Run 11332329

Paper: https://arxiv.org/abs/astro-ph/0212274

Formulas:

1-

0.006 < f_{s a t} < 0.07

2-

R \sim 5 h^{- 1}

3-

0.006 < f_{s a t} < 0.07

4-

τ = τ_{5} {(ν / 5 GHz)}^{α}

5-

τ = τ_{5} {(ν / 5 GHz)}^{α}

6-

\log (f_{o b s} / f_{m o d})

7-

f_{m o d}

8-

f_{o b s}

9-

\log (f_{o b s} / f_{m o d})

10-

f_{o b s}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0803.3883

Formulas:

1-

𝐱 = [r_{1}, p_{1}, r_{2}, p_{2}, \dots, r_{n}, p_{n}] = Tr \hat{𝐱} ϱ

2-

Σ_{i j} = Tr \frac{{\hat{𝐱}}_{𝐢} {\hat{𝐱}}_{𝐣} + {\hat{𝐱}}_{𝐣} {\hat{𝐱}}_{𝐢}}{2} ϱ - 𝐱_{i} 𝐱_{j}

3-

W (\vec{r}, \vec{p}) = \int d^{n} q ⟨ \vec{r} - \frac{\vec{q}}{2} | ϱ | \vec{x} + \frac{\vec{q}}{2} ⟩ e^{\frac{i}{ℏ} \vec{p} \vec{q}}

4-

W (x) = \frac{1}{\sqrt{π^{n} det Σ}} e^{- \frac{1}{2} (x - 𝐱) Σ^{- 1} (x - 𝐱)} .

5-

\dot{𝐱} = 𝒮 \nabla H

6-

\frac{\partial Σ}{\partial t} = 2 (Σ ℋ 𝒮 - 𝒮 ℋ Σ),

7-

ℋ_{i j} = \frac{1}{2} \frac{\partial^{2} H}{\partial 𝐱_{i} \partial 𝐱_{j}}

8-

| α ⟩ + | β ⟩

9-

| α ⟩ ⟨ β |

10-

| β ⟩ ⟨ α |

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1411.3831

Formulas:

1-

\bar{1} \bar{1} 2

2-

(\bar{1} 10)

3-

(\bar{1} 10)

4-

(\bar{1} \bar{1} 2)

5-

[\bar{1} 10]

6-

[\bar{1} \bar{1} 2]

7-

[\bar{1} 10]

8-

(\bar{1} 10)

9-

W_{(111)} = 5.84

10-

(\bar{1} 10)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1512.05259

Formulas:

1-

T < T_{c} = 100

2-

T_{c} = 1043 K

3-

M \sim {(T - T_{c})}^{- β}

4-

Γ \sim {(T - T_{c})}^{- ξ}

5-

(- 1, + 1)

6-

x_{1}, x_{2}, \dots, x_{N}

7-

[x_{1} (t), x_{2} (t), \dots, x_{N} (t)]

8-

2 p (1 - p) K < 1

9-

2 p (1 - p) K > 1

10-

2 p (1 - p) K = 1

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1703.03780

Formulas:

1-

θ = (k_{r})

2-

k_{r} - k_{r - 1} \to \infty

3-

A S C_{θ}

4-

x = (x_{m})

5-

| x_{m} - x_{⟨ m, n ⟩} | < ε,

6-

θ = (k_{r})

7-

h_{r} = k_{r} - k_{r - 1} \to \infty

8-

I_{r} = (k_{r - 1}, k_{r}]

9-

\frac{k_{r}}{k_{r - 1}}, r \geq 1, k_{0} \neq 0

10-

N_{θ} = {x = (x_{m}) \in w : lim_{r \to \infty} \frac{1}{h_{r}} \sum_{m \in I_{r}} | x_{m} - l | = 0 for some l} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0507181

Formulas:

1-

R_{s} \sim 20 r_{*}

2-

\partial_{t} ρ + \nabla \cdot ρ 𝐮 = 0

3-

\partial_{t} ρ 𝐮 + \nabla \cdot ρ 𝐮𝐮 + \nabla p = - ρ G M / r^{2}

4-

\partial_{t} ρ ℰ + \nabla \cdot (ρ ℰ 𝐮 + p 𝐮) = - ℒ

5-

ℰ = \frac{1}{2} 𝐮^{2} + e - G M / r

6-

ρ e = p / (γ - 1)

7-

ℒ = A ρ^{β - α} p^{α} .

8-

\dot{E} \propto ρ T^{6}

9-

\dot{M} = 4 π

10-

u_{s} = \frac{γ - 1}{γ + 1}, ρ_{s} = \frac{γ + 1}{γ - 1}, p_{s} = \frac{2}{γ + 1} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9909135

Formulas:

1-

L_{X} \propto L_{B}^{m}

2-

r_{c o r e} = r_{e} / 11

3-

I_{X} = I_{o} {[1 + {(r / r_{c o r e, X})}^{2}]}^{- 3 β + 0.5}

4-

r_{c o r e, X}

5-

r_{1} = 4 r_{e}

6-

r_{2} = 6.3 r_{e}

7-

k T = μ m_{p} σ^{2},

8-

L_{X} = 37.39 ergs s^{- 1}

9-

\frac{e r g s^{- 1}}{L_{⊙}}

10-

L_{g r a v}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0305222

Formulas:

1-

R (t) = R_{0} + Δ R (t)

2-

Δ R (t) = - p \int_{0}^{t} [V_{rad} (τ) - ⟨ V_{rad} ⟩] 𝑑 τ

3-

⟨ V_{rad} ⟩ = P_{0}^{- 1} \int_{0}^{P_{0}} V_{rad} (τ) 𝑑 τ

4-

\log T_{eff} = f [{(V - K)}_{0}, [Fe / H], \log g]

5-

\log T_{eff} = - 0.131 {(V - K)}_{0} - 0.1 B C + 3.9474

6-

g = G M / R^{2} - p d V_{rad} / d t

7-

{(V - K)}_{0}

8-

R_{V} = 3.07 + 0.28 {(B - V)}_{0} + 0.04 E_{B - V}

9-

R_{I} = 1.82 + 0.205 {(B - V)}_{0} + 0.022 E_{B - V}

10-

A_{V} = R_{V} E_{B - V}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0704.1179

Formulas:

1-

1 - 8 n m

2-

λ_{N N 2}

3-

h_{0} > 2 n m

4-

λ_{N N 2} \propto h_{0}^{1 / 2}

5-

r_{p 2} \propto h_{0}^{1 / 2}

6-

h_{0} \leq 2 n m

7-

1 \leq h_{0} \leq 8 n m

8-

λ_{N N 1}

9-

r_{p 1} \sim h_{0}^{5 / 3}

10-

1 \leq h_{0} \leq 8 n m

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1110.5903

Formulas:

1-

M_{V} = - {8.5}_{- 1.0}^{+ 0.4}

2-

r_{h} = 210_{- 50}^{+ 60}

3-

0.3 < r - i < 0.8

4-

0.3 < {(r - i)}_{0} < 0.8

5-

41 \overset{s}{.} 2

6-

1 \overset{'}{.} 11

7-

0 \overset{'}{.} 21

8-

{(m - M)}_{0}

9-

- {8.5}_{- 1.0}^{+ 0.4}

10-

ψ = {\begin{matrix} 10^{a (m - m_{TRGB}) + b} & m - m_{TRGB} \geq 0, \\ 10^{c (m - m_{TRGB})} & m - m_{TRGB} < 0 . \end{matrix}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0606197

