

U_{e}^{'} = m_{e} c^{2} \int_{γ_{\min}}^{γ_{\max}} N (γ) (γ - 1) 𝑑 γ

7-

U_{e}^{'} = n_{e} < γ > m_{e} c^{2}

8-

P = f^{- 2 α / (3 + α)}

9-

p = U_{B}^{'} / 3 + U_{e}^{'} / 3

10-

20 - 6 \times 10^{10}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1601.01518

Formulas:

1-

δ Y = \frac{V}{U} \cdot δ X

2-

P S F (r)

3-

I (r) = O (r) * P S F (r)

4-

I (r) ⋆ I (r) = (O (r) ⋆ O (r)) * (P S F (r) ⋆ P S F (r))

5-

P S F (r) ⋆ P S F (r)

6-

I (r) ⋆ I (r)

7-

U = 161 m m

8-

U = 65 m m

9-

δ x \approx \sqrt{{(\frac{λ}{D_{b u n d l e}} \cdot U)}^{2} + d_{m o d e}^{2}}

10-

D_{b u n d l e}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1804.00876

Formulas:

1-

P_{0} (n)

2-

P_{0} (n) \sim θ^{n}

3-

P_{0} (n)

4-

A (n) \to 0

5-

A (n) = 0

6-

A (n) = 0

7-

A (n) = c_{1} λ_{1}^{n} + c_{2} λ_{2}^{n}

8-

A (n) \sim n^{- μ}

9-

P_{0} (n)

10-

P_{0} (n)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0906.1531

Formulas:

1-

J \in 𝒥^{l} (M)

2-

d u \circ j = J \circ d u,

3-

d u (ξ_{Σ}) = λ ξ,

4-

u^{*} θ = λ θ_{Σ}

5-

(2 n + 1)

6-

(P, J, θ)

7-

(P, J, θ)

8-

[X, Y] - [J X, J Y] - J ([J X, Y] + [X, J Y]) = 0, for X, Y \in Γ (P),

9-

L (X, Y) = - d θ (J X, Y)

10-

S = {X - \sqrt{- 1} J X ∣ X \in P} \subset ℂ T M

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9807296

Formulas:

1-

E (x | ω) = E_{l 1} e^{i k_{l} x} + E_{l 2} e^{- i k_{l} x} .

2-

k_{l} = ω \sqrt{ϵ_{l}} / c

3-

E (x | ω) = (\begin{matrix} E_{l 1} e^{i k_{l} x} \\ E_{l 2} e^{- i k_{l} x} \end{matrix}) .

4-

T (x_{l - 1}, x_{l})

5-

E (x_{l} | ω) = T (x_{l - 1}, x_{l}) E (x_{l - 1} | ω),

6-

T (x_{l - 1}, x_{l}) = P_{l} (Δ x_{l}) Q_{l - 1, l} P_{l - 1} (Δ x_{l - 1}) .

7-

Δ x_{l} = x_{l} - d_{l - 1, l}

8-

Δ x_{l - 1} = d_{l - 1, l} - x_{l - 1}

9-

x = d_{l - 1, l}

10-

P_{l} (Δ x) = (\begin{matrix} e^{i k_{l} Δ x} & 0 \\ 0 & e^{- i k_{l} Δ x} \end{matrix}),

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/9901023

Formulas:

1-

ℓ_{P} = \sqrt{ℏ G / c^{3}}

2-

ℓ_{P} \approx 10^{- 33}

3-

\exp i \oint A . d ℓ

4-

{\vec{A}}_{i} τ_{i}

5-

{\vec{E}}_{i} τ_{i}

6-

({\vec{A}}_{i}, {\vec{E}}_{i})

7-

Ψ (\vec{x})

8-

Ψ ({\vec{A}}_{i})

9-

\hat{P} \cdot Ψ_{I} = - i ℏ (\partial Ψ / \partial x_{I})

10-

I = 1, 2, 3

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0812.4387

Formulas:

1-

d_{\pm}^{†}, d_{\pm}

2-

J_{x} \equiv (d_{-}^{†} d_{-} - d_{+}^{†} d_{+}) / 2

3-

J_{y} \equiv i (d_{-}^{†} d_{+} - d_{+}^{†} d_{-}) / 2

4-

J_{z} \equiv (d_{+}^{†} d_{-} + d_{-}^{†} d_{+}) / 2

5-

\hat{H} = 2 [2 Λ (N - 1) + \frac{Ω}{2}] J_{z} + 2 (κ - η) J_{x}^{2} + 4 η J_{z}^{2} .

6-

η = κ ϵ^{2}, Λ = κ ϵ^{\frac{3}{2}}

7-

ϵ = ⟨ u_{+} | u_{-} ⟩

8-

Ω^{^{'}} \equiv 2 [2 Λ (N - 1) + Ω / 2]

9-

s u (2)

10-

S U (2)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1810.04865

Formulas:

1-

F m \bar{3} m

2-

F \bar{4} 3 m

3-

M_{t} = Z_{t} -

4-

M_{t} = Z_{t} -

5-

d M / d T

6-

d M / d T

7-

d M / d T

8-

M / H = C / (T - θ) + χ_{0}

9-

μ_{e f f}

10-

σ (T) = σ_{0} + σ_{g} \exp (- E_{g} / k_{B} T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0808.3561

Formulas:

1-

l = 20 ° - 45 °

2-

E (B - V) < 0.2

3-

0.25 ° \times 0.25 °

4-

(X, Y, Z) = (6.5 k p c, - 2.2 k p c, 1.5 k p c)

5-

(6.5 k p c, 0.3 k p c, 1.5 k p c)

6-

(6.5 k p c, 2.2 k p c, 1.5 k p c)

7-

l = 0 ° - 90 °

8-

l = 270 ° - 360 °

9-

l = 25 ° - 45 °

10-

b = 30 ° - 35 °

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9912568

Formulas:

1-

2 G M / c^{2}

2-

f = 1 - Q^{-} / Q^{+}

3-

f = 1 - Q^{-} / (Q^{+} + Q_{nuc})

