Run 11314750

Paper: https://arxiv.org/abs/0907.0514

Formulas:

1-

KK + K ⇌_{k_{2}}^{k_{1}}

2-

KK - K \overset{k_{3}}{\to}

3-

KK + K_{p} ⇌_{k_{5}}^{k_{4}}

4-

KK - K_{p} \overset{k_{6}}{\to}

5-

P + K_{pp} ⇌_{k_{2}}^{k_{1}}

6-

P - K_{pp} \overset{k_{3}}{\to}

7-

P + K_{p} ⇌_{k_{5}}^{k_{4}}

8-

P - K_{p} \overset{k_{6}}{\to}

9-

KK, P^{*} \overset{k_{7}}{\to}

10-

τ_{rel} ≃ 1 / k_{7}

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0811.0001

Formulas:

1-

p (σ_{j}) \propto {[1 + σ_{j} / (1 m s^{- 1})]}^{- 1}

2-

0^{\circ} \leq i_{rel} \leq 30^{\circ}

3-

30^{\circ} \leq i_{rel} \leq 60^{\circ}

4-

60^{\circ} \leq i_{rel} \leq 90^{\circ}

5-

90^{\circ} \leq i_{rel} \leq 120^{\circ}

6-

120^{\circ} \leq i_{rel} \leq 150^{\circ}

7-

150^{\circ} \leq i_{rel} \leq 180^{\circ}

8-

P, K, e, ω, i, Ω, u

9-

a_{\max} - a_{\min} > τ a_{i}

10-

τ \in [0.10, 0.30]

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0408360

Formulas:

1-

y {}^{5}F_{1}^{\circ}

2-

(I - I_{c}) / I_{c}

3-

y {}^{5}F_{1}^{\circ}

4-

2 π ν_{L} g = 8.79 \times 10^{6} B g \approx D

5-

D = 5 \times 10^{8}

6-

D = 5 \times 10^{8}

7-

3 d^{3} ({}^{2}H) 4 s {}^{3}H

8-

3 d^{3} ({}^{2}G) 4 s {}^{1}G

9-

3 d^{3} ({}^{2}H) 4 s {}^{3}H

10-

3 d^{3} ({}^{2}G) 4 s {}^{1}G

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1410.0067

Formulas:

1-

| Ψ | = | Ψ_{\max} |

2-

V = A e^{- z^{2} / (2 σ^{2})}

3-

e^{- i θ} \partial_{t} Ψ (𝐱, t) =

4-

{[1 - V (𝐱)] + \frac{1}{2} \nabla^{2} - {| Ψ (𝐱, t) |}^{2}} Ψ (𝐱, t) + χ (𝐱, t) .

5-

μ \to μ - V (𝐱)

6-

V (𝐱) = (ω_{⟂}^{2} (x^{2} + y^{2}) + ω_{z} z^{2}) / 2

7-

⟨ χ (𝐱, t) χ^{*} (𝐱^{'}, t^{'}) ⟩ = 2 γ T_{χ} δ (𝐱 - 𝐱^{'}) δ (t - t^{'})

8-

γ T_{χ} = 5 \times 10^{- 9}

9-

ω_{z} = 0.019 μ / ℏ

10-

λ = ω_{⟂} / ω_{z} = 6.6

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9906526

Formulas:

1-

\frac{e g}{4 π} = \frac{N}{2} ℏ c,

2-

\frac{e_{1} g_{2} - e_{2} g_{1}}{4 π} = \frac{N}{2} ℏ c,

3-

(e_{1}, g_{1})

4-

(e_{2}, g_{2})

5-

\frac{M}{N} > {\begin{matrix} 610 GeV & for S = 0 \\ 870 GeV & for S = 1 / 2 \\ 1580 GeV & for S = 1 \end{matrix},

6-

q \bar{q} \to γ^{*} \to M \bar{M}

7-

W^{(e g)} = - ϵ_{μ ν σ τ} \int (d x) (d x^{'}) (d x^{''}) J^{μ} (x) \partial^{σ} D_{+} (x - x^{'}) f^{τ} (x^{'} - x^{''}) {}^{*}J^{ν} (x^{''}) .

8-

f_{μ} (x - x^{'})

9-

\partial_{μ} f^{μ} (x - x^{'}) = δ (x - x^{'}) .

10-

f^{μ} (x) = - f^{μ} (- x) .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/0811.4512

Formulas:

1-

U {(1)}_{B}

2-

U {(1)}_{B}

3-

U {(1)}_{B}

4-

g (μ) ≪ 1

5-

ℒ_{eff} = \frac{3}{4 π^{2}} {[{(\partial_{0} φ - μ)}^{2} - {(\partial_{i} φ)}^{2}]}^{2} .

6-

\partial_{ν} (n_{0} v^{ν}) = 0, \partial_{ρ} T_{0}^{ρ σ} = 0 .

7-

n_{0} = {\frac{d P}{d μ} |}_{μ = \bar{μ}} = \frac{3}{π^{2}} {\bar{μ}}^{3}

8-

\bar{μ} = {(D_{ρ} \bar{φ} D^{ρ} \bar{φ})}^{1 / 2}

9-

v_{ρ} = - \frac{D_{ρ} \bar{φ}}{\bar{μ}},

10-

T_{0}^{ρ σ} = (n_{0} \bar{μ}) v^{ρ} v^{σ} - g^{ρ σ} P_{0} = (ρ_{0} + P_{0}) v^{ρ} v^{σ} - η^{ρ σ} P_{0},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1606.08690

Formulas:

1-

M_{n} = 2^{n} - 1

2-

ω (M_{n}) \leq 3

3-

ω (M_{n}) > 2^{(1 - ϵ) \log \log n} - 3

4-

M_{n} = 2^{n} - 1

5-

n \neq 1, 2, 6

6-

d = \gcd (m, n)

7-

\gcd (M_{m}, M_{n}) = M_{d}

8-

m = n q + r

9-

2^{m} - 2^{n q} + 2^{n q} - 1

10-

2^{n q} \cdot 2^{r} - 2^{n q} + 2^{n q} - 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1404.4444

Formulas:

1-

J H K_{s}

2-

J H K_{s}

3-

J_{0} = {15.348}_{- 0.208}^{+ 0.206}

4-

H_{0} = {14.703}_{- 0.180}^{+ 0.176}

5-

K_{s}_{0} = {14.534}_{- 0.146}^{+ 0.142}

6-

{(V - K_{s})}_{0}

7-

J H K_{s}

8-

J H K_{s}

9-

J H K_{s}

10-

J H K_{s}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0902.4094

Formulas:

1-

- d / 2 \leq x \leq d / 2

2-

{\hat{H}}_{sd} = - (J / 2) \hat{𝝈} \cdot \hat{𝐒}

3-

\hat{𝐒} (x) = (0, - \sin θ, \cos θ)

4-

x < - d / 2

5-

(π / d) (x + d / 2)

6-

- d / 2 < x < d / 2

7-

𝐞_{x} = α^{- 1} \hat{𝐒} \times (\partial \hat{𝐒} / \partial x)

8-

𝐞_{y} = - α^{- 1} \partial \hat{𝐒} / \partial x

9-

α = d θ / d x ≪ 2 π / λ_{F}

10-

\hat{f} (x, p_{x}) = [f (x, p_{x}) \hat{1} + 𝐠 (x, p_{x}) \cdot \hat{𝝈}] / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0502148

Formulas:

1-

T_{c} \approx E_{F} \exp [- π / {(2 k_{F} a_{s c})}^{3}]

2-

U (r) = \sum_{i} U_{0, i} \cos^{2} (k_{i} x_{i})

3-

U_{0, z} ≫ \min {U_{0, x}, U_{0, y}}

4-

x_{i} = x, y, or z

5-

k_{i} = 2 π / λ_{i}

6-

H = \sum_{𝐤} ξ (𝐤) a_{𝐤 ↑}^{†} a_{𝐤 ↑} + \frac{1}{2} \sum_{𝐤, 𝐤^{'}, 𝐪} V (𝐤, 𝐤^{'}) b_{𝐤, 𝐪}^{†} b_{𝐤^{'}, 𝐪},

7-

b_{𝐤, 𝐪}^{†} = a_{𝐤 + 𝐪 / 2, ↑}^{†} a_{- 𝐤 + 𝐪 / 2, ↑}^{†}

8-

ξ (𝐤) = ε (𝐤) - \tilde{μ}

9-

ε (𝐤) = - t_{x} \cos k_{x} a_{x} - t_{y} \cos k_{y} a_{y} - t_{z} \cos k_{z} a_{z}

10-

\tilde{μ} = μ + g μ_{B} h

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0605651

Formulas:

1-

F (t) = F Θ (t)

2-

σ (t) \propto E_{T H z} (t) / F

3-

σ_{s} (ω)

4-

σ_{s} (ω)

5-

σ_{s} (ω)

6-

σ_{s} (ω) = \frac{σ_{0}}{1 + ω_{B}^{2} τ_{e} τ_{p}} \cdot \frac{1 - ω_{B}^{2} τ_{e} τ_{p} - i ω τ_{e}}{(ω_{B}^{2} - ω^{2}) τ_{e} τ_{p} + 1 - i ω (τ_{e} + τ_{p})},

7-

\dot{v} (t)

8-

σ (ω) = σ_{0} \frac{1 - i ω τ_{e}}{(ω_{B}^{2} - ω^{2}) τ_{e} τ_{p} + 1 - i ω (τ_{e} + τ_{p})} .