Formulas:

1-

ρ_{DM} \overset{<}{\sim} 3 \times 10^{- 16} kg / m^{3}

2-

μ (x) = x / {(1 + x^{2})}^{1 / 2}

3-

ρ_{DM} \sim 0.2 \times 10^{- 21} kg / m^{3}

4-

μ (| 𝐠 | / a_{0}) 𝐠 = 𝐠_{N}

5-

μ (x) = x

6-

μ (x) = 1

7-

g ≃ \sqrt{g_{N} a_{o}}

8-

a_{0} ≃ 1.2 \times 10^{- 10} m s^{- 2}

9-

μ (x) = x / \sqrt{1 + x^{2}},

10-

μ (x) = x / (1 + x) .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1004.4899

Formulas:

1-

= 0 % - 50 %

2-

\sim 5 % - 12 %

3-

208 km s^{- 1}

4-

N_{*} = 2 \times 10^{5}

5-

N_{gas} = 4 \times 10^{4}

6-

ρ_{d} (R, z) = \frac{M_{d}}{4 π h^{2} z_{0}} \exp (- R / h) {sech}^{2} (\frac{z}{z_{0}}),

7-

ρ_{h} (r) = \frac{M_{h}}{2 π^{3 / 2}} \frac{α}{r_{c}} \frac{\exp (- r^{2} / r_{c}^{2})}{r^{2} + γ^{2}},

8-

α = {1 - \sqrt{π} q \exp (q^{2}) [1 - \erf (q)]}^{- 1}

9-

q = γ / r_{c}

10-

\sim 5 % - 7 %

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0901.1019

Formulas:

1-

10^{- 7} - 10^{- 6}

2-

J_{K_{a} K_{c}}

3-

J_{K_{- 1} K_{+ 1}}

4-

1.8 - 27 \times 10^{13}

5-

0.095 - 2.4 \times 10^{13}

6-

10^{- 4} - 10^{- 3}

7-

(1 - 5) \times 10^{- 4}

8-

Y_{pd} \times I_{UV} \times σ_{gr} \times f_{x} = 4.57 \times 10^{4} \times {(\frac{T}{m_{x}})}^{\frac{1}{2}} \times σ_{gr} \times n_{x}^{g}

9-

f_{x} = \frac{n_{x}^{i}}{n^{i}}

10-

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

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1310.7464

Formulas:

1-

T_{e f f}

2-

ω = \frac{\sqrt{l (l + 1)} \int_{0}^{R} N \frac{d r}{r}}{π (k + l / 2 + α_{g})} .

3-

L P C O D E

4-

P_{h a n d l e r} (s)

5-

P_{k o t a k} (s)

6-

ϕ = \frac{1}{n} \sum (| P_{m o d} - P_{o b s} |) .

7-

P_{m o d}

8-

P_{o b s}

9-

T_{e f f}

10-

P_{o b s} (s)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0204249

Formulas:

1-

Ω_{M, 0} = 0.30 \pm 0.15

2-

Ω_{Λ, 0} = 1 - Ω_{M, 0}

3-

3.6 h_{0}^{- 1}

4-

100 km s^{- 1} {Mpc}^{- 1}

5-

Ω_{M, 0} = 0.3

6-

Ω_{Λ, 0} = 0.7

7-

\sim 100 h_{0}^{- 1} Mpc

8-

3.6 \times 10^{9} h_{0}^{- 1} M_{⊙}

9-

80 h_{0}^{- 1} Mpc

10-

\sim 200 h_{0}^{- 1} Mpc

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9910429

Formulas:

1-

z = (1.43 \pm 0.03)

2-

J^{e x p} = (0.58 \pm 0.06)

3-

z = (1.43 \pm 0.04)

4-

ℋ = J \sum_{< {𝐫𝐫}^{'} >} 𝐒_{𝐫} \cdot 𝐒_{𝐫^{'}},

5-

𝐒_{𝐫} = (S_{𝐫}^{x}, S_{𝐫}^{y}, S_{𝐫}^{z})

6-

S (𝐪, ω)

7-

S^{k} (𝐪, ω) = \sum_{𝐫, 𝐫^{'}} \exp [i 𝐪 \cdot (𝐫 - 𝐫^{'})] \int_{- \infty}^{+ \infty} \exp (i ω t) C^{k} (𝐫 - 𝐫^{'}, t) \frac{d t}{\sqrt{2 π}},

8-

C^{k} (𝐫 - 𝐫^{'}, t)

9-

k = x, y,

10-

C^{k} (𝐫 - 𝐫^{'}, t) = ⟨ S_{𝐫}^{k} (t) S_{𝐫^{'}}^{k} (0) ⟩ - ⟨ S_{𝐫}^{k} (t) ⟩ ⟨ S_{𝐫^{'}}^{k} (0) ⟩ .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0109067

Formulas:

1-

j = l - 1 / 2

2-

j = (l - 2) + 1 / 2

3-

\tilde{η} = η - 1

4-

\tilde{l} = l - 1

5-

{1^{n_{α}^{N}}} {{\tilde{f}}_{α}} {f_{α}} γ_{α} (λ_{α}, μ_{α}) {\tilde{S}}_{α} K_{α}

6-

U (Ω_{α}^{N}) \supset U (Ω_{α}^{N} / 2) \times U (2) \supset S U (3) \times S U (2) \supset

7-

{\tilde{L}}_{α} J_{α}^{N}

8-

S O (3) \times S U (2) \supset S U_{J} (2),

9-

{\tilde{S}}_{π, ν} = 0

10-

H_{s p, π [ν]}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/1101.3203

Formulas:

1-

ψ = ψ_{0} \exp (- r / Λ), r ≫ Λ,

2-

σ_{h} = σ_{h 0} \exp [- {(T_{1} / T)}^{1 / ν}], ν = 1, 2, 3 or 4;

3-

a_{B} ⩽ Λ ⩽ \infty .

4-

ψ_{2} = ψ_{20} \exp (- r / Λ_{2}), r ≫ Λ_{2},

5-

a_{2 B} ⩽ Λ_{2} ⩽ \infty .

6-

Δ σ_{h}^{(2)} = Δ σ_{h} \exp (- Δ_{L} / T) .

7-

Δ / δ ε = Δ g_{F} b^{3} ≫ 1,

8-

δ ε = {(g_{F} b^{3})}^{- 1}

9-

(g_{F} Δ) b^{3} ≫ 1 .