4-

λ / λ_{K} = 1

5-

λ / λ_{K} = 1

6-

Q_{mag} = \frac{3 B^{2}}{16 π x ρ} v

7-

T > \sim 0.5 \times 10^{9}

8-

T \sim 0.01 - 0.2 \times 10^{9}

9-

T \sim 0.02 - 0.5 \times 10^{9}

10-

{\dot{m}}_{d} < \sim 0.1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1403.0694

Formulas:

1-

\sim 0.3 M W / m^{2}

2-

\sim 10^{12} / c m^{2}

3-

z T = \frac{S^{2} σ}{κ_{e l} + κ_{p h}} T,

4-

η_{m a x}

5-

η_{m a x P}

6-

2 M W / m^{2}

7-

0.3 M W / m^{2}

8-

10^{12} / c m^{2}

9-

η_{m a x P} / η_{C}

10-

T_{H} = 330 K

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0812.0402

Formulas:

1-

1 / 7, 1 / 9 \dots

2-

E ({s_{i}}) = J \sum_{< i, j >} s_{i} s_{j} + J^{'} \sum_{≪ i, j ≫} s_{i} s_{j} - h \sum_{i} s_{i},

3-

s_{i} = \pm 1 / 2

4-

J^{'} / J = 1

5-

s_{1}, s_{2}, s_{3}, s_{4}

6-

ϵ^{A} (s_{1}, s_{2}, s_{3}, s_{4})

7-

J (s_{1} s_{2} + s_{2} s_{3} + s_{3} s_{4} + s_{4} s_{1})

8-

J^{'} s_{2} s_{4} - \frac{h}{2} (s_{1} + s_{2} + s_{3} + s_{4}) .

9-

T_{α_{1}, α_{2}, α_{3}, α_{4}}^{A} = \exp [- β ϵ^{A} (s_{1}, s_{2}, s_{3}, s_{4})]

10-

α_{i} \equiv s_{i} + 3 / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1510.01884

Formulas:

1-

R (x, y)

2-

n \geq R (x, y)

3-

R (x, y)

4-

2^{n (n - 1) / 2}

5-

R (x, y)

6-

2^{n (n - 1) / 2}

7-

R (x, y)

8-

L = n (n - 1) / 2

9-

(v, v^{'})

10-

(v, v^{'})

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0708.4033

Formulas:

1-

𝒰 (1) \times S_{4}

2-

θ_{23} ≃ π / 4

3-

θ_{13} (≃ 0.015)

4-

| Δ m_{atm}^{2} | = 2.4 \cdot (1_{- 0.26}^{+ 0.21}) \times 10^{- 3} {eV}^{2}, \sin^{2} θ_{23} = 0.44 \cdot (1_{- 0.22}^{+ 0.41}),

5-

Δ m_{sol}^{2} = 7.92 \cdot (1 \pm 0.09) \times 10^{- 5} {eV}^{2}, \sin^{2} θ_{12} = 0.314 \cdot (1_{- 0.15}^{+ 0.18}) .

6-

θ_{13} \overset{<}{_{\sim}} 0.2,

7-

θ_{13} \overset{<}{_{\sim}} 12^{o}

8-

G_{f} = 𝒰 (1)

9-

\sim \frac{Δ m_{sol}^{2}}{Δ m_{atm}^{2}} \approx 0.03

10-

G_{f} = 𝒰 (1) \times S_{4}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/2009.14028

Formulas:

1-

N = 2, \dots, 9

2-

N = 2, 3, 4, 5

3-

x_{k}, y_{k} \in {0, 1}^{2}

4-

k = 1, \dots, N

5-

b \equiv b_{1} \dots b_{N} \in {0, 1}^{N}

6-

{x_{k}, y_{k}}_{k}

7-

b_{1} = \oplus_{k = 1}^{N} x_{k} and b_{k} = y_{k} \oplus y_{1},

8-

k = 2, \dots, N

9-

| GHZ ⟩ = \frac{1}{\sqrt{2}} ({| 0 ⟩}^{\otimes N} + {| 1 ⟩}^{\otimes N})

10-

| M_{b} ⟩ = σ_{Z}^{b_{1}} \otimes σ_{X}^{b_{2}} \otimes \dots \otimes σ_{X}^{b_{N}} | GHZ ⟩,

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0412270

Formulas:

1-

M_{p, t u r b}

2-

M_{p, d a m p}

3-

\frac{M_{p, t u r b}}{M_{p, d a m p}} \geq 2.5 \sqrt{α_{t u r b}} {(\frac{h}{r_{p}})}^{- 23 / 26} Q^{5 / 13},

4-

Q = Ω c_{s} / (π G Σ)

5-

M_{p, t u r b} / M_{p, d a m p} \sim 100

6-

V_{A} \equiv B / \sqrt{4 π ρ} \sim α_{turb}^{1 / 2} c_{s}

7-

R e_{M} \equiv \frac{V_{A} h}{η},

8-

R e_{M, crit} = 10^{2} - 10^{4}

9-

R e_{M, crit} = 10^{2} - 10^{4}

10-

α_{turb} = 10^{- 3} - 1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1211.6556

Formulas:

1-

R_{V} = A_{V} / E (B - V)

2-

A_{λ} \propto λ^{- α}

3-

- 27^{\circ} 26^{'} 10^{''}

4-

- 25^{\circ} 33^{'} 09^{''}

5-

(λ - K_{s})

6-

A_{λ} / A_{K_{s}}

7-

(K_{s} - λ)

8-

(H - K_{s})

9-

(K_{s} - λ)

10-

(H - K_{s})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1708.04583

Formulas:

1-

\dot{m} = \frac{p_{0} A^{*}}{\sqrt{T_{0}}} \sqrt{\frac{γ}{R} {(\frac{2}{γ + 1})}^{(γ + 1) / (γ - 1)}} .

2-

= f (p_{0}, A^{*}, T_{0}, R, γ)

3-

= φ_{1} (p_{0}) \times φ_{2} (A^{*}) \times φ_{3} (T_{0}) \times φ_{4} (R) \times φ_{5} (γ) .