9-

σ_{s} (ω)

10-

σ_{s} (0) = {d j / d F |}_{F_{0}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1506.01982

Formulas:

1-

J H K_{S}

2-

J H K_{s}

3-

(J - K_{s}) \leq 0.75

4-

(i), (i i)

5-

(i i i)

6-

d (in pc) = 10^{(K_{S} - M_{K_{S}} + 5 - A_{K_{S}}) / 5}

7-

σ (A_{V}) = \sqrt{{4.7}^{2} . σ_{J H}^{2} + {7.9}^{2} . σ_{H K_{S}}^{2} + 2 \times 37 \times c o v (J H, H K_{S})}

8-

σ_{J H}^{2} = σ_{J}^{2} + σ_{H}^{2}

9-

σ_{H K_{S}}^{2} = σ_{H}^{2} + σ_{K_{S}}^{2}

10-

c o v (J H, H K_{S}) = r_{s} \times σ_{J H} σ_{H K_{S}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1611.03859

Formulas:

1-

Σ_{g} = 1700 {(R / 1 au)}^{- 3 / 2} g {cm}^{- 2}

2-

Σ_{d} = 7.1 {(R / 1 au)}^{- 3 / 2} g {cm}^{- 2}

3-

280 {(R / 1 au)}^{- 1 / 2} K

4-

t_{L} = Ω_{K}^{- 1}

5-

t_{η} = {Re}_{t}^{- 1 / 2} t_{L}

6-

a_{0} = 2.5 nm

7-

E_{roll} = 6 π^{2} γ a_{0} ξ = 1.1 \times 10^{- 11} erg

8-

γ = 25 erg {cm}^{- 2}

9-

ρ_{hit} = {(\frac{3}{5})}^{3 / 2} N^{- 1 / 2} ρ_{0},

10-

ρ_{0} = 3 g {cm}^{- 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1811.03907

Formulas:

1-

{(A / A_{\max})}^{2}

2-

t / h = 3.4 GHz

3-

2 t < h ν_{r}

4-

t / h = 4.5 GHz

5-

2 t > h ν_{r}

6-

2 t < h ν_{r}

7-

2 t > h ν_{r}

8-

ν_{r} = 8.33 GHz

9-

κ / 2 π = 101 MHz

10-

(N_{L}, N_{R})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1409.6564

Formulas:

1-

Φ = {(Ψ_{A}^{+}, Ψ_{A}^{-}, Ψ_{B}^{+}, Ψ_{B}^{-})}^{T}

2-

Ψ_{A (B)}^{\pm}

3-

H_{𝐤} = (\begin{matrix} Δ σ_{z} & F_{𝐤} \\ F_{𝐤}^{†} & Δ σ_{z} \end{matrix}), Δ = {| x |}^{2} g_{X} μ_{B} H_{z} / 2,

4-

F_{𝐤} = - (\begin{matrix} f_{𝐤} J & f_{𝐤}^{+} δ J \\ f_{𝐤}^{-} δ J & f_{𝐤} J \end{matrix}),

5-

f_{𝐤} = \sum_{j = 1}^{3} \exp (- i {𝐤𝐝}_{φ_{j}}), f_{𝐤}^{\pm} = \sum_{j = 1}^{3} \exp (- i [{𝐤𝐝}_{φ_{j}} \mp 2 φ_{j}]),

6-

φ_{j} = 2 π (j - 1) / 3

7-

E_{g} \sim 3 δ J

8-

| Δ | \sim δ J

9-

n_{m} = \frac{1}{2 π} \iint_{BZ} 𝐁_{𝐤, m} d^{2} 𝐤,

10-

| Φ_{𝐤, m} ⟩

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1007.0796

Formulas:

1-

ℒ_{0} (𝐏_{𝟎}) \propto \exp (- χ^{2} (𝐏_{𝟎}) / 2)

2-

α = \min {1, ℒ_{1} / ℒ_{0}}

3-

N (γ) \propto γ^{- p} \exp (- γ / γ_{c})

4-

E_{c} = γ_{c} m_{e} c^{2}

5-

N (γ) \propto γ^{- p} \exp - {(γ / γ_{c})}^{1 / 2} .

6-

N (γ) \propto γ^{- p} \exp [- 9 χ (γ) / U^{2} T_{life}],

7-

χ (γ) = η {\begin{matrix} {(γ / γ_{c})}^{β} γ_{c} m_{e} c^{3} / (3 e B) & for γ \leq γ_{c}, \\ γ^{2} m_{e} c^{3} / (3 e B γ_{c}) & for γ \geq γ_{c}, \end{matrix}

8-

l_{d} = γ_{c} m_{e} c^{2} / e B

9-

η / T_{life} U^{2}

10-

η (1.6 kyr / T_{life}) {(4 M m s^{- 1} / U)}^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0707.4651

Formulas:

1-

𝐆 (𝐗 - 𝐗_{0}) \geq 0,

2-

| 𝐗 | < | 𝐗_{0} |

3-

{𝐀𝐆𝐁𝐁}^{- 1} (𝐗 - 𝐗_{0}) \geq 0,

4-

𝐆^{'} 𝐗^{'} \geq 𝐡^{'}

5-

𝐆^{'} := 𝐀𝐆𝐁 and 𝐡^{'} := {𝐀𝐆𝐗}_{0} .

6-

𝐗^{'} = 𝐁^{- 1} 𝐗

7-

\tilde{𝐡} = 𝐡 - 𝐂

8-

\tilde{𝐡} = 𝐡 + 𝐃

9-

𝐆 = [\begin{matrix} - 89.20509815216064 & 0.0000000000000000 \\ 74.79768991470337 & 0.0000000000000000 \\ 66.23740792274475 & 0.0000000000000000 \\ - 18.51919293403625 & 0.0000000000000000 \end{matrix}] 𝐡 = [\begin{matrix} - 12073.43407295207 \\ 10123.19482867013 \\ 8350.549301112449 \\ - 24612.94532321187 \end{matrix}]

10-

𝐆 = [\begin{matrix} 81.82253837585449 & 0.0000000000000000 \\ - 74.02672171592712 & 0.0000000000000000 \\ 0.0000000000000000 & - 17.36225485801697 \\ - 89.47155475616455 & 0.0000000000000000 \end{matrix}] 𝐡 = [\begin{matrix} - 77004.09890544150 \\ 69248.37468031116 \\ 11241.52852765946 \\ 84233.37742495652 \end{matrix}]

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1304.4923

Formulas:

1-

SWAP | ϕ ⟩ | ψ ⟩ = | ψ ⟩ | ϕ ⟩ .

2-

CNOT | x ⟩ | y ⟩ = | x ⟩ | x \oplus y ⟩ .

3-

| x ⟩ | y ⟩ \overset{{CNOT}_{2, 1}}{⟶} | x \oplus y ⟩ | y ⟩,

4-

| x \oplus y ⟩ | y ⟩ \overset{{CNOT}_{1, 2}}{⟶} | x \oplus y ⟩ | y \oplus x \oplus y ⟩ = | x \oplus y ⟩ | x ⟩,

5-

| x \oplus y ⟩ | x ⟩ \overset{{CNOT}_{2, 1}}{⟶} | x \oplus y \oplus x ⟩ | x ⟩ = | y ⟩ | x ⟩ .

6-

| ϕ ⟩ | ψ ⟩

7-

| ψ ⟩ | ϕ ⟩

8-

{| 0 ⟩, | 1 ⟩, \dots, | d - 1 ⟩}

9-

C \tilde{X} | x ⟩ | y ⟩ = | x ⟩ | - x - y ⟩ .

10-

| - x - y ⟩

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1309.5924

Formulas:

1-

σ_{\max} = σ (X_{\max})

2-

X_{\max} = X_{\max} (A)

3-

⟨ X_{\max} ∣ A ⟩ = C + D Log (\frac{E}{E_{0} A}) .

4-

D = \frac{d ⟨ x_{\max} ⟩}{dLog E}

5-

⟨ x_{\max} ⟩ = ⟨ X_{\max} ∣ A = 1 ⟩

6-

C = ⟨ x_{\max} ⟩ (E_{0})

7-

σ^{2} (X_{\max} ∣ A) = σ_{fr}^{2} + σ_{sh}^{2},

8-

σ_{fr}^{2} = σ_{fr}^{2} (A, E)

9-

σ_{sh}^{2} = σ_{sh}^{2} (A, E)

10-

⟨ X_{\max} ⟩ = ⟨ ⟨ X_{\max} ∣ A ⟩ ⟩ = ⟨ x_{\max} ⟩ - d ⟨ \ln A ⟩,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0312558

Formulas:

1-

ν_{e} : ν_{μ} : ν_{τ} = 1 : 2 : 0

2-

\frac{d N}{d E} \propto E^{- (2 \div 2.5)}

3-

E_{\max} \sim β Z (\frac{B}{1 μ G}) (\frac{R}{1 k p c}) 10^{18}

4-

\frac{d Φ}{d E_{ν} d Ω_{ν}}

5-

N_{μ} = \int \int 𝑑 E_{ν} 𝑑 Ω_{ν} A_{ν}^{e f f} (E_{ν}, Ω_{ν}) \frac{d Φ}{d E_{ν} d Ω_{ν}}

6-

E^{- (2 \div 2.5)}

7-

ν_{μ} + {\bar{ν}}_{μ}

8-

E_{μ m i n}

9-

ν_{μ} + {\bar{ν}}_{μ}

10-

E_{ν} \sim 6 - 10^{3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ex

Result: incorrect

Paper: https://arxiv.org/abs/0909.4772

Formulas:

1-

h \to b \bar{b}

2-

m_{\tilde{g}}, m_{\tilde{q}} > m_{{\tilde{χ}}^{0}}, m_{{\tilde{χ}}^{\pm}} > m_{{\tilde{ℓ}}_{L}} > m_{h}, m_{{\tilde{ℓ}}_{R}},

3-

\tilde{χ} \to {\tilde{ℓ}}_{R} ℓ

4-

\tilde{χ} \to {\tilde{ℓ}}_{L} ℓ \to {\tilde{ℓ}}_{R} ℓ ℓ^{'} ℓ^{''}

5-

< 1 {fb}^{- 1}

6-

h \to b \bar{b}

7-

n_{ℓ} = 4, n_{jet} \geq 2

8-

| η_{ℓ, jet} | < 2.5, p_{T}^{ℓ} > 10

9-

Δ R_{jj, ℓ ℓ, j ℓ} > 0.4

10-

ℒ = 1 {fb}^{- 1}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/0904.3014

Formulas:

1-

α = {({90.7}_{- 2.9}^{+ 4.5})}^{\circ}

2-

α = {(89.0 \pm 0.2)}^{\circ}

3-

α = {({90.7}_{- 2.9}^{+ 4.5})}^{\circ}

4-

\sim 90^{\circ} \pm 20^{\circ}

5-

\sin^{2} θ_{C} ≃ \frac{m_{d}}{m_{s}} + \frac{m_{u}}{m_{c}}

6-

K_{α i} := Re (V_{β j} V_{γ k} V_{β k}^{*} V_{γ j}^{*})

7-

𝒮 := \sum_{γ k} K_{γ k};

8-

ℛ^{'} := \sum_{γ k} (K_{α i} K_{β j} + K_{α j} K_{β i});

9-

𝒦 := Det K;

10-

3 J^{2} = \sum_{α} \sum_{i \neq j} K_{α i} K_{α j} = \sum_{i} \sum_{α \neq β} K_{α i} K_{β i} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/9906053

Formulas:

1-

F = G \frac{M m}{r (r - δ)} (1)

2-

h^{2} u^{2} (\frac{d^{2} u}{d θ^{2}} + u) = - \frac{F}{m} (2)

3-

h^{2} u^{2} (\frac{d^{2} u}{d θ^{2}} + u) = \frac{G M u^{2}}{(1 - δ u)} (4)

4-

h^{2} u^{2} (\frac{d^{2} u}{d θ^{2}} + u) = G M u^{2} (1 + δ u) (5)

5-

\frac{d^{2} u}{d θ^{2}} + (1 - D δ) u = D (6)

6-

D = \frac{G M}{h^{2}}

7-

u = A \cos \sqrt{1 - D δ} θ + B \sin \sqrt{1 - D δ} θ + \frac{D}{1 - D δ} (5)

8-

A = - \frac{D}{1 - D δ}

9-

y = r \sin θ

10-

\frac{1}{y} = \frac{D}{1 - D δ} \frac{1 - \cos \sqrt{1 - D δ} θ}{\sin θ} + B \frac{\sin \sqrt{1 - D δ} θ}{\sin θ} (6)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/physics/0308111

Formulas:

1-

\nabla^{2} 𝐄 + k^{2} 𝐄 = 0,

2-

𝐌_{n m}^{(1, 2)} (k r)

3-

N_{n} h_{n}^{(1, 2)} (k r) 𝐂_{n m} (θ, ϕ)

4-

𝐍_{n m}^{(1, 2)} (k r)

5-

\frac{h_{n}^{(1, 2)} (k r)}{k r N_{n}} 𝐏_{n m} (θ, ϕ) +

6-

N_{n} (h_{n - 1}^{(1, 2)} (k r) - \frac{n h_{n}^{(1, 2)} (k r)}{k r}) 𝐁_{n m} (θ, ϕ)

7-

h_{n}^{(1, 2)} (k r)

8-

N_{n} = 1 / \sqrt{n (n + 1)}

9-

𝐁_{n m} (θ, ϕ)

10-

𝐂_{n m} (θ, ϕ)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0904.4029

Formulas:

1-

w_{d e} < - 1

2-

w_{d e} = - 1

3-

ω (z) = - 1

4-

r = a (τ)

5-

d S_{5 \pm}^{2} = - (k - \frac{η_{\pm}}{r^{2}} + \frac{r^{2}}{ℓ^{2}}) d t^{2} + \frac{1}{k - \frac{η_{\pm}}{r^{2}} + \frac{r^{2}}{ℓ^{2}}} d r^{2} + r^{2} γ_{i j} d x^{i} d x^{j},

6-

k = - 1, 0, 1

7-

\frac{{\dot{a}}^{2}}{a^{2}} + \frac{k}{a^{2}} = \frac{ρ}{3} + \frac{η}{a^{4}} + \frac{ℓ^{2}}{36} ρ^{2},

8-

\frac{\ddot{a}}{a} = - \frac{ρ}{6} (1 + 3 w) - \frac{η}{a^{4}} - \frac{ℓ^{2}}{36} ρ^{2} (2 + 3 w),

9-

η_{+} = η_{-} \equiv η

10-

S_{φ} = \int d^{4} x \sqrt{- g} [\frac{1}{k_{4}^{2}} α (φ) R [g] - \frac{1}{2} g^{μ ν} \nabla_{μ} φ \nabla_{ν} φ - V (φ)] .

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1807.09276

Formulas:

1-

R_{NLR} \sim 10^{1 - 4}

2-

\sim 10^{4 - 8} M_{⊙}

3-

\sim 10^{2 - 4} M_{⊙}

4-

L_{AGN} \propto M_{BH} \propto M_{host}

5-

M_{⋆} > 10^{11} M_{⊙}

6-

λ_{obs} = 4900 - 9500

7-

\log (M_{⋆} / M_{⊙}) < 11

8-

\log (M_{⋆} / M_{⊙}) < 11.0

9-

11.0 < \log (M_{⋆} / M_{⊙}) < 11.5

10-

\log (M_{⋆} / M_{⊙}) > 11.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1401.1265

Formulas:

1-

R = λ / δ λ = 40, 000

2-

{(g - r)}_{0}

3-

g r i z

4-

g r i z

5-

g r i z

6-

E (B - V)

7-

E (B - V)

8-

\log (E W / λ) < - 4.7

9-

\log (E W / λ) < - 4.7

10-

\log (E W / λ) \leq - 4.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1803.09445

Formulas:

1-

a = b q^{m}

2-

U = U (a, b; q) = \frac{{(- a; q)}_{\infty} {(b; q)}_{\infty} - {(a; q)}_{\infty} {(- b; q)}_{\infty}}{{(- a; q)}_{\infty} {(b; q)}_{\infty} + {(a; q)}_{\infty} {(- b; q)}_{\infty}} =

3-

= \frac{a - b}{1 - q +} \frac{(a - b q) (a q - b)}{1 - q^{3} +} \frac{q (a - b q^{2}) (a q^{2} - b)}{1 - q^{5} +} \frac{q^{2} (a - b q^{3}) (a q^{3} - b)}{1 - q^{7} +} \dots

4-

{(- 1 + \frac{2}{1 - U (a, b; q)})}^{2} = \frac{(- 1 + \frac{2}{1 - u_{0} (a, q)})}{(- 1 + \frac{2}{1 - u_{0} (b, q)})},

5-

\frac{P - 1}{P + 1} = u_{0} (a, q) .

6-

P := {(\frac{{(- a; q)}_{\infty}}{{(a; q)}_{\infty}})}^{2} .

7-

\log (- 1 + \frac{2}{1 - u_{0} (a, q)}) = \log P

8-

| a / q | < 1

9-

u_{0} (a, q) = \frac{2 a}{1 - q +} \frac{a^{2} {(1 + q)}^{2}}{1 - q^{3} +} \frac{a^{2} q {(1 + q^{2})}^{2}}{1 - q^{5} +} \frac{a^{2} q^{2} {(1 + q^{3})}^{2}}{1 - q^{7} +} \dots

10-

\log P = \log (- 1 + \frac{2}{1 - u_{0} (A, q)}) = 4 \sum_{n = 0}^{\infty} \frac{A^{2 n + 1}}{(2 n + 1) (1 - q^{2 n + 1})} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/math/0611616

Formulas:

1-

Γ = (V, E)

2-

δ_{X} (v)

3-

Γ = (V, E)

4-

δ_{S} (v) \geq δ_{\bar{S}} (v) + k,

5-

γ_{k}^{a} (Γ)

6-

γ_{k}^{a} (Γ)

7-

Γ = (V, E)

8-

N_{X} (v)

9-

N_{X} (v) := {u \in X : u \sim v},

10-

δ_{X} (v) = | N_{X} (v) | .

Formulas (html):

Correct Categorie: math

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/hep-lat/0106017

Formulas:

1-

R = \sum_{x, μ} {Tr}_{A} [{}^{g}U_{μ} (x)]

2-

{}^{g}U_{μ} (x)

3-

R_{M} = \sum_{x, μ} Tr [M^{T} (x) U_{A μ} (x) M (x + \hat{μ})]

4-

U_{A μ} (x)

5-

\frac{1}{𝒱} \sum_{x} \sum_{j} M_{i j}^{T} (x) M_{j k} (x) = δ_{i k}

6-

g_{A} (x)

7-

U_{μ} (x)

8-

g (x) g^{†} (x + \hat{μ})

9-

\frac{1}{4 𝒱} \sum_{x, μ} Tr [(U_{μ} (x) - g (x) g^{†} (x + \hat{μ})) {(U_{μ} (x) - g (x) g^{†} (x + \hat{μ}))}^{†}]

10-

\frac{1}{4 𝒱} \sum_{x, μ} 2 Tr [I - g^{†} (x) U_{μ} (x) g (x + \hat{μ})]

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/hep-th/9809023

Formulas:

1-

{(n p)}^{- α}, α \geq 1

2-

S_{gf} = \frac{1}{2} \int d^{d} x n_{μ} A_{μ}^{a} \frac{1}{ξ n^{2}} n_{ν} A_{ν}^{a}

3-

\frac{1}{4} \int d^{d} x F_{μ ν}^{a} F_{μ ν}^{a}

4-

\partial_{μ} A_{ν}^{a} - \partial_{ν} A_{μ}^{a} + g f_{b c}^{a} A_{μ}^{b} A^{c}_{ν}

5-

δ^{a b} \partial_{μ} + g f^{a c b} A_{μ}^{c}, [t^{b}, t^{c}] = f_{a}^{b c} t^{a} .

6-

a_{1} \frac{δ_{μ ν}}{p^{2}} + a_{2} \frac{p_{μ} p_{ν}}{p^{4}} + a_{3} \frac{n_{μ} p_{ν} + n_{ν} p_{μ}}{p^{2} (n p)} + a_{4} \frac{n_{μ} n_{ν}}{n^{2} p^{2}} .