10-

Φ (𝐫) = | Φ | \exp (i φ (𝐫)),

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1212.1929

Formulas:

1-

R \sim \frac{1}{1 - p}

2-

b l k s i z e

3-

b l k s i z e

4-

n u m b l k s

5-

n u m b l k s

6-

n u m b l k s

7-

n u m b l k s

8-

c u r r b l k

9-

n u m b l k s \times b l k s i z e

10-

c u r r b l k

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0808.3954

Formulas:

1-

\begin{matrix} x_{n + 1} & = x_{n} + y_{n} (m o d 2 π) \\ y_{n + 1} & = (1 - ν) y_{n} + f_{0} \sin (x_{n} + y_{n}), \end{matrix}

2-

x \in [0, 2 π]

3-

ν \in [0, 1]

4-

N_{p 1} = 2 I (\frac{f_{0}}{2 π ν}) + 1,

5-

𝚪 = [0, 2 π] \times ℝ

6-

| y_{k + 1} | \leq (1 - ν) | y_{k} | + f_{0}

7-

| y_{k} | > f_{0} / ν \Rightarrow | y_{k + 1} | < | y_{k} |

8-

𝚲 = [0, 2 π] \times [- y_{m a x}, y_{m a x}]

9-

y_{m a x} = f_{0} / ν

10-

(x_{0}, y_{0}) \in 𝚲

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9401021

Formulas:

1-

ψ = 𝒜 {\prod_{i = 1}^{N} {(u - z_{i})}^{m} e^{- {| u |}^{2} / 4} ψ_{W C} (z_{1}, \dots, z_{N})}

2-

ψ_{W C} (z_{1}, \dots, z_{N})

3-

z_{i} = x_{i} - i y_{i}

4-

l_{o} = {(ℏ c / e B)}^{1 / 2}

5-

ϵ = E_{N + 1}^{i} - E_{N + 1}^{W C}

6-

E_{N + 1}^{W C}

7-

ρ_{x y} = h / ν e^{2}

8-

{| ψ |}^{2} = e^{- β Φ}

9-

Φ = - 2 m \sum_{i = 1}^{N} \ln (| u - z_{i} |) + \frac{{| u |}^{2}}{2} + \sum_{i = 1}^{N} \frac{{| z_{i} - R_{i}^{o} |}^{2}}{2}

10-

R_{i}^{o} = R_{i x}^{o} - i R_{i y}^{o}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0406465

Formulas:

1-

σ^{*} ≃ P^{\frac{1}{10}} N^{- \frac{3}{5}}

2-

E ≃ k T / a^{3} P

3-

σ = a^{2} Σ

4-

ϕ_{a v} ≃ σ N a / L

5-

(L a^{2} / σ) (k T (1 - ϕ_{a v}) l n [1 - ϕ_{a v}] / a^{3} P) ≃ k T (σ N a / L - 1) (N / P)

6-

(L a^{2} / σ) (k T ϕ_{a v}^{2} / a^{3} P) ≃ k T (σ N^{2} a / L P)

7-

ϕ_{a v} / L

8-

\frac{F}{k T} ≃ \frac{L^{2}}{a^{2} N} + \frac{a^{2} N}{L^{2}} + \frac{σ N^{2} a}{P L}

9-

L ϕ_{a v}^{2}

10-

σ = 1 / P^{\frac{1}{2}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1411.4050

Formulas:

1-

25 km s^{- 1}

2-

v_{⊙} = 220 km s^{- 1}

3-

3 \times 10^{19} cm

4-

\sim 10^{5} M_{⊙}

5-

180 km s^{- 1}

6-

T = 2 \times 10^{6} K

7-

T < 10^{4.2} K

8-

t_{delay} = 46 Myr

9-

- 200 < v_{LSR} < - 75 km s^{- 1}

10-

90 km s^{- 1}

Formulas (html):