4-

C_{L} = C_{L α} (α - α_{0}) + C_{L δ_{e}} δ_{e} \frac{S_{𝐻𝑇}}{S_{𝑟𝑒𝑓}},

5-

(α, α_{0})

6-

f (C_{L α}, α, α_{0}, C_{L δ_{e}}, δ_{e}, S_{𝐻𝑇}, S_{𝑟𝑒𝑓})

7-

φ_{1} (C_{L α}) \times φ_{2} (α, α_{0})

8-

+ φ_{3} (C_{L δ_{e}}) \times φ_{4} (δ_{e}) \times φ_{5} (S_{𝐻𝑇}) \times φ_{6} (S_{𝑟𝑒𝑓}) .

9-

ψ = (V_{\infty} r \sin θ) (1 - \frac{R^{2}}{r^{2}}) + \frac{Γ}{2 π} \ln \frac{r}{R},

10-

f (V_{\infty}, \sin θ, R, r, Γ)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1801.01747

Formulas:

1-

σ \sim {(ε N)}^{- d}

2-

ε_{b r} \sim q_{0}^{4} / N^{4}

3-

ε_{c} \propto 1 / N

4-

σ \sim {(ε N)}^{- d}

5-

T^{e q} \sim ε^{- a}

6-

T^{e q} \sim \exp (1 / ε^{δ})

7-

H = \frac{1}{2} \sum_{k = 1}^{N} y_{k}^{2} + \frac{1}{2} \sum_{k = 0}^{N} {(x_{k + 1} - x_{k})}^{2} + \frac{α}{3} \sum_{k = 0}^{N} {(x_{k + 1} - x_{k})}^{3},

8-

x_{0} = x_{N + 1} = 0

9-

H_{T} = \sum_{k = 1}^{N} y_{k}^{2} + \frac{1}{4 α^{2}} \sum_{k = 0}^{N} e^{2 α (x_{k + 1} - x_{k})} - \frac{N + 1}{4 α^{2}}

10-

(x_{k}, y_{k})

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0501087

Formulas:

1-

M_{WD} \sim 0.8 M_{⊙}

2-

M_{WD} \sim 0.5 M_{⊙}

3-

n_{e} = {1.0}_{- 0.5}^{+ 2.0} \times 10^{14} {cm}^{- 3}

4-

K_{1} = 58.2 \pm 3.7 km s^{- 1}

5-

M_{WD} = 0.49 \pm 0.13 M_{⊙}

6-

M_{WD} = 0.91 \pm 0.05 M_{⊙}

7-

ϕ_{98} = 0.30 - 0.39

8-

R X T E

9-

R X T E

10-

T = 2437699.94179 + 0.068233846 (4) E

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1205.5537

Formulas:

1-

γ (\vec{C_{m}} □ \vec{C_{n}})

2-

γ (\vec{C_{m}} □ \vec{C_{n}})

3-

(m o d 3)

4-

γ (\vec{C_{m}} □ \vec{C_{n}})

5-

(m o d 3)

6-

(m o d 3)

7-

D = (V, A)

8-

D_{1} = (V_{1}, A_{1})

9-

D_{2} = (V_{2}, A_{2})

10-

D_{1} □ D_{2}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0305657

Formulas:

1-

q = (1 - r) / 2

2-

T_{0} = 2 \sqrt{3} \cdot 10^{2}

3-

g = d = m = 1

4-

n_{c} = 2 / (\sqrt{3} d^{2})

5-

\frac{d p}{d y} + m n g = 0, \frac{d}{d y} (κ \frac{d}{d y} T) - I = 0,

6-

p = p (n, T)

7-

p = n T \frac{n_{c} + n}{n_{c} - n}, κ = \frac{μ n {(α l + d)}^{2} T^{1 / 2}}{m^{1 / 2} l}

8-

I = 4 (μ / γ l) q n m^{- 1 / 2} T^{3 / 2}

9-

l = \frac{1}{\sqrt{8} n d} \frac{n_{c} - n}{n_{c} - a n},

10-

a = 1 - {(3 / 8)}^{1 / 2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0309288

Formulas:

1-

Δ ν = 274 \pm 11

2-

ν_{b u r s t}

3-

ν_{b u r s t}

4-

ν_{b u r s t} / 2

5-

ν_{l} = 308 \pm 5

6-

ν_{u} = 582 \pm 10

7-

Δ ν = 274 \pm 11

8-

ν_{l} = 385 \pm 9

9-

ν_{u} = 616 \pm 19

10-

Δ ν = 231 \pm 21

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2004.08331

Formulas:

1-

W_{s} (p, q)

2-

W_{s} (p, q) = \int_{- \infty}^{+ \infty} s (t) p^{- 1 / 2} ψ^{*} (\frac{t - q}{p}) 𝑑 t

3-

ω_{s} (p, q)

4-

ω_{s} (p, q) = \frac{- i}{W_{s} (p, q)} \frac{\partial}{\partial q} W_{s} (p, q)

5-

W_{x} (p, q)

6-

p_{k} - p_{k - 1} = {(δ p)}_{k}

7-

T_{s} (ω, q)

8-

[ω_{n} - δ ω / 2, ω_{n} + δ ω / 2]

9-

δ ω = ω_{n} - ω_{n - 1}

10-

T_{s} (ω_{n}, q) = \frac{1}{δ ω} \sum_{p_{k}} W_{s} (p_{k}, q) p_{k}^{- 3 / 2} {(δ p)}_{k}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0606198

Formulas:

1-

\sim 10^{10} M_{⊙}

2-

M_{⋆} > M_{12} = (M_{1} + M_{2})

3-

\tilde{M} = 2 M_{2} / M_{12}

4-

q = M_{2} / M_{1}

5-

\tilde{M} = 2 q / (1 + q)

6-

\tilde{r} = r / (r + r_{c})

7-

\tilde{r} = 1 / 2

8-

\tilde{r} \leq 1 / 2

9-

\tilde{r} > 1 / 2

10-

r_{c} = r_{⋆} {(M_{12} / M_{⋆})}^{1 / 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/physics/9807021

Formulas:

1-

t_{s} = ω_{0}^{- 1} {(m_{i} / Z m_{e})}^{1 / 2}

2-

2.0 \cdot 10^{18} {W/cm}^{2}

3-

1.0 \cdot 10^{17} {W/cm}^{2}

4-

23 μ m \times 23 μ m

5-

6 μ m \times 12 μ m

6-

x = 6 μ m

7-

x = 0 μ m

8-

z (x, y, t)