7-

a_{1, \dots, 4}

8-

S = S_{A} + S_{gf}

9-

(1 + ξ p^{2}) / s^{2}

10-

{(n p)}^{2} / n^{2} p^{2} .

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/2002.00176

Formulas:

1-

θ (e_{i})

2-

τ (e_{i})

3-

(e_{i}, e_{j})

4-

{rc}_{i, j, 1}, \dots, {rc}_{i, j, n}

5-

(e_{i}, e_{j})

6-

{rs}_{i, j, k}

7-

{rc}_{i, j, k}

8-

(e_{i}, e_{j})

9-

{sc}_{i, j} = f (θ (e_{i}), τ (e_{j})),

10-

θ (e_{i})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1010.3474

Formulas:

1-

v \sin i = 13.9 \pm 0.03

2-

λ = 5_{- 5}^{+ 6}

3-

\sqrt{1 - e^{2}} \cos i

4-

g (x) = \frac{1}{\sqrt{2 π} s} e^{- \frac{x^{2}}{2 s^{2}}}

5-

f (x) = \frac{6 ((1 - u) \sqrt{1 - x^{2}} - π u (x^{2} - 1) / 4)}{π (u - 3)}

6-

- 1 < x < 1

7-

T_{e f f} = 6500

8-

\log g_{*} = 4.5

9-

[M / H] = 0

10-

T_{e f f} = 6400 \pm 100

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1407.0248

Formulas:

1-

M_{Ch} \approx 1.39 M_{⊙}

2-

M_{u} + Δ M_{u}^{hy}

3-

M_{i} > M_{u} + Δ M_{u}^{hy}

4-

α_{MLT} = Λ / λ_{P} = 1.91

5-

D_{CBM} (r) = D_{MLT} (r_{0}) \exp (- \frac{| r - r_{0} |}{f λ_{P}}),

6-

D_{MLT} (r_{0}) = (Λ v_{conv}) / 3

7-

M_{i} = 6.3 M_{⊙}

8-

{\dot{M}}_{acc} > {\dot{M}}_{stable}

9-

{\dot{M}}_{acc} > {\dot{M}}_{RG}

10-

{\dot{M}}_{acc} > {\dot{M}}_{RG}

Formulas (html):