<math><mrow id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mpadded width="+1.7pt" id="S2.p1.2.m2.1.1.2" xref="S2.p1.2.m2.1.1.2.cmml"><mn id="S2.p1.2.m2.1.1.2a" xref="S2.p1.2.m2.1.1.2.cmml">25</mn></mpadded><mo id="S2.p1.2.m2.1.1.1" xref="S2.p1.2.m2.1.1.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.p1.2.m2.1.1.3" xref="S2.p1.2.m2.1.1.3.cmml"><mi id="S2.p1.2.m2.1.1.3a" xref="S2.p1.2.m2.1.1.3.cmml">km</mi></mpadded><mo id="S2.p1.2.m2.1.1.1a" xref="S2.p1.2.m2.1.1.1.cmml">⁢</mo><msup id="S2.p1.2.m2.1.1.4" xref="S2.p1.2.m2.1.1.4.cmml"><mi mathvariant="normal" id="S2.p1.2.m2.1.1.4.2" xref="S2.p1.2.m2.1.1.4.2.cmml">s</mi><mrow id="S2.p1.2.m2.1.1.4.3" xref="S2.p1.2.m2.1.1.4.3.cmml"><mo id="S2.p1.2.m2.1.1.4.3.1" xref="S2.p1.2.m2.1.1.4.3.1.cmml">-</mo><mn id="S2.p1.2.m2.1.1.4.3.2" xref="S2.p1.2.m2.1.1.4.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><msub id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml"><mi id="S2.p1.4.m4.1.1.2.2" xref="S2.p1.4.m4.1.1.2.2.cmml">v</mi><mo id="S2.p1.4.m4.1.1.2.3" xref="S2.p1.4.m4.1.1.2.3.cmml">⊙</mo></msub><mo id="S2.p1.4.m4.1.1.1" xref="S2.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.p1.4.m4.1.1.3.2" xref="S2.p1.4.m4.1.1.3.2.cmml"><mn id="S2.p1.4.m4.1.1.3.2a" xref="S2.p1.4.m4.1.1.3.2.cmml">220</mn></mpadded><mo id="S2.p1.4.m4.1.1.3.1" xref="S2.p1.4.m4.1.1.3.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.p1.4.m4.1.1.3.3" xref="S2.p1.4.m4.1.1.3.3.cmml"><mi id="S2.p1.4.m4.1.1.3.3a" xref="S2.p1.4.m4.1.1.3.3.cmml">km</mi></mpadded><mo id="S2.p1.4.m4.1.1.3.1a" xref="S2.p1.4.m4.1.1.3.1.cmml">⁢</mo><msup id="S2.p1.4.m4.1.1.3.4" xref="S2.p1.4.m4.1.1.3.4.cmml"><mi mathvariant="normal" id="S2.p1.4.m4.1.1.3.4.2" xref="S2.p1.4.m4.1.1.3.4.2.cmml">s</mi><mrow id="S2.p1.4.m4.1.1.3.4.3" xref="S2.p1.4.m4.1.1.3.4.3.cmml"><mo id="S2.p1.4.m4.1.1.3.4.3.1" xref="S2.p1.4.m4.1.1.3.4.3.1.cmml">-</mo><mn id="S2.p1.4.m4.1.1.3.4.3.2" xref="S2.p1.4.m4.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S2.F1.4.m2.1.1" xref="S2.F1.4.m2.1.1.cmml"><mrow id="S2.F1.4.m2.1.1.2" xref="S2.F1.4.m2.1.1.2.cmml"><mn id="S2.F1.4.m2.1.1.2.2" xref="S2.F1.4.m2.1.1.2.2.cmml">3</mn><mo id="S2.F1.4.m2.1.1.2.1" xref="S2.F1.4.m2.1.1.2.1.cmml">×</mo><mpadded width="+1.7pt" id="S2.F1.4.m2.1.1.2.3" xref="S2.F1.4.m2.1.1.2.3.cmml"><msup id="S2.F1.4.m2.1.1.2.3b" xref="S2.F1.4.m2.1.1.2.3.cmml"><mn id="S2.F1.4.m2.1.1.2.3.2" xref="S2.F1.4.m2.1.1.2.3.2.cmml">10</mn><mn id="S2.F1.4.m2.1.1.2.3.3" xref="S2.F1.4.m2.1.1.2.3.3.cmml">19</mn></msup></mpadded></mrow><mo id="S2.F1.4.m2.1.1.1" xref="S2.F1.4.m2.1.1.1.cmml">⁢</mo><mi id="S2.F1.4.m2.1.1.3" xref="S2.F1.4.m2.1.1.3.cmml">cm</mi></mrow></math>, <math><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mi id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml"/><mo id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">∼</mo><mrow id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.p2.1.m1.1.1.3.2" xref="S2.p2.1.m1.1.1.3.2.cmml"><msup id="S2.p2.1.m1.1.1.3.2a" xref="S2.p2.1.m1.1.1.3.2.cmml"><mn id="S2.p2.1.m1.1.1.3.2.2" xref="S2.p2.1.m1.1.1.3.2.2.cmml">10</mn><mn id="S2.p2.1.m1.1.1.3.2.3" xref="S2.p2.1.m1.1.1.3.2.3.cmml">5</mn></msup></mpadded><mo id="S2.p2.1.m1.1.1.3.1" xref="S2.p2.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.p2.1.m1.1.1.3.3" xref="S2.p2.1.m1.1.1.3.3.cmml"><mi mathvariant="normal" id="S2.p2.1.m1.1.1.3.3.2" xref="S2.p2.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S2.p2.1.m1.1.1.3.3.3" xref="S2.p2.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml"><mpadded width="+1.7pt" id="S2.p4.2.m2.1.1.2" xref="S2.p4.2.m2.1.1.2.cmml"><mn id="S2.p4.2.m2.1.1.2a" xref="S2.p4.2.m2.1.1.2.cmml">180</mn></mpadded><mo id="S2.p4.2.m2.1.1.1" xref="S2.p4.2.m2.1.1.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.p4.2.m2.1.1.3" xref="S2.p4.2.m2.1.1.3.cmml"><mi id="S2.p4.2.m2.1.1.3a" xref="S2.p4.2.m2.1.1.3.cmml">km</mi></mpadded><mo id="S2.p4.2.m2.1.1.1a" xref="S2.p4.2.m2.1.1.1.cmml">⁢</mo><msup id="S2.p4.2.m2.1.1.4" xref="S2.p4.2.m2.1.1.4.cmml"><mi mathvariant="normal" id="S2.p4.2.m2.1.1.4.2" xref="S2.p4.2.m2.1.1.4.2.cmml">s</mi><mrow id="S2.p4.2.m2.1.1.4.3" xref="S2.p4.2.m2.1.1.4.3.cmml"><mo id="S2.p4.2.m2.1.1.4.3.1" xref="S2.p4.2.m2.1.1.4.3.1.cmml">-</mo><mn id="S2.p4.2.m2.1.1.4.3.2" xref="S2.p4.2.m2.1.1.4.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml"><mi id="S2.p4.3.m3.1.1.2" xref="S2.p4.3.m3.1.1.2.cmml">T</mi><mo id="S2.p4.3.m3.1.1.1" xref="S2.p4.3.m3.1.1.1.cmml">=</mo><mrow id="S2.p4.3.m3.1.1.3" xref="S2.p4.3.m3.1.1.3.cmml"><mrow id="S2.p4.3.m3.1.1.3.2" xref="S2.p4.3.m3.1.1.3.2.cmml"><mn id="S2.p4.3.m3.1.1.3.2.2" xref="S2.p4.3.m3.1.1.3.2.2.cmml">2</mn><mo id="S2.p4.3.m3.1.1.3.2.1" xref="S2.p4.3.m3.1.1.3.2.1.cmml">×</mo><mpadded width="+1.7pt" id="S2.p4.3.m3.1.1.3.2.3" xref="S2.p4.3.m3.1.1.3.2.3.cmml"><msup id="S2.p4.3.m3.1.1.3.2.3a" xref="S2.p4.3.m3.1.1.3.2.3.cmml"><mn id="S2.p4.3.m3.1.1.3.2.3.2" xref="S2.p4.3.m3.1.1.3.2.3.2.cmml">10</mn><mn id="S2.p4.3.m3.1.1.3.2.3.3" xref="S2.p4.3.m3.1.1.3.2.3.3.cmml">6</mn></msup></mpadded></mrow><mo id="S2.p4.3.m3.1.1.3.1" xref="S2.p4.3.m3.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.p4.3.m3.1.1.3.3" xref="S2.p4.3.m3.1.1.3.3.cmml">K</mi></mrow></mrow></math>, <math><mrow id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml"><mi id="S2.p4.5.m5.1.1.2" xref="S2.p4.5.m5.1.1.2.cmml">T</mi><mo id="S2.p4.5.m5.1.1.1" xref="S2.p4.5.m5.1.1.1.cmml"><</mo><mrow id="S2.p4.5.m5.1.1.3" xref="S2.p4.5.m5.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.p4.5.m5.1.1.3.2" xref="S2.p4.5.m5.1.1.3.2.cmml"><msup id="S2.p4.5.m5.1.1.3.2a" xref="S2.p4.5.m5.1.1.3.2.cmml"><mn id="S2.p4.5.m5.1.1.3.2.2" xref="S2.p4.5.m5.1.1.3.2.2.cmml">10</mn><mn id="S2.p4.5.m5.1.1.3.2.3" xref="S2.p4.5.m5.1.1.3.2.3.cmml">4.2</mn></msup></mpadded><mo id="S2.p4.5.m5.1.1.3.1" xref="S2.p4.5.m5.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.p4.5.m5.1.1.3.3" xref="S2.p4.5.m5.1.1.3.3.cmml">K</mi></mrow></mrow></math>, <math><mrow id="S2.F2.6.m3.1.1" xref="S2.F2.6.m3.1.1.cmml"><msub id="S2.F2.6.m3.1.1.2" xref="S2.F2.6.m3.1.1.2.cmml"><mi id="S2.F2.6.m3.1.1.2.2" xref="S2.F2.6.m3.1.1.2.2.cmml">t</mi><mi id="S2.F2.6.m3.1.1.2.3" xref="S2.F2.6.m3.1.1.2.3.cmml">delay</mi></msub><mo id="S2.F2.6.m3.1.1.1" xref="S2.F2.6.m3.1.1.1.cmml">=</mo><mrow id="S2.F2.6.m3.1.1.3" xref="S2.F2.6.m3.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.F2.6.m3.1.1.3.2" xref="S2.F2.6.m3.1.1.3.2.cmml"><mn id="S2.F2.6.m3.1.1.3.2b" xref="S2.F2.6.m3.1.1.3.2.cmml">46</mn></mpadded><mo id="S2.F2.6.m3.1.1.3.1" xref="S2.F2.6.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.F2.6.m3.1.1.3.3" xref="S2.F2.6.m3.1.1.3.3.cmml">Myr</mi></mrow></mrow></math>, <math><mrow id="S2.p6.3.m3.1.1" xref="S2.p6.3.m3.1.1.cmml"><mrow id="S2.p6.3.m3.1.1.2" xref="S2.p6.3.m3.1.1.2.cmml"><mo id="S2.p6.3.m3.1.1.2.1" xref="S2.p6.3.m3.1.1.2.1.cmml">-</mo><mn id="S2.p6.3.m3.1.1.2.2" xref="S2.p6.3.m3.1.1.2.2.cmml">200</mn></mrow><mo id="S2.p6.3.m3.1.1.3" xref="S2.p6.3.m3.1.1.3.cmml"><</mo><msub id="S2.p6.3.m3.1.1.4" xref="S2.p6.3.m3.1.1.4.cmml"><mi id="S2.p6.3.m3.1.1.4.2" xref="S2.p6.3.m3.1.1.4.2.cmml">v</mi><mi id="S2.p6.3.m3.1.1.4.3" xref="S2.p6.3.m3.1.1.4.3.cmml">LSR</mi></msub><mo id="S2.p6.3.m3.1.1.5" xref="S2.p6.3.m3.1.1.5.cmml"><</mo><mrow id="S2.p6.3.m3.1.1.6" xref="S2.p6.3.m3.1.1.6.cmml"><mo id="S2.p6.3.m3.1.1.6.1" xref="S2.p6.3.m3.1.1.6.1.cmml">-</mo><mrow id="S2.p6.3.m3.1.1.6.2" xref="S2.p6.3.m3.1.1.6.2.cmml"><mpadded width="+1.7pt" id="S2.p6.3.m3.1.1.6.2.2" xref="S2.p6.3.m3.1.1.6.2.2.cmml"><mn id="S2.p6.3.m3.1.1.6.2.2a" xref="S2.p6.3.m3.1.1.6.2.2.cmml">75</mn></mpadded><mo id="S2.p6.3.m3.1.1.6.2.1" xref="S2.p6.3.m3.1.1.6.2.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.p6.3.m3.1.1.6.2.3" xref="S2.p6.3.m3.1.1.6.2.3.cmml"><mi id="S2.p6.3.m3.1.1.6.2.3a" xref="S2.p6.3.m3.1.1.6.2.3.cmml">km</mi></mpadded><mo id="S2.p6.3.m3.1.1.6.2.1a" xref="S2.p6.3.m3.1.1.6.2.1.cmml">⁢</mo><msup id="S2.p6.3.m3.1.1.6.2.4" xref="S2.p6.3.m3.1.1.6.2.4.cmml"><mi mathvariant="normal" id="S2.p6.3.m3.1.1.6.2.4.2" xref="S2.p6.3.m3.1.1.6.2.4.2.cmml">s</mi><mrow id="S2.p6.3.m3.1.1.6.2.4.3" xref="S2.p6.3.m3.1.1.6.2.4.3.cmml"><mo id="S2.p6.3.m3.1.1.6.2.4.3.1" xref="S2.p6.3.m3.1.1.6.2.4.3.1.cmml">-</mo><mn id="S2.p6.3.m3.1.1.6.2.4.3.2" xref="S2.p6.3.m3.1.1.6.2.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow></mrow></math>, <math><mrow id="S3.F3.12.m4.1.1" xref="S3.F3.12.m4.1.1.cmml"><mpadded width="+1.7pt" id="S3.F3.12.m4.1.1.2" xref="S3.F3.12.m4.1.1.2.cmml"><mn id="S3.F3.12.m4.1.1.2b" xref="S3.F3.12.m4.1.1.2.cmml">90</mn></mpadded><mo id="S3.F3.12.m4.1.1.1" xref="S3.F3.12.m4.1.1.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S3.F3.12.m4.1.1.3" xref="S3.F3.12.m4.1.1.3.cmml"><mi id="S3.F3.12.m4.1.1.3b" xref="S3.F3.12.m4.1.1.3.cmml">km</mi></mpadded><mo id="S3.F3.12.m4.1.1.1b" xref="S3.F3.12.m4.1.1.1.cmml">⁢</mo><msup id="S3.F3.12.m4.1.1.4" xref="S3.F3.12.m4.1.1.4.cmml"><mi mathvariant="normal" id="S3.F3.12.m4.1.1.4.2" xref="S3.F3.12.m4.1.1.4.2.cmml">s</mi><mrow id="S3.F3.12.m4.1.1.4.3" xref="S3.F3.12.m4.1.1.4.3.cmml"><mo id="S3.F3.12.m4.1.1.4.3.1" xref="S3.F3.12.m4.1.1.4.3.1.cmml">-</mo><mn id="S3.F3.12.m4.1.1.4.3.2" xref="S3.F3.12.m4.1.1.4.3.2.cmml">1</mn></mrow></msup></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1711.04331