9-

(x_{0}, y_{0})

10-

0, E_{y}^{B} = \frac{1}{\bar{γ}} z (x_{r}, y_{r}, t), B_{z}^{B} = \frac{1}{c \bar{γ}} z (x_{r}, y_{r}, t),

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0906.3150

Formulas:

1-

S O_{4}^{2 -}

2-

σ_{0} + θ_{0} \leq 1

3-

A + C \to C,

4-

A + C \to 0,

5-

t^{1 / 2} / L

6-

θ \sim \exp [- a ρ^{2 / 3} {(D t)}^{1 / 3}],

7-

a \equiv 3 / 2 {(2 π^{2})}^{1 / 3}

8-

σ = σ_{0} - p θ_{0}

9-

σ_{0} > p θ_{0}

10-

σ_{0} < p θ_{0}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0407311

Formulas:

1-

{(1 + z)}^{3}

2-

E_{m a x} = k e Z (u / c) B L

3-

E_{m a x} = 0.9 Z (B L)

4-

E_{r e s} = 4 {[m_{ν} (eV)]}^{- 1}

5-

≃ [E_{ν} / (100 E e V)]

6-

c_{i} - c_{j} \equiv δ_{i j}

7-

T_{C B R} = 2.73 K

8-

δ_{p π} > 5 \times 10^{- 24} {(ϵ / T_{C B R})}^{2} .

9-

δ_{e p} > 5 \times 10^{- 19} {(ϵ / T_{C B R})}^{2} .

10-

δ_{e γ} < 1.3 \times 10^{- 15}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1606.05780

Formulas:

1-

H = \frac{p^{2}}{2 m (x)} + V (x, α),

2-

\hat{H} = - (\frac{1}{2 m (x)}) \frac{d^{2}}{d x^{2}} - {(\frac{1}{2 m (x)})}^{^{'}} \frac{d}{d x} + V (x, α),

3-

\hat{H} = {\hat{A}}^{†} (α) \hat{A} (α) + E_{0},

4-

\hat{A} (α), {\hat{A}}^{†} (α)

5-

\hat{A} (α)

6-

\frac{1}{\sqrt{2 m (x)}} \frac{d}{d x} + W (x, α),

7-

\hat{A^{†}} (α)

8-

\frac{- 1}{\sqrt{2 m (x)}} \frac{d}{d x} - {(\frac{1}{\sqrt{2 m (x)}})}^{^{'}} + W (x, α),

9-

W (x, α)

10-

[\hat{A} (α), {\hat{A}}^{†} (α)]

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0101208

Formulas:

1-

p_{x} \pm i p_{y}

2-

𝑸 = (1 \pm δ, 1 \pm δ, 0) π / a

3-

χ^{z} \equiv χ_{z z} ≫ χ_{x x} = χ_{y y} \equiv χ^{⟂}

4-

χ_{x y} = 0

5-

χ^{z} (𝒒) = \sum_{δ_{x, y} = \pm δ} \frac{χ_{Q} / 4}{1 + 4 ξ_{sfl}^{2} (\cos^{2} \frac{q_{x} - δ_{x}}{2} + \cos^{2} \frac{q_{y} - δ_{y}}{2})} .

6-

𝑸_{A F} = (1, 1, 0) π / a

7-

𝑸_{F M} = (0, 0, 0)

8-

ξ_{sfl} \approx 4.0 a

9-

g (𝒑) χ^{i} (𝒑 - 𝒑^{'}) g (𝒑^{'})

10-

ϵ_{𝒑}^{i} = 2 t_{x}^{i} \cos p_{x} + 2 t_{y}^{i} \cos p_{y} - 4 t^{', i} \cos p_{x} \cos p_{y} - μ^{i} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1502.05358

Formulas:

1-

δ 𝑿 (𝒙_{0}, t, τ) = 𝑿 (𝒙_{0}, t) - 𝑿 (𝒙_{0}, t - τ)

2-

𝑿 (𝒙_{0}, t)

3-

d 𝑿 / d t = 𝐮

4-

Θ (t, τ)

5-

\cos (Θ (t, τ)) = \frac{δ 𝑿 (𝒙_{0}, t, τ) \cdot δ 𝑿 (𝒙_{0}, t + τ, τ)}{| δ 𝑿 (𝒙_{0}, t, τ) | | δ 𝑿 (𝒙_{0}, t + τ, τ) |} .

6-

θ (τ) \equiv ⟨ | Θ (t, τ) | ⟩ .

7-

Θ (τ, t)

8-

κ (t) = lim_{τ \to 0} \frac{| Θ (t, τ) |}{2 τ ∥ 𝒖 (t) ∥},

9-

\tilde{θ} (τ) \equiv θ (τ) σ_{u} / {(ϵ τ)}^{1 / 2}

10-

τ_{K} = {(ν / ϵ)}^{1 / 2} = 0.036

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1904.12910

Formulas:

1-

\partial Ω \in C^{2 + p}

2-

K_{i} (x)

3-

C^{1 + p} (\bar{Ω})

4-

r (x) \geq 0

5-

r (x) > 0

6-

u (t, x)

7-

v (t, x)

8-

\frac{\partial u (t, x)}{\partial t} = \nabla \cdot [a (x) \nabla (\frac{u (t, x)}{P (x)})] + r (x) u (t, x) (1 - \frac{u (t, x) + v (t, x)}{K (x)}) - α r (x) u (t, x),

9-

\frac{\partial v (t, x)}{\partial t} = \nabla \cdot [b (x) \nabla (\frac{v (t, x)}{Q (x)})] + r (x) v (t, x) (1 - \frac{u (t, x) + v (t, x)}{K (x)}) - β r (x) v (t, x),

10-

t > 0, x \in Ω,

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/2006.13432

Formulas:

1-

B_{1}, \dots, B_{K}

2-

w_{i}^{m i n}

3-

w_{i}^{m a x}

4-

r_{i} \leq j \leq d_{i}

5-

𝒜 = {A_{1}, A_{2}, \dots, A_{n}}

6-

B_{1}, B_{2}, \dots, B_{K}

7-

\sum_{A_{i} \in B_{j}} s_{i} \leq L

8-

\sum_{A_{i} \in 𝒜^{'}} s_{i} w_{i}

9-

\begin{matrix} \max \sum_{j = 1}^{K} \sum_{i = 1}^{n} s_{i} x_{i j} \\ subject to & \sum_{i = 1}^{n} s_{i} x_{i j} \leq L, & j = 1, 2, \dots, K \\ \sum_{j = 1}^{K} x_{i j} = w_{i} y_{i}, & i = 1, 2, \dots, n \\ x_{i j} \in {0, 1}, y_{i} \in {0, 1} \end{matrix}

10-

s_{1} > s_{2} > s_{3} > \dots

Formulas (html):

<math><mrow id="Sx1.p1.4.m4.3.3.2" xref="Sx1.p1.4.m4.3.3.3.cmml"><msub id="Sx1.p1.4.m4.2.2.1.1" xref="Sx1.p1.4.m4.2.2.1.1.cmml"><mi id="Sx1.p1.4.m4.2.2.1.1.2" xref="Sx1.p1.4.m4.2.2.1.1.2.cmml">B</mi><mn id="Sx1.p1.4.m4.2.2.1.1.3" xref="Sx1.p1.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="Sx1.p1.4.m4.3.3.2.3" xref="Sx1.p1.4.m4.3.3.3.cmml">,</mo><mi mathvariant="normal" id="Sx1.p1.4.m4.1.1" xref="Sx1.p1.4.m4.1.1.cmml">…</mi><mo id="Sx1.p1.4.m4.3.3.2.4" xref="Sx1.p1.4.m4.3.3.3.cmml">,</mo><msub id="Sx1.p1.4.m4.3.3.2.2" xref="Sx1.p1.4.m4.3.3.2.2.cmml"><mi id="Sx1.p1.4.m4.3.3.2.2.2" xref="Sx1.p1.4.m4.3.3.2.2.2.cmml">B</mi><mi id="Sx1.p1.4.m4.3.3.2.2.3" xref="Sx1.p1.4.m4.3.3.2.2.3.cmml">K</mi></msub></mrow></math>, <math><msubsup id="Sx1.p1.17.m17.1.1" xref="Sx1.p1.17.m17.1.1.cmml"><mi id="Sx1.p1.17.m17.1.1.2.2" xref="Sx1.p1.17.m17.1.1.2.2.cmml">w</mi><mi id="Sx1.p1.17.m17.1.1.3" xref="Sx1.p1.17.m17.1.1.3.cmml">i</mi><mrow id="Sx1.p1.17.m17.1.1.2.3" xref="Sx1.p1.17.m17.1.1.2.3.cmml"><mi id="Sx1.p1.17.m17.1.1.2.3.2" xref="Sx1.p1.17.m17.1.1.2.3.2.cmml">m</mi><mo id="Sx1.p1.17.m17.1.1.2.3.1" xref="Sx1.p1.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="Sx1.p1.17.m17.1.1.2.3.3" xref="Sx1.p1.17.m17.1.1.2.3.3.cmml">i</mi><mo id="Sx1.p1.17.m17.1.1.2.3.1a" xref="Sx1.p1.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="Sx1.p1.17.m17.1.1.2.3.4" xref="Sx1.p1.17.m17.1.1.2.3.4.cmml">n</mi></mrow></msubsup></math>, <math><msubsup id="Sx1.p1.18.m18.1.1" xref="Sx1.p1.18.m18.1.1.cmml"><mi id="Sx1.p1.18.m18.1.1.2.2" xref="Sx1.p1.18.m18.1.1.2.2.cmml">w</mi><mi id="Sx1.p1.18.m18.1.1.3" xref="Sx1.p1.18.m18.1.1.3.cmml">i</mi><mrow id="Sx1.p1.18.m18.1.1.2.3" xref="Sx1.p1.18.m18.1.1.2.3.cmml"><mi id="Sx1.p1.18.m18.1.1.2.3.2" xref="Sx1.p1.18.m18.1.1.2.3.2.cmml">m</mi><mo id="Sx1.p1.18.m18.1.1.2.3.1" xref="Sx1.p1.18.m18.1.1.2.3.1.cmml">⁢</mo><mi id="Sx1.p1.18.m18.1.1.2.3.3" xref="Sx1.p1.18.m18.1.1.2.3.3.cmml">a</mi><mo id="Sx1.p1.18.m18.1.1.2.3.1a" xref="Sx1.p1.18.m18.1.1.2.3.1.cmml">⁢</mo><mi id="Sx1.p1.18.m18.1.1.2.3.4" xref="Sx1.p1.18.m18.1.1.2.3.4.cmml">x</mi></mrow></msubsup></math>, <math><mrow id="Sx1.p1.20.m20.1.1" xref="Sx1.p1.20.m20.1.1.cmml"><msub id="Sx1.p1.20.m20.1.1.2" xref="Sx1.p1.20.m20.1.1.2.cmml"><mi id="Sx1.p1.20.m20.1.1.2.2" xref="Sx1.p1.20.m20.1.1.2.2.cmml">r</mi><mi id="Sx1.p1.20.m20.1.1.2.3" xref="Sx1.p1.20.m20.1.1.2.3.cmml">i</mi></msub><mo id="Sx1.p1.20.m20.1.1.3" xref="Sx1.p1.20.m20.1.1.3.cmml">≤</mo><mi id="Sx1.p1.20.m20.1.1.4" xref="Sx1.p1.20.m20.1.1.4.cmml">j</mi><mo id="Sx1.p1.20.m20.1.1.5" xref="Sx1.p1.20.m20.1.1.5.cmml">≤</mo><msub id="Sx1.p1.20.m20.1.1.6" xref="Sx1.p1.20.m20.1.1.6.cmml"><mi id="Sx1.p1.20.m20.1.1.6.2" xref="Sx1.p1.20.m20.1.1.6.2.cmml">d</mi><mi id="Sx1.p1.20.m20.1.1.6.3" xref="Sx1.p1.20.m20.1.1.6.3.cmml">i</mi></msub></mrow></math>, <math><mrow id="S1.p3.1.m1.4.4" xref="S1.p3.1.m1.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.4.4.5" xref="S1.p3.1.m1.4.4.5.cmml">𝒜</mi><mo id="S1.p3.1.m1.4.4.4" xref="S1.p3.1.m1.4.4.4.cmml">=</mo><mrow id="S1.p3.1.m1.4.4.3.3" xref="S1.p3.1.m1.4.4.3.4.cmml"><mo stretchy="false" id="S1.p3.1.m1.4.4.3.3.4" xref="S1.p3.1.m1.4.4.3.4.cmml">{</mo><msub id="S1.p3.1.m1.2.2.1.1.1" xref="S1.p3.1.m1.2.2.1.1.1.cmml"><mi id="S1.p3.1.m1.2.2.1.1.1.2" xref="S1.p3.1.m1.2.2.1.1.1.2.cmml">A</mi><mn id="S1.p3.1.m1.2.2.1.1.1.3" xref="S1.