<math><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><msub id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml"><mi id="S1.p1.1.m1.1.1.2.2" xref="S1.p1.1.m1.1.1.2.2.cmml">M</mi><mi id="S1.p1.1.m1.1.1.2.3" xref="S1.p1.1.m1.1.1.2.3.cmml">Ch</mi></msub><mo id="S1.p1.1.m1.1.1.1" xref="S1.p1.1.m1.1.1.1.cmml">≈</mo><mrow id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml"><mpadded width="+1.7pt" id="S1.p1.1.m1.1.1.3.2" xref="S1.p1.1.m1.1.1.3.2.cmml"><mn id="S1.p1.1.m1.1.1.3.2a" xref="S1.p1.1.m1.1.1.3.2.cmml">1.39</mn></mpadded><mo id="S1.p1.1.m1.1.1.3.1" xref="S1.p1.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S1.p1.1.m1.1.1.3.3" xref="S1.p1.1.m1.1.1.3.3.cmml"><mi id="S1.p1.1.m1.1.1.3.3.2" xref="S1.p1.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S1.p1.1.m1.1.1.3.3.3" xref="S1.p1.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><msub id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml"><mi id="S1.p5.2.m2.1.1.2.2" xref="S1.p5.2.m2.1.1.2.2.cmml">M</mi><mi mathvariant="normal" id="S1.p5.2.m2.1.1.2.3" xref="S1.p5.2.m2.1.1.2.3.cmml">u</mi></msub><mo id="S1.p5.2.m2.1.1.1" xref="S1.p5.2.m2.1.1.1.cmml">+</mo><mrow id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml"><mi mathvariant="normal" id="S1.p5.2.m2.1.1.3.2" xref="S1.p5.2.m2.1.1.3.2.cmml">Δ</mi><mo id="S1.p5.2.m2.1.1.3.1" xref="S1.p5.2.m2.1.1.3.1.cmml">⁢</mo><msubsup id="S1.p5.2.m2.1.1.3.3" xref="S1.p5.2.m2.1.1.3.3.cmml"><mi id="S1.p5.2.m2.1.1.3.3.2.2" xref="S1.p5.2.m2.1.1.3.3.2.2.cmml">M</mi><mi mathvariant="normal" id="S1.p5.2.m2.1.1.3.3.2.3" xref="S1.p5.2.m2.1.1.3.3.2.3.cmml">u</mi><mi id="S1.p5.2.m2.1.1.3.3.3" xref="S1.p5.2.m2.1.1.3.3.3.cmml">hy</mi></msubsup></mrow></mrow></math>, <math><mrow id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml"><msub id="S1.p5.4.m4.1.1.2" xref="S1.p5.4.m4.1.1.2.cmml"><mi id="S1.p5.4.m4.1.1.2.2" xref="S1.p5.4.m4.1.1.2.2.cmml">M</mi><mi mathvariant="normal" id="S1.p5.4.m4.1.1.2.3" xref="S1.p5.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S1.p5.4.m4.1.1.1" xref="S1.p5.4.m4.1.1.1.cmml">></mo><mrow id="S1.p5.4.m4.1.1.3" xref="S1.p5.4.m4.1.1.3.cmml"><msub id="S1.p5.4.m4.1.1.3.2" xref="S1.p5.4.m4.1.1.3.2.cmml"><mi id="S1.p5.4.m4.1.1.3.2.2" xref="S1.p5.4.m4.1.1.3.2.2.cmml">M</mi><mi mathvariant="normal" id="S1.p5.4.m4.1.1.3.2.3" xref="S1.p5.4.m4.1.1.3.2.3.cmml">u</mi></msub><mo id="S1.p5.4.m4.1.1.3.1" xref="S1.p5.4.m4.1.1.3.1.cmml">+</mo><mrow id="S1.p5.4.m4.1.1.3.3" xref="S1.p5.4.m4.1.1.3.3.cmml"><mi mathvariant="normal" id="S1.p5.4.m4.1.1.3.3.2" xref="S1.p5.4.m4.1.1.3.3.2.cmml">Δ</mi><mo id="S1.p5.4.m4.1.1.3.3.1" xref="S1.p5.4.m4.1.1.3.3.1.cmml">⁢</mo><msubsup id="S1.p5.4.m4.1.1.3.3.3" xref="S1.p5.4.m4.1.1.3.3.3.cmml"><mi id="S1.p5.4.m4.1.1.3.3.3.2.2" xref="S1.p5.4.m4.1.1.3.3.3.2.2.cmml">M</mi><mi mathvariant="normal" id="S1.p5.4.m4.1.1.3.3.3.2.3" xref="S1.p5.4.m4.1.1.3.3.3.2.3.cmml">u</mi><mi id="S1.p5.4.m4.1.1.3.3.3.3" xref="S1.p5.4.m4.1.1.3.3.3.3.cmml">hy</mi></msubsup></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><msub id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2.2" xref="S2.SS1.p2.3.m3.1.1.2.2.cmml">α</mi><mi id="S2.SS1.p2.3.m3.1.1.2.3" xref="S2.SS1.p2.3.m3.1.1.2.3.cmml">MLT</mi></msub><mo id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">=</mo><mrow id="S2.SS1.p2.3.m3.1.1.4" xref="S2.SS1.p2.3.m3.1.1.4.cmml"><mi mathvariant="normal" id="S2.SS1.p2.3.m3.1.1.4.2" xref="S2.SS1.p2.3.m3.1.1.4.2.cmml">Λ</mi><mo id="S2.SS1.p2.3.m3.1.1.4.1" xref="S2.SS1.p2.3.m3.1.1.4.1.cmml">/</mo><msub id="S2.SS1.p2.3.m3.1.1.4.3" xref="S2.SS1.p2.3.m3.1.1.4.3.cmml"><mi id="S2.SS1.p2.3.m3.1.1.4.3.2" xref="S2.SS1.p2.3.m3.1.1.4.3.2.cmml">λ</mi><mi id="S2.SS1.p2.3.m3.1.1.4.3.3" xref="S2.SS1.p2.3.m3.1.1.4.3.3.cmml">P</mi></msub></mrow><mo id="S2.SS1.p2.3.m3.1.1.5" xref="S2.SS1.p2.3.m3.1.1.5.cmml">=</mo><mn id="S2.SS1.p2.3.m3.1.1.6" xref="S2.SS1.p2.3.m3.1.1.6.cmml">1.91</mn></mrow></math>, <math><mrow id="S2.E1.m1.4.4.1" xref="S2.E1.m1.4.4.1.1.cmml"><mrow id="S2.E1.m1.4.4.1.1" xref="S2.E1.m1.4.4.1.1.cmml"><mrow id="S2.E1.m1.4.4.1.1.4" xref="S2.E1.m1.4.4.1.1.4.cmml"><msub id="S2.E1.m1.4.4.1.1.4.2" xref="S2.E1.m1.4.4.1.1.4.2.cmml"><mi id="S2.E1.m1.4.4.1.1.4.2.2" xref="S2.E1.m1.4.4.1.1.4.2.2.cmml">D</mi><mi id="S2.E1.m1.4.4.1.1.4.2.3" xref="S2.E1.m1.4.4.1.1.4.2.3.cmml">CBM</mi></msub><mo id="S2.E1.m1.4.4.1.1.4.1" xref="S2.E1.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.1.1.4.3.2" xref="S2.E1.m1.4.4.1.1.4.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.1.1.4.3.2.1" xref="S2.E1.m1.4.4.1.1.4.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">r</mi><mo stretchy="false" id="S2.E1.m1.4.4.1.1.4.3.2.2" xref="S2.E1.m1.4.4.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.4.1.1.3" xref="S2.E1.m1.4.4.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.4.4.1.1.2" xref="S2.E1.m1.4.4.1.1.2.cmml"><msub id="S2.E1.m1.4.4.1.1.2.4" xref="S2.E1.m1.4.4.1.1.2.4.cmml"><mi id="S2.E1.m1.4.4.1.1.2.4.2" xref="S2.E1.m1.4.4.1.1.2.4.2.cmml">D</mi><mi id="S2.E1.m1.4.4.1.1.2.4.3" xref="S2.E1.m1.4.4.1.1.2.4.3.cmml">MLT</mi></msub><mo id="S2.E1.m1.4.4.1.1.2.3" xref="S2.E1.m1.4.4.1.1.2.3.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.1.1.1.1.1.2" xref="S2.E1.m1.4.4.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.4.4.1.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.4.4.1.1.1.1.1.1.2" xref="S2.E1.m1.4.4.1.1.1.1.1.1.2.cmml">r</mi><mn id="S2.E1.m1.4.4.1.1.1.1.1.1.3" xref="S2.E1.m1.4.4.1.1.1.1.1.1.3.cmml">0</mn></msub><mo stretchy="false" id="S2.E1.m1.4.4.1.1.1.1.1.3" xref="S2.E1.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.4.4.1.1.2.3a" xref="S2.E1.m1.4.4.1.1.2.3.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.1.1.2.2.1" xref="S2.E1.m1.4.4.1.1.2.2.2.cmml"><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">exp</mi><mo id="S2.E1.m1.4.4.1.1.2.2.1a" xref="S2.E1.m1.4.4.1.1.2.2.2.cmml">⁡</mo><mrow id="S2.E1.m1.4.4.1.1.2.2.1.1" xref="S2.E1.m1.4.4.1.1.2.2.2.cmml"><mo id="S2.E1.m1.4.4.1.1.2.2.1.1.2" xref="S2.E1.m1.4.4.1.1.2.2.2.cmml">(</mo><mrow id="S2.E1.m1.4.4.1.1.2.2.1.1.1" xref="S2.E1.m1.4.4.1.1.2.2.1.1.1.cmml"><mo id="S2.E1.m1.4.4.1.1.2.2.1.1.1.1" xref="S2.E1.m1.4.4.1.1.2.2.1.1.1.1.cmml">-</mo><mstyle displaystyle="true" id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml"><mfrac id="S2.E1.m1.1.1a" xref="S2.E1.m1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.2.cmml"><mo stretchy="false" id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.2.1.cmml">|</mo><mrow id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.2.cmml">r</mi><mo id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml">-</mo><msub id="S2.E1.m1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.3.cmml"><mi id="S2.E1.m1.1.1.1.1.1.3.2" xref="S2.E1.m1.1.1.1.1.1.3.2.cmml">r</mi><mn id="S2.E1.m1.1.1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo stretchy="false" id="S2.E1.m1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S2.E1.m1.1.1.3" xref="S2.E1.m1.1.1.3.cmml"><mi id="S2.E1.m1.1.1.3.2" xref="S2.E1.m1.1.1.3.2.cmml">f</mi><mo id="S2.E1.m1.1.1.3.1" xref="S2.E1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.E1.m1.1.1.3.3" xref="S2.E1.m1.1.1.3.3.cmml"><mi id="S2.E1.m1.1.1.3.3.2" xref="S2.E1.m1.1.1.3.3.2.cmml">λ</mi><mi id="S2.E1.m1.1.1.3.3.3" xref="S2.E1.m1.1.1.3.3.3.cmml">P</mi></msub></mrow></mfrac></mstyle></mrow><mo id="S2.E1.m1.4.4.1.1.2.2.1.1.3" xref="S2.E1.m1.4.4.1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E1.m1.4.4.1.2" xref="S2.E1.m1.4.4.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p2.4.m1.2.2" xref="S2.SS1.p2.4.m1.2.2.cmml"><mrow id="S2.SS1.p2.4.m1.1.1.1" xref="S2.SS1.p2.4.m1.1.1.1.cmml"><msub id="S2.SS1.p2.4.m1.1.1.1.3" xref="S2.SS1.p2.4.m1.1.1.1.3.cmml"><mi id="S2.SS1.p2.4.m1.1.1.1.3.2" xref="S2.SS1.p2.4.m1.1.1.1.3.2.cmml">D</mi><mi id="S2.SS1.p2.4.m1.1.1.1.3.3" xref="S2.SS1.p2.4.m1.1.1.1.3.3.cmml">MLT</mi></msub><mo id="S2.SS1.p2.4.m1.1.1.1.2" xref="S2.SS1.p2.4.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m1.1.1.1.1.1" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p2.4.m1.1.1.1.1.1.2" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p2.4.m1.1.1.1.1.1.1" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.4.m1.1.1.1.1.1.1.2" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.2.cmml">r</mi><mn id="S2.SS1.p2.4.m1.1.1.1.1.1.1.3" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo stretchy="false" id="S2.SS1.p2.4.m1.1.1.1.1.1.3" xref="S2.SS1.p2.4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.4.m1.2.2.3" xref="S2.SS1.p2.4.m1.2.2.3.cmml">=</mo><mrow id="S2.SS1.p2.4.m1.2.2.2" xref="S2.SS1.p2.4.m1.2.2.2.cmml"><mrow id="S2.SS1.p2.4.m1.2.2.2.1.1" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p2.4.m1.2.2.2.1.1.2" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.4.m1.2.2.2.1.1.1" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S2.SS1.p2.4.m1.2.2.2.1.1.1.2" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.2.cmml">Λ</mi><mo id="S2.SS1.p2.4.m1.2.2.2.1.1.1.1" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p2.4.m1.2.2.2.1.1.1.3" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.3.cmml"><mi id="S2.SS1.p2.4.m1.2.2.2.1.1.1.3.2" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.3.2.cmml">v</mi><mi id="S2.SS1.p2.4.m1.2.2.2.1.1.1.3.3" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.3.3.cmml">conv</mi></msub></mrow><mo stretchy="false" id="S2.SS1.p2.4.m1.2.2.2.1.1.3" xref="S2.SS1.p2.4.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p2.4.m1.2.2.2.2" xref="S2.SS1.p2.4.m1.2.2.2.2.cmml">/</mo><mn id="S2.SS1.p2.4.m1.2.2.2.3" xref="S2.SS1.p2.4.m1.2.2.2.3.cmml">3</mn></mrow></mrow></math>, <math><mrow id="S2.F1.2.1.m1.1.1" xref="S2.F1.2.1.m1.1.1.cmml"><msub id="S2.F1.2.1.m1.1.1.2" xref="S2.F1.2.1.m1.1.1.2.cmml"><mi id="S2.F1.2.1.m1.1.1.2.2" xref="S2.F1.2.1.m1.1.1.2.2.cmml">M</mi><mi mathvariant="normal" id="S2.F1.2.1.m1.1.1.2.3" xref="S2.F1.2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.F1.2.1.m1.1.1.1" xref="S2.F1.2.1.m1.1.1.1.cmml">=</mo><mrow id="S2.F1.2.1.m1.1.1.3" xref="S2.F1.2.1.m1.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.F1.2.1.m1.1.1.3.2" xref="S2.F1.2.1.m1.1.1.3.2.cmml"><mn id="S2.F1.2.1.m1.1.1.3.2b" xref="S2.F1.2.1.m1.1.1.3.2.cmml">6.3</mn></mpadded><mo id="S2.F1.2.1.m1.1.1.3.1" xref="S2.F1.2.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.F1.2.1.m1.1.1.3.3" xref="S2.F1.2.1.m1.1.1.3.3.cmml"><mi id="S2.F1.2.1.m1.1.1.3.3.2" xref="S2.F1.2.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S2.F1.2.1.m1.1.1.3.3.3" xref="S2.F1.2.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml"><msub id="S2.SS2.p1.3.m3.1.1.2" xref="S2.SS2.p1.3.m3.1.1.2.cmml"><mover accent="true" id="S2.SS2.p1.3.m3.1.1.2.2" xref="S2.SS2.p1.3.m3.1.1.2.2.cmml"><mi id="S2.SS2.p1.3.m3.1.1.2.2.2" xref="S2.SS2.p1.3.m3.1.1.2.2.2.cmml">M</mi><mo id="S2.SS2.p1.3.m3.1.1.2.2.1" xref="S2.SS2.p1.3.m3.1.1.2.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p1.3.m3.1.1.2.3" xref="S2.SS2.p1.3.m3.1.1.2.3.cmml">acc</mi></msub><mo id="S2.SS2.p1.3.m3.1.1.1" xref="S2.SS2.p1.3.m3.1.1.1.cmml">></mo><msub id="S2.SS2.p1.3.m3.1.1.3" xref="S2.SS2.p1.3.m3.1.1.3.cmml"><mover accent="true" id="S2.SS2.p1.3.m3.1.1.3.2" xref="S2.SS2.p1.3.m3.1.1.3.2.cmml"><mi id="S2.SS2.p1.3.m3.1.1.3.2.2" xref="S2.SS2.p1.3.m3.1.1.3.2.2.cmml">M</mi><mo id="S2.SS2.p1.3.m3.1.1.3.2.1" xref="S2.SS2.p1.3.m3.1.1.3.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p1.3.m3.1.1.3.3" xref="S2.SS2.p1.3.m3.1.1.3.3.cmml">stable</mi></msub></mrow></math>, <math><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><msub id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml"><mover accent="true" id="S2.SS2.p2.2.m2.1.1.2.2" xref="S2.SS2.p2.2.m2.1.1.2.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2.2.2" xref="S2.SS2.p2.2.m2.1.1.2.2.2.cmml">M</mi><mo id="S2.SS2.p2.2.m2.1.1.2.2.1" xref="S2.SS2.p2.2.m2.1.1.2.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p2.2.m2.1.1.2.3" xref="S2.SS2.p2.2.m2.1.1.2.3.cmml">acc</mi></msub><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">></mo><msub id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml"><mover accent="true" id="S2.SS2.p2.2.m2.1.1.3.2" xref="S2.SS2.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.3.2.2" xref="S2.SS2.p2.2.m2.1.1.3.2.2.cmml">M</mi><mo id="S2.SS2.p2.2.m2.1.1.3.2.1" xref="S2.SS2.p2.2.m2.1.1.3.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p2.2.m2.1.1.3.3" xref="S2.SS2.p2.2.m2.1.1.3.3.cmml">RG</mi></msub></mrow></math>, <math><mrow id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml"><msub id="S2.SS2.p2.4.m4.1.1.2" xref="S2.SS2.p2.4.m4.1.1.2.cmml"><mover accent="true" id="S2.SS2.p2.4.m4.1.1.2.2" xref="S2.SS2.p2.4.m4.1.1.2.2.cmml"><mi id="S2.SS2.p2.4.m4.1.1.2.2.2" xref="S2.SS2.p2.4.m4.1.1.2.2.2.cmml">M</mi><mo id="S2.SS2.p2.4.m4.1.1.2.2.1" xref="S2.SS2.p2.4.m4.1.1.2.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p2.4.m4.1.1.2.3" xref="S2.SS2.p2.4.m4.1.1.2.3.cmml">acc</mi></msub><mo id="S2.SS2.p2.4.m4.1.1.1" xref="S2.SS2.p2.4.m4.1.1.1.cmml">></mo><msub id="S2.SS2.p2.4.m4.1.1.3" xref="S2.SS2.p2.4.m4.1.1.3.cmml"><mover accent="true" id="S2.SS2.p2.4.m4.1.1.3.2" xref="S2.SS2.p2.4.m4.1.1.3.2.cmml"><mi id="S2.SS2.p2.4.m4.1.1.3.2.2" xref="S2.SS2.p2.4.m4.1.1.3.2.2.cmml">M</mi><mo id="S2.SS2.p2.4.m4.1.1.3.2.1" xref="S2.SS2.p2.4.m4.1.1.3.2.1.cmml">˙</mo></mover><mi id="S2.SS2.p2.4.m4.1.1.3.3" xref="S2.SS2.p2.4.m4.1.1.3.3.cmml">RG</mi></msub></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0707.4211