Formulas:

1-

𝒪 (1 - 10)

2-

ℒ \supset \frac{1}{M} {\bar{χ}}_{2} σ_{μ ν} χ_{1} F^{μ ν} - m_{1} {\bar{χ}}_{1} χ_{1} - m_{2} {\bar{χ}}_{2} χ_{2},

3-

χ_{1} γ \to χ_{2} \to χ_{1} γ

4-

E_{0} \overset{!}{=} \frac{m_{2}^{2} - m_{1}^{2}}{2 m_{1}} = 3.54 keV

5-

σ_{BW} (s) = \frac{2 π}{p_{CM}^{2}} \frac{{(m_{2} Γ)}^{2}}{{(s - m_{2}^{2})}^{2} + {(m_{2} Γ)}^{2}},

6-

p_{CM}^{2} = \frac{m_{1}^{2} E^{2}}{m_{1}^{2} + 2 m_{1} E}

7-

s ≃ m_{1}^{2} + 2 m_{1} E_{γ} (1 - β_{DM} \cos θ),

8-

β_{d m} = v_{DM} / c

9-

β_{DM} ≃ 10^{- 3}

10-

σ_{DM}^{2} ≃ 10^{- 6} c^{2}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1801.03882

Formulas:

1-

y_{t o p}

2-

Δ \geq \frac{δ_{1 - loop} m_{H}^{2}}{m_{H}^{2}} = \frac{3 y_{t o p}^{2}}{8 π^{2}} {(\frac{Λ_{SM}}{m_{H}})}^{2} ≃ {(\frac{Λ_{SM}}{450 GeV})}^{2} .

3-

SO (5) / SO (4)

4-

SO (4) ≃ SU {(2)}_{L} \times SU {(2)}_{R}

5-

SO {(3)}_{c}

6-

ξ = v^{2} / f^{2} ≲ 0.1

7-

SO (5) \to SO (4)

8-

κ_{V} = \sqrt{1 - ξ}

9-

κ_{F} = (1 - 2 ξ) / \sqrt{1 - ξ}

10-

κ_{F} = \sqrt{1 - ξ}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1310.5585

Formulas:

1-

d x / d t = b (x; α, β)

2-

x_{1}^{*} < x_{3}^{*} < x_{2}^{*}

3-

b (x, α, β) = 0

4-

b (x; α, β) = 0

5-

\vec{N} = (n_{1}, n_{2} \dots)

6-

\vec{x} (t) = \vec{N} (t) / V

7-

\vec{N}, V \to \infty

8-

d x (t) = b (x) d t + V^{- \frac{1}{2}} d B_{t}

9-

f_{V}^{s t} (x) = 𝒞_{V} e^{- V ϕ (x)},

10-

f_{V}^{s t} (x)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1711.02494

Formulas:

1-

_{[\pm 1, \pm 1]}

2-

σ = \partial E_{c l} / \partial d

3-

C_{11} C_{22} > C_{12} C_{21}

4-

Y (θ) = \frac{C_{11} C_{22} - C_{12}^{2}}{C_{22} \cos^{4} θ + A \cos^{2} θ \sin^{2} θ + C_{11} \sin^{4} θ},

5-

ν (θ) = \frac{C_{12} \cos^{4} θ - B \cos^{2} θ \sin^{2} θ + C_{12} \sin^{4} θ}{C_{22} \cos^{4} θ + A \cos^{2} θ \sin^{2} θ + C_{11} \sin^{4} θ} .