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.p3.1.m1.4.4.3.3.5" xref="S1.p3.1.m1.4.4.3.4.cmml">,</mo><msub id="S1.p3.1.m1.3.3.2.2.2" xref="S1.p3.1.m1.3.3.2.2.2.cmml"><mi id="S1.p3.1.m1.3.3.2.2.2.2" xref="S1.p3.1.m1.3.3.2.2.2.2.cmml">A</mi><mn id="S1.p3.1.m1.3.3.2.2.2.3" xref="S1.p3.1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S1.p3.1.m1.4.4.3.3.6" xref="S1.p3.1.m1.4.4.3.4.cmml">,</mo><mi mathvariant="normal" id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">…</mi><mo id="S1.p3.1.m1.4.4.3.3.7" xref="S1.p3.1.m1.4.4.3.4.cmml">,</mo><msub id="S1.p3.1.m1.4.4.3.3.3" xref="S1.p3.1.m1.4.4.3.3.3.cmml"><mi id="S1.p3.1.m1.4.4.3.3.3.2" xref="S1.p3.1.m1.4.4.3.3.3.2.cmml">A</mi><mi id="S1.p3.1.m1.4.4.3.3.3.3" xref="S1.p3.1.m1.4.4.3.3.3.3.cmml">n</mi></msub><mo stretchy="false" id="S1.p3.1.m1.4.4.3.3.8" xref="S1.p3.1.m1.4.4.3.4.cmml">}</mo></mrow></mrow></math>, <math><mrow id="S1.p4.3.m3.4.4.3" xref="S1.p4.3.m3.4.4.4.cmml"><msub id="S1.p4.3.m3.2.2.1.1" xref="S1.p4.3.m3.2.2.1.1.cmml"><mi id="S1.p4.3.m3.2.2.1.1.2" xref="S1.p4.3.m3.2.2.1.1.2.cmml">B</mi><mn id="S1.p4.3.m3.2.2.1.1.3" xref="S1.p4.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.p4.3.m3.4.4.3.4" xref="S1.p4.3.m3.4.4.4.cmml">,</mo><msub id="S1.p4.3.m3.3.3.2.2" xref="S1.p4.3.m3.3.3.2.2.cmml"><mi id="S1.p4.3.m3.3.3.2.2.2" xref="S1.p4.3.m3.3.3.2.2.2.cmml">B</mi><mn id="S1.p4.3.m3.3.3.2.2.3" xref="S1.p4.3.m3.3.3.2.2.3.cmml">2</mn></msub><mo id="S1.p4.3.m3.4.4.3.5" xref="S1.p4.3.m3.4.4.4.cmml">,</mo><mi mathvariant="normal" id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">…</mi><mo id="S1.p4.3.m3.4.4.3.6" xref="S1.p4.3.m3.4.4.4.cmml">,</mo><msub id="S1.p4.3.m3.4.4.3.3" xref="S1.p4.3.m3.4.4.3.3.cmml"><mi id="S1.p4.3.m3.4.4.3.3.2" xref="S1.p4.3.m3.4.4.3.3.2.cmml">B</mi><mi id="S1.p4.3.m3.4.4.3.3.3" xref="S1.p4.3.m3.4.4.3.3.3.cmml">K</mi></msub></mrow></math>, <math><mrow id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml"><mrow id="S1.p4.7.m7.1.1.2" xref="S1.p4.7.m7.1.1.2.cmml"><msub id="S1.p4.7.m7.1.1.2.1" xref="S1.p4.7.m7.1.1.2.1.cmml"><mo largeop="true" symmetric="true" id="S1.p4.7.m7.1.1.2.1.2" xref="S1.p4.7.m7.1.1.2.1.2.cmml">∑</mo><mrow id="S1.p4.7.m7.1.1.2.1.3" xref="S1.p4.7.m7.1.1.2.1.3.cmml"><msub id="S1.p4.7.m7.1.1.2.1.3.2" xref="S1.p4.7.m7.1.1.2.1.3.2.cmml"><mi id="S1.p4.7.m7.1.1.2.1.3.2.2" xref="S1.p4.7.m7.1.1.2.1.3.2.2.cmml">A</mi><mi id="S1.p4.7.m7.1.1.2.1.3.2.3" xref="S1.p4.7.m7.1.1.2.1.3.2.3.cmml">i</mi></msub><mo id="S1.p4.7.m7.1.1.2.1.3.1" xref="S1.p4.7.m7.1.1.2.1.3.1.cmml">∈</mo><msub id="S1.p4.7.m7.1.1.2.1.3.3" xref="S1.p4.7.m7.1.1.2.1.3.3.cmml"><mi id="S1.p4.7.m7.1.1.2.1.3.3.2" xref="S1.p4.7.m7.1.1.2.1.3.3.2.cmml">B</mi><mi id="S1.p4.7.m7.1.1.2.1.3.3.3" xref="S1.p4.7.m7.1.1.2.1.3.3.3.cmml">j</mi></msub></mrow></msub><msub id="S1.p4.7.m7.1.1.2.2" xref="S1.p4.7.m7.1.1.2.2.cmml"><mi id="S1.p4.7.m7.1.1.2.2.2" xref="S1.p4.7.m7.1.1.2.2.2.cmml">s</mi><mi id="S1.p4.7.m7.1.1.2.2.3" xref="S1.p4.7.m7.1.1.2.2.3.cmml">i</mi></msub></mrow><mo id="S1.p4.7.m7.1.1.1" xref="S1.p4.7.m7.1.1.1.cmml">≤</mo><mi id="S1.p4.7.m7.1.1.3" xref="S1.p4.7.m7.1.1.3.cmml">L</mi></mrow></math>, <math><mrow id="S1.p4.8.m8.1.1" xref="S1.p4.8.m8.1.1.cmml"><msub id="S1.p4.8.m8.1.1.1" xref="S1.p4.8.m8.1.1.1.cmml"><mo largeop="true" symmetric="true" id="S1.p4.8.m8.1.1.1.2" xref="S1.p4.8.m8.1.1.1.2.cmml">∑</mo><mrow id="S1.p4.8.m8.1.1.1.3" xref="S1.p4.8.m8.1.1.1.3.cmml"><msub id="S1.p4.8.m8.1.1.1.3.2" xref="S1.p4.8.m8.1.1.1.3.2.cmml"><mi id="S1.p4.8.m8.1.1.1.3.2.2" xref="S1.p4.8.m8.1.1.1.3.2.2.cmml">A</mi><mi id="S1.p4.8.m8.1.1.1.3.2.3" xref="S1.p4.8.m8.1.1.1.3.2.3.cmml">i</mi></msub><mo id="S1.p4.8.m8.1.1.1.3.1" xref="S1.p4.8.m8.1.1.1.3.1.cmml">∈</mo><msup id="S1.p4.8.m8.1.1.1.3.3" xref="S1.p4.8.m8.