Formulas:

1-

\sim 2^{\circ} \times 2^{\circ}

2-

Δ ξ_{cc} \sim \frac{4 q}{{(s - s^{- 1})}^{2}},

3-

Δ ξ_{cc} \propto {\begin{matrix} s^{- 2} & for s ≫ 1.0, \\ s^{2} & for s ≪ 1.0 . \end{matrix}

4-

𝐫_{pc} = 𝐬 (1 - \frac{1}{s^{2}}) .

5-

sign (𝐫_{pc}) = sign (𝐬)

6-

sign (𝐫_{pc}) = - sign (𝐬)

7-

Δ ξ_{pc} \propto {\begin{matrix} q^{1 / 2} {[s {(s^{2} - 1)}^{1 / 2}]}^{- 1} & for s > 1, \\ q^{1 / 2} (κ_{0} - κ_{0}^{- 1} + κ_{0} s^{- 2}) \cos θ_{0} & for s < 1, \end{matrix}

8-

κ (θ) = {[\cos 2 θ \pm {(s^{4} - \sin^{2} 2 θ)}^{1 / 2}] / (s^{2} - s^{- 2})}^{1 / 2}

9-

θ_{0} = [π \pm \sin^{- 1} (3^{1 / 2} s^{2} / 2)] / 2

10-

κ_{0} = κ (θ_{0})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1311.1856

Formulas:

1-

E (S) = S^{T} U + S^{T} M S, S \in {0, 1}^{Ω}

2-

S = {s_{p} | p \in Ω}

3-

U = {u_{p} \in ℛ | p \in Ω}

4-

M = {m_{p q} \in ℛ | p, q \in Ω}

5-

m_{p q} = 0

6-

\min_{S \in {0, 1}^{Ω}} E (S) .

7-

m_{p q} \leq 0 \forall (p, q)

8-

S_{i n t}

9-

S_{i n t}

10-

S_{r l x}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/math/0301162

Formulas:

1-

ℱ \to ℋ o m (ℋ o m (ℱ, ω), ω)

2-

D^{'} \sim D + m H

3-

ω = {Ext}_{P}^{r} (A, P)

4-

M \to {Hom}_{A} ({Hom}_{A} (M, ω), ω)

5-

M \to {Hom}_{A} ({Hom}_{A} (M, ω), ω)

6-

{Hom}_{A} (M, ω)

7-

H_{𝔪}^{0} (M) = M

8-

ω_{A} = {Ext}_{P}^{r} (A, P)

9-

{Ext}_{P}^{r} (\cdot, P)

10-

dim A / x A < dim A

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9906319

Formulas:

1-

α = 13^{h} 38^{m} {07.11}^{s}

2-

δ = - 80^{\circ} 29^{'} 48.2 "

3-

ℓ^{I I} = 304.942

4-

b^{I I} = - 17.8035

5-

F_{ν} \propto t^{- 0.85}

6-

F_{ν} \propto t^{- 1.3}

7-

F_{ν} \propto t^{- 1.8}

8-

F_{ν} \propto t^{- 2.5}

9-

S (ϕ) = \frac{\frac{I (ϕ) / I (ϕ + 90^{\circ})}{⟨ I_{u} (ϕ) / I_{u} (ϕ + 90^{\circ}) ⟩} - 1}{\frac{I (ϕ) / I (ϕ + 90^{\circ})}{⟨ I_{u} (ϕ) / I_{u} (ϕ + 90^{\circ}) ⟩} + 1}

10-

I (ϕ + 90^{\circ})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1203.5662

Formulas:

1-

Y_{L} = N_{L} / N_{H}

2-

\frac{d Y_{L}^{I S M}}{d t}

3-

F_{p, a}^{G C R} σ_{p a + C N O} Y_{C N O}^{I S M}

4-

F_{C N O}^{G C R} σ_{p a + C N O} Y_{p, a}^{I S M} P_{L}

5-

F_{a}^{G C R} σ_{a + a} Y_{a}^{I S M} P_{L},

6-

Y^{I S M}

7-

F_{C N O}^{G C R} \propto Y_{C N O}^{G C R}

8-

F_{p}^{G C R} \sim

9-

σ_{p, a + C N O ⟶ B e} \sim

10-

Y_{C N O} \sim

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1607.01828

Formulas:

1-

[0 - 9] +

2-

𝚃𝚁𝙰𝙽𝚂𝙻𝙰𝚃𝙴 (s, ’ 023456789 ’, ’ 111111111 ’)

3-

𝚁𝙴𝙿𝙻𝙰𝙲𝙴 (𝚁𝙴𝙿𝙻𝙰𝙲𝙴 (s, ’ 1^{ℓ_{1}} ’, ’ 1^{r_{1}} ’), ’ 1^{ℓ_{2}} ’, ’ 1^{r_{2}} ’),

4-

1 \leq ℓ_{1} \leq 3

5-

1^{2 r_{1} + 1}

6-

1^{2 r_{1} + 1}

7-

ℓ_{2} = 2 r_{1} + 1 \geq 3

8-

1^{2 r_{1} + 1}

9-

ℓ_{2} = 2, r_{2} = 1

10-

𝚁𝙴𝙿𝙻𝙰𝙲𝙴 (s, ’ 11 ’, ’ 1 ’)

Formulas (html):