6-

A = (C_{11} C_{22} - C_{12}^{2}) / C_{66} - 2 C_{12}

7-

B = C_{11} + C_{22} - (C_{11} C_{22} - C_{12}^{2}) / C_{66}

8-

{| χ^{(2)} |}_{\max}

9-

{| χ^{(2)} |}_{\max}

10-

_{[- 1, - 1]}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/math/0611826

Formulas:

1-

F (x) = u (x) f (x) + v (x) f^{'} (x)

2-

{1, 2, \dots, n}

3-

π = a_{1} a_{2} \dots a_{n} \in S_{n}

4-

A_{n} (x)

5-

A_{0} (x) = 1, A_{n} (x) = \sum_{π \in S_{n}} x^{exc (π) + 1} for n \geq 1,

6-

A_{n + 1} (x) = (n + 1) x A_{n} (x) + x (1 - x) A_{n}^{'} (x)

7-

A_{0} (x; q) = 1, A_{n} (x; q) = \sum_{π \in 𝒮_{n}} x^{exc (π)} q^{c (π)} for n \geq 1,

8-

A_{0} (x; q) = 1, A_{1} (x; q) = q, A_{2} (x; q) = q (x + q), A_{3} (x; q) = q [x^{2} + (3 q + 1) x + q^{2}] .

9-

A_{n} (x) = x A_{n} (x; 1)

10-

A_{n + 1} (x; q) = (n x + q) A_{n} (x; q) + x (1 - x) \frac{d}{d x} A_{n} (x; q)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1311.6644

Formulas:

1-

L_{bol} \sim 10^{4} - 10^{6}

2-

E_{u} / k_{B} \sim 580 - 2400

3-

E_{u} / k_{B} = 1071

4-

L^{0.6} M_{env}^{- 1}

5-

L^{0.6} M_{env}^{- 1}

6-

9 \overset{''}{.} 4 \times 9 \overset{''}{.} 4

7-

\sim 47^{''} \times 47^{''}

8-

R = λ / Δ λ \approx

9-

J = 9 / 2 - 7 / 2

10-

1_{1}^{-} - 1_{1}^{+}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1510.08756

Formulas:

1-

U_{j} (x^{'})

2-

{\hat{U}}_{j} (k_{x})

3-

a (k_{x})

4-

(k_{x}, k_{z})

5-

{\hat{V}}_{e} (k_{x}, k_{z})

6-

𝐫 = (r_{x}, r_{y}, r_{z})

7-

𝐤 = (k_{x}, k_{y}, k_{z})

8-

V (𝐫) = \frac{k}{4 π} (n^{2} (𝐫) - n_{b}^{2}) .

9-

k = 2 π / λ

10-

n (𝐫) = n_{r} (𝐫) + i n_{i} (𝐫)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0812.1314

Formulas:

1-

\sum_{i = 1}^{k - 1} \frac{2^{i}}{2^{i} - 1} x_{i} = (- 1 + \sum_{i = 1}^{k - 1} \frac{2^{i}}{2^{i} - 1}) x_{0}

2-

\sum_{i = 0}^{k - 2} 2^{i} x_{i} = 2^{k - 1} x_{k - 1}

3-

\sum_{i = 1}^{k - 1} \frac{2^{i}}{2^{i} - 1} x_{i} = (- 1 + \sum_{i = 1}^{k - 1} \frac{2^{i}}{2^{i} - 1}) x_{0},

4-

{ord}_{2} (r)

5-

r = 2^{m} \frac{a}{b}

6-

{ord}_{2} (0) = \infty

7-

{ord}_{2} (x y) = {ord}_{2} (x) + {ord}_{2} (y)

8-

{ord}_{2} (x + y) \geq \min ({ord}_{2} (x), {ord}_{2} (y))

9-

{ord}_{2} (x)

10-

{ord}_{2} (y)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0311080

Formulas:

1-

H = - J \sum_{⟨ i j ⟩} B_{i j} - K \sum_{⟨ i j k l ⟩} P_{i j k l} - h \sum_{i} S_{i}^{z},

2-

B_{i j} = S_{i}^{+} S_{j}^{-} + S_{i}^{-} S_{j}^{+}

3-

P_{i j k l} = S_{i}^{+} S_{j}^{-} S_{k}^{+} S_{l}^{-} + S_{i}^{-} S_{j}^{+} S_{k}^{-} S_{l}^{+}

4-

⟨ i j k l ⟩

5-

T_{K T} / J \approx 0.68

6-

m = ⟨ S^{z} ⟩

7-

7.9 ≲ K / J ≲ 14.5

8-

⟨ P_{i j k l} ⟩

9-

E = - ⟨ n ⟩ / β

10-

ρ_{s} = \frac{\partial^{2} E}{\partial ϕ^{2}},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0210405

Formulas:

1-

R_{23} = ([O ii] 3727 + [O iii] 4959, 5007) / H β

2-

[N ii] 6583 / H α

3-

[N ii] 6583 / H α

4-

σ_{F} = σ_{c} D \sqrt{2 N_{p i x} + E W / D}

5-

σ_{E W} = \frac{E W}{F} σ_{c} D \sqrt{E W / D + 2 N_{p i x} + {(E W / D)}^{2} / N_{p i x}}

6-

N_{p i x}

7-

E W / σ (E W)

8-

F_{line}^{cor} = F_{line}^{obs} (1 + E W_{line}^{abs} / E W_{line}^{obs}),

9-

τ_{λ} = τ_{V} {(λ / 5500 Å)}^{- 0.7}

10-

E_{B - V}^{i n t}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/0405070

Formulas:

1-

h (t) = \frac{L_{x} (t) - L_{y} (t)}{L_{0}}

2-

L_{x} (t)

3-

L_{y} (t)

4-

h (t) = \int R (t - t^{'}) q (t^{'}) 𝑑 t^{'},

5-

h (f) = R (f) q (f),

6-

C = α (t) C_{0}

7-

D = β (t) D_{0}

8-

C = α (t) C_{0}

9-

D = β (t) D_{0}

10-

f_{p} \approx 0.75 Hz

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0804.0612

Formulas:

1-

t_{b a t h}

2-

t_{c o l l}

3-

t_{d y n} \sim 1

4-

t_{c o l l} \sim N^{δ}

5-

t_{b a t h} \sim 1 / γ

6-

H_{H M F} = \sum_{i = 1}^{N} \frac{l_{i}^{2}}{2} + \frac{1}{2 N} \sum_{i, j = 1}^{N} [1 - \cos (θ_{i} - θ_{j})],

7-

θ_{i} \in [0, 2 π)

8-

l_{i} = {\dot{θ}}_{i} \in ℝ

9-

T \equiv \sum_{i = 1}^{N} l_{i}^{2} / N

10-

m \equiv | \sum_{i = 1}^{N} (\cos θ_{i}, \sin θ_{i}) | / N

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0810.1746

Formulas:

1-

m \frac{d v}{d t} + \int_{0}^{t} 𝑑 t_{1} γ (t - t_{1}) v (t_{1}) + k x = p θ (t),

2-

x_{0} = x (0)

3-

v_{0} = v (0)

4-

\bar{x} (s) = \frac{x_{0} (m s + \bar{γ}) + m v_{0} + p / s}{m s^{2} + s \bar{γ} + k},

5-

\bar{x} (s) = (x_{0} (m s + \bar{γ}) + m v_{0} + p / s) \bar{G} (s)

6-

\bar{G} (s) = \frac{1}{m s^{2} + s \bar{γ} + k} .

7-

x_{0} = v_{0} = 0

8-

\bar{x} (s) = \frac{p}{s} \bar{G} (s) .