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.8.m8.1.1.1.3.3.2" xref="S1.p4.8.m8.1.1.1.3.3.2.cmml">𝒜</mi><mo id="S1.p4.8.m8.1.1.1.3.3.3" xref="S1.p4.8.m8.1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><mrow id="S1.p4.8.m8.1.1.2" xref="S1.p4.8.m8.1.1.2.cmml"><msub id="S1.p4.8.m8.1.1.2.2" xref="S1.p4.8.m8.1.1.2.2.cmml"><mi id="S1.p4.8.m8.1.1.2.2.2" xref="S1.p4.8.m8.1.1.2.2.2.cmml">s</mi><mi id="S1.p4.8.m8.1.1.2.2.3" xref="S1.p4.8.m8.1.1.2.2.3.cmml">i</mi></msub><mo id="S1.p4.8.m8.1.1.2.1" xref="S1.p4.8.m8.1.1.2.1.cmml">⁢</mo><msub id="S1.p4.8.m8.1.1.2.3" xref="S1.p4.8.m8.1.1.2.3.cmml"><mi id="S1.p4.8.m8.1.1.2.3.2" xref="S1.p4.8.m8.1.1.2.3.2.cmml">w</mi><mi id="S1.p4.8.m8.1.1.2.3.3" xref="S1.p4.8.m8.1.1.2.3.3.cmml">i</mi></msub></mrow></mrow></math>, <math><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S1.Ex1.m1.16.16" xref="S1.Ex1.m1.16.16.cmml"><mtr id="S1.Ex1.m1.16.16a" xref="S1.Ex1.m1.16.16.cmml"><mtd columnalign="left" id="S1.Ex1.m1.16.16b" xref="S1.Ex1.m1.16.16.cmml"><mrow id="S1.Ex1.m1.16.16.17.1.1" xref="S1.Ex1.m1.16.16.17.1.1.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.2" xref="S1.Ex1.m1.16.16.17.1.1.2.cmml">max</mi><mo id="S1.Ex1.m1.16.16.17.1.1.1" xref="S1.Ex1.m1.16.16.17.1.1.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.16.16.17.1.1.3" xref="S1.Ex1.m1.16.16.17.1.1.3.cmml"><munderover id="S1.Ex1.m1.16.16.17.1.1.3.1" xref="S1.Ex1.m1.16.16.17.1.1.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S1.Ex1.m1.16.16.17.1.1.3.1.2.2" xref="S1.Ex1.m1.16.16.17.1.1.3.1.2.2.cmml">∑</mo><mrow id="S1.Ex1.m1.16.16.17.1.1.3.1.3" xref="S1.Ex1.m1.16.16.17.1.1.3.1.3.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.3.1.3.2" xref="S1.Ex1.m1.16.16.17.1.1.3.1.3.2.cmml">j</mi><mo id="S1.Ex1.m1.16.16.17.1.1.3.1.3.1" xref="S1.Ex1.m1.16.16.17.1.1.3.1.3.1.cmml">=</mo><mn id="S1.Ex1.m1.16.16.17.1.1.3.1.3.3" xref="S1.Ex1.m1.16.16.17.1.1.3.1.3.3.cmml">1</mn></mrow><mi id="S1.Ex1.m1.16.16.17.1.1.3.1.2.3" xref="S1.Ex1.m1.16.16.17.1.1.3.1.2.3.cmml">K</mi></munderover><mrow id="S1.Ex1.m1.16.16.17.1.1.3.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.cmml"><munderover id="S1.Ex1.m1.16.16.17.1.1.3.2.1" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S1.Ex1.m1.16.16.17.1.1.3.2.1.2.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.2.2.cmml">∑</mo><mrow id="S1.Ex1.m1.16.16.17.1.1.3.2.1.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.2.cmml">i</mi><mo id="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.1" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.1.cmml">=</mo><mn id="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.3.3.cmml">1</mn></mrow><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.1.2.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.1.2.3.cmml">n</mi></munderover><mrow id="S1.Ex1.m1.16.16.17.1.1.3.2.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.cmml"><msub id="S1.Ex1.m1.16.16.17.1.1.3.2.2.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.2.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.2.2.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.2.2.cmml">s</mi><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.2.2.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.2.3.cmml">i</mi></msub><mo id="S1.Ex1.m1.16.16.17.1.1.3.2.2.1" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.1.cmml">⁢</mo><msub id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.2.cmml">x</mi><mrow id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.cmml"><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.2" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.2.cmml">i</mi><mo id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.1" xref="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.1.cmml">⁢</mo><mi id="S1.Ex1.m1.16.16.17.1.1.3.2.2.3.3.3" 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Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0104233