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id="S2.Ex2.m1.2.2.1.1.3.3.3.3.3.3" xref="S2.Ex2.m1.2.2.1.1.3.3.3.3.3.3.cmml">2</mn></msub></msup><mo id="S2.Ex2.m1.2.2.1.1.3.3.3.1a" xref="S2.Ex2.m1.2.2.1.1.3.3.3.1.cmml">⁢</mo><mtext id="S2.Ex2.m1.2.2.1.1.3.3.3.4" xref="S2.Ex2.m1.2.2.1.1.3.3.3.4a.cmml">’</mtext></mrow><mo stretchy="false" id="S2.Ex2.m1.2.2.1.1.3.3.7" xref="S2.Ex2.m1.2.2.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.2.2.1.2" xref="S2.Ex2.m1.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.p7.7.m7.1.1" xref="S2.p7.7.m7.1.1.cmml"><mn id="S2.p7.7.m7.1.1.2" xref="S2.p7.7.m7.1.1.2.cmml">1</mn><mo id="S2.p7.7.m7.1.1.3" xref="S2.p7.7.m7.1.1.3.cmml">≤</mo><msub id="S2.p7.7.m7.1.1.4" xref="S2.p7.7.m7.1.1.4.cmml"><mi mathvariant="normal" id="S2.p7.7.m7.1.1.4.2" xref="S2.p7.7.m7.1.1.4.2.cmml">ℓ</mi><mn id="S2.p7.7.m7.1.1.4.3" xref="S2.p7.7.m7.1.1.4.3.cmml">1</mn></msub><mo id="S2.p7.7.m7.1.1.5" xref="S2.p7.7.m7.1.1.5.cmml">≤</mo><mn id="S2.p7.7.m7.1.1.6" xref="S2.p7.7.m7.1.1.6.cmml">3</mn></mrow></math>, <math><msup id="S2.p9.5.m5.1.1" xref="S2.p9.5.m5.1.1.cmml"><mn id="S2.p9.5.m5.1.1.2" xref="S2.p9.5.m5.1.1.2.cmml">1</mn><mrow id="S2.p9.5.m5.1.1.3" xref="S2.p9.5.m5.1.1.3.cmml"><mrow id="S2.p9.5.m5.1.1.3.2" xref="S2.p9.5.m5.1.1.3.2.cmml"><mn id="S2.p9.5.m5.1.1.3.2.2" xref="S2.p9.5.m5.1.1.3.2.2.cmml">2</mn><mo id="S2.p9.5.m5.1.1.3.2.1" xref="S2.p9.5.m5.1.1.3.2.1.cmml">⁢</mo><msub id="S2.p9.5.m5.1.1.3.2.3" xref="S2.p9.5.m5.1.1.3.2.3.cmml"><mi id="S2.p9.5.m5.1.1.3.2.3.2" xref="S2.p9.5.m5.1.1.3.2.3.2.cmml">r</mi><mn id="S2.p9.5.m5.1.1.3.2.3.3" xref="S2.p9.5.m5.1.1.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.p9.5.m5.1.1.3.1" xref="S2.p9.5.m5.1.1.3.1.cmml">+</mo><mn id="S2.p9.5.m5.1.1.3.3" xref="S2.p9.5.m5.1.1.3.3.cmml">1</mn></mrow></msup></math>, <math><msup id="S2.p9.11.m11.1.1" xref="S2.p9.11.m11.1.1.cmml"><mn id="S2.p9.11.m11.1.1.2" xref="S2.p9.11.m11.1.1.2.cmml">1</mn><mrow id="S2.p9.11.m11.1.1.3" xref="S2.p9.11.m11.1.1.3.cmml"><mrow id="S2.p9.11.m11.1.1.3.2" xref="S2.p9.11.m11.1.1.3.2.cmml"><mn id="S2.p9.11.m11.1.1.3.2.2" xref="S2.p9.11.m11.1.1.3.2.2.cmml">2</mn><mo id="S2.p9.11.m11.1.1.3.2.1" xref="S2.p9.11.m11.1.1.3.2.1.cmml">⁢</mo><msub id="S2.p9.11.m11.1.1.3.2.3" xref="S2.p9.11.m11.1.1.3.2.3.cmml"><mi id="S2.p9.11.m11.1.1.3.2.3.2" xref="S2.p9.11.m11.1.1.3.2.3.2.cmml">r</mi><mn id="S2.p9.11.m11.1.1.3.2.3.3" xref="S2.p9.11.m11.1.1.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.p9.11.m11.1.1.3.1" xref="S2.p9.11.m11.1.1.3.1.cmml">+</mo><mn id="S2.p9.11.m11.1.1.3.3" xref="S2.p9.11.m11.1.1.3.3.cmml">1</mn></mrow></msup></math>, <math><mrow id="S2.p9.14.m14.1.1" xref="S2.p9.14.m14.1.1.cmml"><msub id="S2.p9.14.m14.1.1.2" xref="S2.p9.14.m14.1.1.2.cmml"><mi mathvariant="normal" id="S2.p9.14.m14.1.1.2.2" xref="S2.p9.14.m14.1.1.2.2.cmml">ℓ</mi><mn id="S2.p9.14.m14.1.1.2.3" xref="S2.p9.14.m14.1.1.2.3.cmml">2</mn></msub><mo id="S2.p9.14.m14.1.1.3" xref="S2.p9.14.m14.1.1.3.cmml">=</mo><mrow id="S2.p9.14.m14.1.1.4" xref="S2.p9.14.m14.1.1.4.cmml"><mrow id="S2.p9.14.m14.1.1.4.2" xref="S2.p9.14.m14.1.1.4.2.cmml"><mn id="S2.p9.14.m14.1.1.4.2.2" xref="S2.p9.14.m14.1.1.4.2.2.cmml">2</mn><mo id="S2.p9.14.m14.1.1.4.2.1" xref="S2.p9.14.m14.1.1.4.2.1.cmml">⁢</mo><msub id="S2.p9.14.m14.1.1.4.2.3" xref="S2.p9.14.m14.1.1.4.2.3.cmml"><mi id="S2.p9.14.m14.1.1.4.2.3.2" xref="S2.p9.14.m14.1.1.4.2.3.2.cmml">r</mi><mn id="S2.p9.14.m14.1.1.4.2.3.3" xref="S2.p9.14.m14.1.1.4.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.p9.14.m14.1.1.4.1" xref="S2.p9.14.m14.1.1.4.1.cmml">+</mo><mn id="S2.p9.14.m14.1.1.4.3" xref="S2.p9.14.m14.1.1.4.3.cmml">1</mn></mrow><mo id="S2.p9.14.m14.1.1.5" xref="S2.p9.14.m14.1.1.5.cmml">≥</mo><mn id="S2.p9.14.m14.1.1.6" xref="S2.p9.14.m14.1.1.6.cmml">3</mn></mrow></math>, <math><msup id="S2.p9.15.m15.1.1" xref="S2.p9.15.m15.1.1.cmml"><mn id="S2.p9.15.m15.1.1.2" xref="S2.p9.15.m15.1.1.2.cmml">1</mn><mrow id="S2.p9.15.m15.1.1.3" xref="S2.p9.15.m15.1.1.3.cmml"><mrow id="S2.p9.15.m15.1.1.3.2" xref="S2.p9.15.m15.1.1.3.2.cmml"><mn id="S2.p9.15.m15.1.1.3.2.2" xref="S2.p9.15.m15.1.1.3.2.2.cmml">2</mn><mo id="S2.p9.15.m15.1.1.3.2.1" xref="S2.p9.15.m15.1.1.3.2.1.cmml">⁢</mo><msub id="S2.p9.15.m15.1.1.3.2.3" xref="S2.p9.15.m15.1.1.3.2.3.cmml"><mi id="S2.p9.15.m15.1.1.3.2.3.2" xref="S2.p9.15.m15.1.1.3.2.3.2.cmml">r</mi><mn id="S2.p9.15.m15.1.1.3.2.3.3" xref="S2.p9.15.m15.1.1.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.p9.15.m15.1.1.3.1" xref="S2.p9.15.m15.1.1.3.1.cmml">+</mo><mn id="S2.p9.15.m15.1.1.3.3" xref="S2.p9.15.m15.1.1.3.3.cmml">1</mn></mrow></msup></math>, <math><mrow id="S2.p10.3.m3.2.2.2" xref="S2.p10.3.m3.2.2.3.cmml"><mrow id="S2.p10.3.m3.1.1.1.1" xref="S2.p10.3.m3.1.1.1.1.cmml"><msub id="S2.p10.3.m3.1.1.1.1.2" xref="S2.p10.3.m3.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S2.p10.3.m3.1.1.1.1.2.2" xref="S2.p10.3.m3.1.1.1.1.2.2.cmml">ℓ</mi><mn id="S2.p10.3.m3.1.1.1.1.2.3" xref="S2.p10.3.m3.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S2.p10.3.m3.1.1.1.1.1" xref="S2.p10.3.m3.1.1.1.1.1.cmml">=</mo><mn id="S2.p10.3.m3.1.1.1.1.3" xref="S2.p10.3.m3.1.1.1.1.3.cmml">2</mn></mrow><mo id="S2.p10.3.m3.2.2.2.3" xref="S2.p10.3.m3.2.2.3a.cmml">,</mo><mrow id="S2.p10.3.m3.2.2.2.2" xref="S2.p10.3.m3.2.2.2.2.cmml"><msub id="S2.p10.3.m3.2.2.2.2.2" xref="S2.p10.3.m3.2.2.2.2.2.cmml"><mi id="S2.p10.3.m3.2.2.2.2.2.2" xref="S2.p10.3.m3.2.2.2.2.2.2.cmml">r</mi><mn id="S2.p10.3.m3.2.2.2.2.2.3" xref="S2.p10.3.m3.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p10.3.m3.2.2.2.2.1" xref="S2.p10.3.m3.2.2.2.2.1.cmml">=</mo><mn id="S2.p10.3.m3.2.2.2.2.3" xref="S2.p10.3.m3.2.2.2.2.3.cmml">1</mn></mrow></mrow></math>, <math><mrow id="S2.Ex3.m1.3.3" xref="S2.Ex3.m1.3.3.cmml"><mtext id="S2.Ex3.m1.3.3.4" xref="S2.Ex3.m1.3.3.4a.cmml">𝚁𝙴𝙿𝙻𝙰𝙲𝙴</mtext><mo id="S2.Ex3.m1.3.3.3" xref="S2.Ex3.m1.3.3.3.cmml">⁢</mo><mrow id="S2.Ex3.m1.3.3.2.2" xref="S2.Ex3.m1.3.3.2.3.cmml"><mo stretchy="false" id="S2.Ex3.m1.3.3.2.2.3" xref="S2.Ex3.m1.3.3.2.3.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">s</mi><mo id="S2.Ex3.m1.3.3.2.2.4" xref="S2.Ex3.m1.3.3.2.3.cmml">,</mo><mrow id="S2.Ex3.m1.2.2.1.1.1" xref="S2.Ex3.m1.2.2.1.1.1.cmml"><mtext id="S2.Ex3.m1.2.2.1.1.1.2" xref="S2.Ex3.m1.2.2.1.1.1.2a.cmml">’</mtext><mo id="S2.Ex3.m1.2.2.1.1.1.1" xref="S2.Ex3.m1.2.2.1.1.1.1.cmml">⁢</mo><mn id="S2.Ex3.m1.2.2.1.1.1.3" xref="S2.Ex3.m1.2.2.1.1.1.3.cmml">11</mn><mo id="S2.Ex3.m1.2.2.1.1.1.1a" xref="S2.Ex3.m1.2.2.1.1.1.1.cmml">⁢</mo><mtext id="S2.Ex3.m1.2.2.1.1.1.4" xref="S2.Ex3.m1.2.2.1.1.1.4a.cmml">’</mtext></mrow><mo id="S2.Ex3.m1.3.3.2.2.5" xref="S2.Ex3.m1.3.3.2.3.cmml">,</mo><mrow id="S2.Ex3.m1.3.3.2.2.2" xref="S2.Ex3.m1.3.3.2.2.2.cmml"><mtext id="S2.Ex3.m1.3.3.2.2.2.2" xref="S2.Ex3.m1.3.3.2.2.2.2a.cmml">’</mtext><mo id="S2.Ex3.m1.3.3.2.2.2.1" xref="S2.Ex3.m1.3.3.2.2.2.1.cmml">⁢</mo><mn id="S2.Ex3.m1.3.3.2.2.2.3" xref="S2.Ex3.m1.3.3.2.2.2.3.cmml">1</mn><mo id="S2.Ex3.m1.3.3.2.2.2.1a" xref="S2.Ex3.m1.3.3.2.2.2.1.cmml">⁢</mo><mtext id="S2.Ex3.m1.3.3.2.2.2.4" xref="S2.Ex3.m1.3.3.2.2.2.4a.cmml">’</mtext></mrow><mo stretchy="false" id="S2.Ex3.m1.3.3.2.2.6" xref="S2.Ex3.m1.3.3.2.3.cmml">)</mo></mrow></mrow></math>