9-

\bar{v} (s)

10-

\bar{v} (s) = p \bar{G} (s) .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1801.10029

Formulas:

1-

U (r) = {(σ / r)}^{n}

2-

U (r) = ε {(1 - r / σ)}^{α} .

3-

U (r) = \frac{ε}{α} {(1 - r / σ)}^{α}

4-

ρ_{m i n} = 0.6

5-

ρ_{m a x} = 10

6-

10^{- 3} - 10^{- 2}

7-

ρ_{m i n}

8-

ρ_{m a x}

9-

Ψ_{6} (𝐫_{𝐢}) = \frac{1}{n (i)} \sum_{j = 1}^{n (i)} e^{i n θ_{i j}},

10-

ψ_{6} = \frac{1}{N} ⟨ | \sum_{i} Ψ_{6} (𝐫_{i}) | ⟩ .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1012.1658

Formulas:

1-

T_{1} = {T o l l, T L R}

2-

T_{i} = S i g (M_{i - 1}) \cup T_{i - 1}

3-

\cup_{k = 1}^{n} S i g (M_{k, i}) = \cup_{k = 1}^{n} S i g (M_{k, i + 1})

4-

M a t c h (T) = T

5-

S_{1} \subseteq S i g (O_{1})

6-

S_{2} \subseteq S i g (O_{2})

7-

[0 \dots 1]

8-

O_{1 \dots n}

9-

\forall_{i \neq j} O_{i, j} \equiv M_{i} \cup M_{j}

10-

O_{1 \dots n}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1906.12033

Formulas:

1-

\in {1, N}

2-

\frac{d {\vec{r}}_{i}}{d t} = v_{0} {\hat{n}}_{θ, i} + μ \sum_{j \in n_{i}} \vec{F} (\vec{r_{i}}, \vec{r_{j}}), \frac{d θ_{i}}{d t} = \sqrt{2 D_{r}} η_{_{i}},

3-

\vec{F} (\vec{r_{i}}, \vec{r_{j}})

4-

η_{i} (t)

5-

⟨ η_{i} (t) ⟩ = 0

6-

⟨ η_{i} (t) η_{j} (t^{'}) ⟩ = 2 δ_{i, j} δ (t - t^{'})

7-

ξ_{i j} = d - | {\vec{r}}_{i j} |

8-

{\vec{r}}_{i j} = {\vec{r}}_{j} - {\vec{r}}_{i}

9-

{\vec{F}}_{i j}^{JKR} = \frac{1}{2} [Y_{i j}^{eff} d^{1 / 2} ξ_{i j}^{3 / 2} - \sqrt{3 π σ_{i j} Y_{i j}^{eff}} d^{3 / 4} ξ_{i j}^{3 / 4}] \frac{{\vec{r}}_{i j}}{| {\vec{r}}_{i j} |},

10-

\frac{1}{Y_{i j}^{eff}} = \frac{3}{4} (\frac{1 - ν_{i}^{2}}{Y_{i}} + \frac{1 - ν_{j}^{2}}{Y_{j}})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1902.01841

Formulas:

1-

K_{i j} = \frac{\partial I_{i}}{\partial T_{j}}

2-

K_{i j}^{C H_{4}}

3-

K_{i j}^{C H_{4}} = K_{i j} - K_{i j}^{c o n t}

4-

K_{i j}^{c o n t}

5-

K_{i j}^{C H_{4}}

6-

K_{i j}^{C H_{4}}

7-

K_{i j}^{c o n t}

8-

K_{i j}^{C H_{4}}

9-

K_{i j}^{c o n t}

10-

K_{i j}^{C H_{4}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1609.09108

Formulas:

1-

M = 10^{9} M_{⊙}

2-

\frac{\partial f (p, t)}{\partial t} = \frac{1}{p^{2}} \frac{\partial}{\partial p} [p^{2} D (p, t) \frac{\partial f (p, t)}{\partial p}],

3-

f (p, t)

4-

D (p, t)

5-

N (p, t) = 4 π p^{2} f (p, t)

6-

\frac{\partial N (γ, t)}{\partial t}

7-

\frac{\partial}{\partial γ} {[C (γ, t) - A (γ, t)] N (γ, t) + D (γ, t) \frac{\partial N (γ, t)}{\partial γ}}

8-

Q (γ, t) - E (γ, t),

9-

C (γ, t) = {(d γ / d t)}_{syn} + {(d γ / d t)}_{IC}

10-

{(d γ / d t)}_{syn} = (4 / 3) (σ_{T} c / m_{e} c^{2}) U_{B} (t) γ^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1905.00247

Formulas:

1-

P = {M, E, Y}

2-

O = {p, d, f, i, v}

3-

S = {P, O}

4-

I_{L} = S \times (\frac{1}{L} - 1)

5-

I = I_{d} + I_{L}

6-

F = \frac{R}{S \times 8} \times L \times T

7-

E_{R} = \frac{S \times 8}{R}

8-

E_{L} = \frac{E_{R}}{L}

9-

E_{R} = 123.04 μ s

10-

E_{L} = 492.16 μ s

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1702.02986

Formulas:

1-

H (A) = 0

2-

{Hom}_{R} (P, Q)

3-

{Hom}_{R} (E, I)

4-

splf R = sup {{proj . \dim}_{R} F ∣ F is a flat R -module}

5-

{proj . \dim}_{R} (\oplus_{λ \in Λ} F_{λ}) = {sup}_{λ \in Λ} {{proj . \dim}_{R} F_{λ}}

6-

(i) \Leftrightarrow (i i) \Leftrightarrow (i i i) ⟹ (i v) .

7-

{inj . \dim}_{R} R

8-

(i i i)

9-

(i i i)

10-

splf R \leq n + 1

Formulas (html):