Formulas:

1-

𝐀 = [𝐈 - g (r) (𝐈 - \hat{𝐫} \hat{𝐫})] / (1 - 2 g (r) / 3) \cdot \tilde{𝐀}

2-

P_{B} (k) d k \sim k^{- 1} d k

3-

𝐀 = [𝐈 - g (r) (𝐈 - \hat{𝐫} \hat{𝐫})] \cdot \tilde{𝐀},

4-

g (r) = {(r / R)}^{2}

5-

g (r) = 1

6-

C (t) = V_{0} / V (t),

7-

C (t) = {(P (t) / P_{0})}^{1 / γ_{rp}},

8-

γ_{rp} = 4 / 3

9-

p = P_{e} / (m_{e} c)

10-

\frac{1}{p_{*} (t)} = a_{0} \int_{t_{0}}^{t} 𝑑 t^{'} (u_{B} (t^{'}) + u_{C} (t^{'})) {(\frac{C (t^{'})}{C (t)})}^{\frac{1}{3}} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1911.12165

Formulas:

1-

(\sqrt{3} \times \sqrt{3}) R 30^{\circ}

2-

(\sqrt{3} \times \sqrt{3}) R 30^{\circ}

3-

R_{M T}^{A u} =

4-

R_{M T}^{S} =

5-

R_{M T}^{O} =

6-

R_{M T}^{C} =

7-

R_{M T}^{H} =

8-

G_{m a x}

9-

R_{M T} G_{m a x} =

10-

R_{M T} G_{m a x} =

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1906.09036

Formulas:

1-

H = - \sum_{⟨ i, j ⟩} s_{i} s_{j}

2-

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

3-

σ = 0, \dots, q - 1

4-

H = - \sum_{⟨ i, j ⟩} \cos (θ_{i} - θ_{j})

5-

θ = 2 π n / p

6-

n \in {0, \dots, p - 1}

7-

β \equiv β_{R} + i β_{I}

8-

| Z (β) |

9-

\tilde{Z} (β) \equiv \frac{Z (β)}{Z (β_{R})} = \sum_{E} P (E; β_{R}) e^{- i β_{I} E} = {⟨ e^{- i β_{I} E} ⟩}_{β_{R}}

10-

P (E; β_{R})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0903.4066

Formulas:

1-

| I_{n} ⟩ := {(\frac{| 0 ⟩ + | 1 ⟩}{\sqrt{2}})}^{\otimes n} .

2-

H_{d} = \frac{π J (t)}{2} \sum_{⟨ a, a^{'} ⟩} (1 l + σ_{z}^{(a)}) (1 l - σ_{z}^{(a^{'})})

3-

\int_{0}^{T} J (t) 𝑑 t = \frac{1}{2}

4-

⟨ a, a^{'} ⟩

5-

H = H_{d} + \sum_{j} u_{j} H_{c}^{(j)},

6-

H_{c}^{(2 j - 1)} = \frac{1}{2} σ_{x}^{(j)}, H_{c}^{(2 j)} = \frac{1}{2} σ_{y}^{(j)}

7-

j \in {1, 2, \dots, 2 n}

8-

H_{c}^{(1)} = F_{x} := \frac{1}{2} \sum_{j = 1}^{n} σ_{x}^{(j)} .

9-

| T_{3} ⟩ := \exp (- i \frac{1}{2 J} H_{d}) | I_{3} ⟩,

10-

H_{d} = \frac{π J}{2} (σ_{z}^{(1)} σ_{z}^{(2)} + σ_{z}^{(2)} σ_{z}^{(3)} + σ_{z}^{(1)} σ_{z}^{(3)}) .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1612.02997

Formulas:

1-

A \propto (κ_{V} - κ_{t})

2-

- 3 \leq κ_{t} \leq 3

3-

0.5 \leq κ_{V} \leq 1.5

4-

κ_{t} = κ_{V} = 1

5-

p_{T} > 30 (25) GeV

6-

| η | < 2.1 (2.4)

7-

p_{T} > 30 GeV

8-

p_{T} > 30 GeV

9-

p_{T} > 30 GeV

10-

113.7 (98.6) \times σ_{SM}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ex

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9605067

Formulas:

1-

σ_{δ} \equiv {⟨ {[(ρ - \bar{ρ}) / \bar{ρ}]}^{2} ⟩}^{1 / 2} \approx 1

2-

P (k) \propto k^{- 3}

3-

𝐫 (𝐪, t) = a (t) [𝐪 - D (t) \nabla Φ (𝐪)]

4-

𝐯_{0} \propto - \nabla Φ (𝐪)

5-

𝐱 = 𝐫 / a (t)

6-

𝐯_{p} = 𝐮 - H (t) 𝐫

7-

H (t) = \dot{a} / a

8-

D (t),

9-

ρ (𝐱, t) = a^{- 3} η (𝐱, D (t)), 𝐯_{p} (𝐱, t) = (a \dot{D}) 𝐯 (𝐱, D (t)) .

10-

\frac{\partial v_{i}}{\partial D} + v_{k} \frac{\partial v_{i}}{\partial x_{k}} = \frac{3}{2} \frac{Ω_{0}}{D \cdot {(\frac{d \ln D}{d \ln a})}^{2}} (\frac{\partial φ}{\partial x_{i}} + v_{i}),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0307178

Formulas:

1-

ℋ = \sum_{⟨ i, j ⟩} J_{i j} S_{i} S_{j} - \sum_{i} h_{i} S_{i} - H \sum_{i} S_{i} .

2-

J_{i j} = 1

3-

f_{i} = \sum_{j} J_{i j} S_{j} - H - h_{i} .

4-

f_{i} \cdot S_{i} < 0

5-

M (H, H_{R})

6-

ρ (H, H_{R}) = - \frac{1}{2} [\partial^{2} M / \partial H \partial H_{R}]

7-

H_{c} = (H - H_{R}) / 2

8-

H_{b} = (H + H_{R}) / 2

9-

ρ (H_{b}, H_{c})

10-

σ_{R} = 5.0 > σ_{crit} \approx 2.16

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Run 11369297 (Agent384)