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1703.03799

Formulas:

1-

\deg_{G} (v)

2-

v_{1} e_{1} v_{2} e_{2} \dots v_{k} e_{k} v_{k + 1}

3-

v_{k + 1} = v_{1}

4-

K = K_{1} # K_{2}

5-

K_{0} # K = K

6-

K # K_{0} = K

7-

2 k = (k - 2) l

8-

b l u e

9-

x \in {r, b}

10-

N_{x} (v)

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/9605019

Formulas:

1-

- G {\vec{S}}^{†} \cdot \vec{S},

2-

\sum_{j, m > 0} {[a_{m}^{†} a_{\bar{m}}^{†}]}_{M}^{T = 1},

3-

S O (5)

4-

| Ω, 𝒩, T, T_{z} ⟩

5-

T_{z} = 1 / 2 (N - Z)

6-

𝒩_{p p} = \frac{S_{1}^{†} S_{1}}{Ω}, 𝒩_{n n} = \frac{S_{- 1}^{†} S_{- 1}}{Ω}, 𝒩_{n p} = \frac{S_{0}^{†} S_{0}}{Ω},

7-

S U (3)

8-

S O (5)

9-

⟨ 𝒩_{n p} ⟩

10-

\frac{(𝒩 - T) (1 - \frac{𝒩 - T - 3}{2 Ω})}{2 T + 3},

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/0704.2465

Formulas:

1-

A d A d A

2-

M_{4} = M_{3} \times ℜ

3-

I [g_{μ ν}, A_{μ}] = \frac{1}{16 π G} \int d^{3} x [\sqrt{- g} (R + \frac{2}{l^{2}} - \frac{1}{4} F_{μ ν} F^{μ ν}) - α ϵ^{μ ν ρ} A_{μ} \partial_{ν} A_{ρ}] .

4-

- d t^{2} - 4 α r d t d φ + [2 r - (α^{2} l^{2} - 1) \frac{2 r^{2}}{l^{2}}] d φ^{2} + {(2 r + (α^{2} l^{2} + 1) \frac{2 r^{2}}{l^{2}})}^{- 1} d r^{2}

5-

\sqrt{α^{2} l^{2} - 1} \frac{2 r}{l} d φ .

6-

{g_{μ ν}, ψ_{μ}}

7-

{A_{μ}, λ}

8-

δ λ = F^{μ ν} γ_{μ ν} ϵ .

9-

γ_{0} ϵ = 0

10-

{A_{μ}, ϕ, λ}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/2007.02053

Formulas:

1-

θ_{x y} \approx 80

2-

λ_{s} θ_{x y} \sim 8 – 10

3-

P 2_{1} / m

4-

P m n 2_{1}

5-

λ_{s} \sim 1 μ

6-

λ_{s} θ_{x y} \sim 0.1 – 0.2

7-

= \sum_{i, s} (Δ + 4 m_{d} + δ) c_{i, s}^{†} c_{i, s} - \sum_{⟨ i j ⟩, s} (m_{p} + m_{d}) c_{i, s}^{†} c_{j, s}

8-

+ \sum_{i, s} (Δ - 4 m_{d} - δ) d_{i, s}^{†} d_{i, s} - \sum_{⟨ i j ⟩, s} (m_{p} - m_{d}) d_{i, s}^{†} d_{j, s}

9-

- \sum_{⟨ i j ⟩, s} \frac{β}{2} ({\hat{𝒍}}_{i j} \cdot \hat{𝒚}) c_{i, s}^{†} d_{j, s} + \sum_{i, s} η c_{i, s}^{†} d_{i, s}

10-

- \sum_{⟨ i j ⟩} \sum_{s s^{'}} \frac{i}{2} (𝚲_{s s^{'}} \times {\hat{𝒍}}_{i j}) \cdot (\hat{𝒚} + \hat{𝒛}) c_{i, s}^{†} d_{j, s^{'}} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1902.00413

Formulas:

1-

H_{0} (\hat{𝐤}) = (\begin{matrix} H (\hat{𝐤}) & 0 \\ 0 & 𝒯 H (\hat{𝐤}) 𝒯^{- 1} \end{matrix}) .

2-

(ψ_{A, ↑}, ψ_{A, ↓}, ψ_{B, ↑}, ψ_{B, ↓})

3-

H (\hat{𝐤}) = v {\hat{k}}_{x} s_{x} + v {\hat{k}}_{y} s_{y} + κ {\hat{𝐤}}^{2} s_{z}

4-

v \equiv ℏ v_{F}

5-

\hat{𝐤} = ({\hat{k}}_{x}, {\hat{k}}_{y}) \equiv - i (\partial_{x}, \partial_{y})

6-

𝒔 = (s_{x}, s_{y}, s_{z})

7-

κ | 𝐤 | ≪ v

8-

𝒯 = i s_{y} 𝒞

9-

μ_{L} μ_{R} < 0

10-

B_{L} B_{R} > 0

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1705.10759

Formulas:

1-

- \nabla^{2} h + h = C,

2-

h (ρ, φ)

3-

0 < φ < 2 π

4-

h = C + R (ρ) Φ (φ)

5-

h (ρ, φ) = C + \sum_{m = 0}^{\infty} (A_{m} \cos m φ + B_{m} \sin m φ) J_{m} (ρ),

6-

J_{m} (ρ)

7-

h (ρ, φ) \approx C + A_{0} J_{0} (ρ) + A_{2} J_{2} (ρ) \cos 2 φ + A_{4} J_{4} (ρ) \cos 4 φ + A_{6} J_{6} (ρ) \cos 6 φ,

8-

(C, A_{0}, A_{2}, A_{4}, A_{6})

9-

h_{m a x}

10-

h (\frac{w_{x}}{2}, 0) = 0, h (\frac{w_{y}}{2}, \frac{π}{2}) = 0, h (0, 0) = h_{m a x}, \frac{\partial h}{\partial x} (\frac{w_{x}}{2}, 0) = θ_{r}, \frac{\partial h}{\partial y} (\frac{w_{y}}{2}, \frac{π}{2}) = θ_{a},

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1708.02771

Formulas:

1-

Δ e = (D_{0} - k_{B} T \ln g)

2-

k_{B} T \ln g

3-

= ℋ_{0} + ℋ_{nn} + ℋ_{nnn}

4-

ℋ_{0} = (\frac{D_{0}}{2} - \frac{T}{2} \ln g) \sum_{i} s_{i}

5-

ℋ_{nn} = \frac{k_{1}}{2} \sum_{⟨ i, j ⟩} {[⟨ 𝒓_{i} - 𝒓_{j} ⟩ - (R_{i} + R_{j})]}^{2}

6-

ℋ_{nnn} = \frac{k_{2}}{2} \sum_{⟨ ⟨ i, j ⟩ ⟩} {[⟨ 𝒓_{i} - 𝒓_{j} ⟩ - \sqrt{2} (R_{i} + R_{j})]}^{2} .

7-

N = 2 L^{2}

8-

⟨ ⟨ i, j ⟩ ⟩

9-

W = ℋ + P V

10-

ℋ + P l^{3}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1902.00975

Formulas:

1-

K = {q_{0}, q_{1}, q_{2}, q_{3}}

2-

Σ = {#, a, 0, 1}

3-

(K \cup {h}) \times {R, L, N}^{2} \times Σ^{2}

4-

δ (q_{0}, a, #)

5-

(q_{1}, R, R, a, 0)

6-

δ (q_{0}, a, 0)

7-

(q_{0}, R, R, a, 0)

8-

δ (q_{1}, a, #)

9-

(q_{2}, R, L, a, 0)

10-

δ (q_{1}, b, #)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/2003.02771

Formulas:

1-

𝐒 = 1 / 2 (𝚿^{†} σ 𝚿)

2-

σ_{x, y, z}

3-

𝚿 = (Ψ_{+}, Ψ_{-})

4-

\begin{matrix} i ℏ \partial_{t} Ψ_{\pm} = (α_{1} {| Ψ_{\pm} |}^{2} + α_{2} {| Ψ_{\mp} |}^{2} + α_{r} n) Ψ_{\pm} + \\ + \frac{i ℏ}{2} (R n - γ_{c}) Ψ_{\pm} + \frac{δ_{l}}{2} Ψ_{\mp} + \sqrt{D} η_{\pm} (t), \end{matrix}

5-

𝚿 = (Ψ_{+}, Ψ_{-})

6-

η_{\pm} (t)

7-

D = ⟨ η_{i} (t) η_{j}^{*} (t^{'}) ⟩ = 1 / 2 R n δ (t - t^{'}) δ_{i j}

8-

i, j = +, -

9-

\partial_{t} n = P - γ_{r} n - R n ({| Ψ_{+} |}^{2} + {| Ψ_{-} |}^{2}),

10-

Q_{x} (0)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9901099

Formulas:

1-

{(x / R_{x})}^{2} + {(y / R_{y})}^{2} + {(z / R_{z})}^{2} = 1

2-

R_{x} = R_{y} \neq R_{z}

3-

R_{s y s} = {(R_{x} R_{y} R_{z})}^{1 / 3}

4-

ℛ = R_{x} / R_{z}

5-

E_{d e f}

6-

E_{d e f}

7-

𝐋^{2} / 2 J_{x} \equiv L_{x}^{2} / 2 J_{x}

8-

τ_{r o t} = J_{x} / L_{x}

9-

τ_{r o t} ≪ τ_{c}

10-

0 \leq ψ \leq 2 π

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

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