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id="S1.p1.12.m12.1.1" xref="S1.p1.12.m12.1.1.cmml">E</mi><mo id="S1.p1.12.m12.3.3.1.2.2" xref="S1.p1.12.m12.3.3.2.cmml">,</mo><mi id="S1.p1.12.m12.2.2" xref="S1.p1.12.m12.2.2.cmml">I</mi><mo stretchy="false" id="S1.p1.12.m12.3.3.1.2.3" xref="S1.p1.12.m12.3.3.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p2.1.m1.5.5" xref="S1.p2.1.m1.5.5.cmml"><mrow id="S1.p2.1.m1.5.5.4" xref="S1.p2.1.m1.5.5.4.cmml"><mi id="S1.p2.1.m1.5.5.4.1" xref="S1.p2.1.m1.5.5.4.1.cmml">splf</mi><mo id="S1.p2.1.m1.5.5.4a" xref="S1.p2.1.m1.5.5.4.cmml">⁡</mo><mi id="S1.p2.1.m1.5.5.4.2" xref="S1.p2.1.m1.5.5.4.2.cmml">R</mi></mrow><mo id="S1.p2.1.m1.5.5.3" xref="S1.p2.1.m1.5.5.3.cmml">=</mo><mrow id="S1.p2.1.m1.5.5.2" xref="S1.p2.1.m1.5.5.2.cmml"><mo id="S1.p2.1.m1.5.5.2.3" xref="S1.p2.1.m1.5.5.2.3.cmml">sup</mo><mo id="S1.p2.1.m1.5.5.2a" xref="S1.p2.1.m1.5.5.2.cmml">⁡</mo><mrow id="S1.p2.1.m1.5.5.2.2.2" xref="S1.p2.1.m1.5.5.2.2.3.cmml"><mo stretchy="false" id="S1.p2.1.m1.5.5.2.2.2.3" xref="S1.p2.1.m1.5.5.2.2.3.1.cmml">{</mo><mrow id="S1.p2.1.m1.4.4.1.1.1.1" xref="S1.p2.1.m1.4.4.1.1.1.1.cmml"><msub id="S1.p2.1.m1.4.4.1.1.1.1.1" xref="S1.p2.1.m1.4.4.1.1.1.1.1.cmml"><mrow id="S1.p2.1.m1.3.3.4" xref="S1.p2.1.m1.3.3.3.cmml"><mi id="S1.p2.1.m1.2.2.1" xref="S1.p2.1.m1.2.2.1.cmml">proj</mi><mo id="S1.p2.1.m1.3.3.4.1" xref="S1.p2.1.m1.3.3.3a.cmml">.</mo><mi id="S1.p2.1.m1.3.3.2" xref="S1.p2.1.m1.3.3.2.cmml">dim</mi></mrow><mi id="S1.p2.1.m1.4.4.1.1.1.1.1.2" xref="S1.p2.1.m1.4.4.1.1.1.1.1.2.cmml">R</mi></msub><mo id="S1.p2.1.m1.4.4.1.1.1.1a" xref="S1.p2.1.m1.4.4.1.1.1.1.cmml">⁡</mo><mi id="S1.p2.1.m1.4.4.1.1.1.1.2" xref="S1.p2.1.m1.4.4.1.1.1.1.2.cmml">F</mi></mrow><mo id="S1.p2.1.m1.5.5.2.2.2.4" xref="S1.p2.1.m1.5.5.2.2.3.1.cmml">∣</mo><mrow id="S1.p2.1.m1.5.5.2.2.2.2" xref="S1.p2.1.m1.5.5.2.2.2.2.cmml"><mpadded width="+5pt" id="S1.p2.1.m1.5.5.2.2.2.2.2" xref="S1.p2.1.m1.5.5.2.2.2.2.2.cmml"><mi id="S1.p2.1.m1.5.5.2.2.2.2.2a" xref="S1.p2.1.m1.5.5.2.2.2.2.2.cmml">F</mi></mpadded><mo id="S1.p2.1.m1.5.5.2.2.2.2.1" xref="S1.p2.1.m1.5.5.2.2.2.2.1.cmml">⁢</mo><mrow id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1c.cmml"><mtext id="S1.p2.1.m1.1.1.1a" xref="S1.p2.1.m1.1.1.1c.cmml">is a flat </mtext><mi id="S1.p2.1.m1.1.1.1.m1.1.1" xref="S1.p2.1.m1.1.1.1.m1.1.1.cmml">R</mi><mtext id="S1.p2.1.m1.1.1.1b" xref="S1.p2.1.m1.1.1.1c.cmml">-module</mtext></mrow></mrow><mo stretchy="false" id="S1.p2.1.m1.5.5.2.2.2.5" xref="S1.p2.1.m1.5.5.2.2.3.1.cmml">}</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p2.3.m3.7.7" xref="S1.p2.3.m3.7.7.cmml"><mrow id="S1.p2.3.m3.6.6.2.2" xref="S1.p2.3.m3.6.6.2.3.cmml"><msub id="S1.p2.3.m3.5.5.1.1.1" xref="S1.p2.3.m3.5.5.1.1.1.cmml"><mrow id="S1.p2.3.m3.2.2.4" xref="S1.p2.3.m3.2.2.3.cmml"><mi id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">proj</mi><mo id="S1.p2.3.m3.2.2.4.1" xref="S1.p2.3.m3.2.2.3a.cmml">.</mo><mi id="S1.p2.3.m3.2.2.2" xref="S1.p2.3.m3.2.2.2.cmml">dim</mi></mrow><mi id="S1.p2.3.m3.5.5.1.1.1.2" xref="S1.p2.3.m3.5.5.1.1.1.2.cmml">R</mi></msub><mo id="S1.p2.3.m3.6.6.2.2a" xref="S1.p2.3.m3.6.6.2.3.cmml">⁡</mo><mrow id="S1.p2.3.m3.6.6.2.2.2" xref="S1.p2.3.m3.6.6.2.3.cmml"><mo stretchy="false" id="S1.p2.3.m3.6.6.2.2.2.2" xref="S1.p2.3.m3.6.6.2.3.cmml">(</mo><mrow id="S1.p2.3.m3.6.6.2.2.2.1" xref="S1.p2.3.m3.6.6.2.2.2.1.cmml"><msub id="S1.p2.3.m3.6.6.2.2.2.1.1" xref="S1.p2.3.m3.6.6.2.2.2.1.1.cmml"><mo largeop="true" mathsize="160%" stretchy="false" symmetric="true" id="S1.p2.3.m3.6.6.2.2.2.1.1.2" xref="S1.p2.3.m3.6.6.2.2.2.1.1.2.cmml">⊕</mo><mrow id="S1.p2.3.m3.6.6.2.2.2.1.1.3" xref="S1.p2.3.m3.6.6.2.2.2.1.1.3.cmml"><mi id="S1.p2.3.m3.6.6.2.2.2.1.1.3.2" xref="S1.p2.3.m3.6.6.2.2.2.1.1.3.2.cmml">λ</mi><mo id="S1.p2.3.m3.6.6.2.2.2.1.1.3.1" xref="S1.p2.3.m3.6.6.2.2.2.1.1.3.1.cmml">∈</mo><mi mathvariant="normal" id="S1.p2.3.m3.6.6.2.2.2.1.1.3.3" xref="S1.p2.3.m3.6.6.2.2.2.1.1.3.3.cmml">Λ</mi></mrow></msub><msub id="S1.p2.3.m3.6.6.2.2.2.1.2" xref="S1.p2.3.m3.6.6.2.2.2.1.2.cmml"><mi id="S1.p2.3.m3.6.6.2.2.2.1.2.2" xref="S1.p2.3.m3.6.6.2.2.2.1.2.2.cmml">F</mi><mi id="S1.p2.3.m3.6.6.2.2.2.1.2.3" xref="S1.p2.3.m3.6.6.2.2.2.1.2.3.cmml">λ</mi></msub></mrow><mo stretchy="false" id="S1.p2.3.m3.6.6.2.2.2.3" xref="S1.p2.3.m3.6.6.2.3.cmml">)</mo></mrow></mrow><mo id="S1.p2.3.m3.7.7.4" xref="S1.p2.3.m3.7.7.4.cmml">=</mo><mrow id="S1.p2.3.m3.7.7.3" xref="S1.p2.3.m3.7.7.3.cmml"><msub id="S1.p2.3.m3.7.7.3.2" xref="S1.p2.3.m3.7.7.3.2.cmml"><mo id="S1.p2.3.m3.7.7.3.2.2" xref="S1.p2.3.m3.7.7.3.2.2.cmml">sup</mo><mrow id="S1.p2.3.m3.7.7.3.2.3" xref="S1.p2.3.m3.7.7.3.2.3.cmml"><mi id="S1.p2.3.m3.7.7.3.2.3.2" xref="S1.p2.3.m3.7.7.3.2.3.2.cmml">λ</mi><mo id="S1.p2.3.m3.7.7.3.2.3.1" xref="S1.p2.3.m3.7.7.3.2.3.1.cmml">∈</mo><mi mathvariant="normal" id="S1.p2.3.m3.7.7.3.2.3.3" xref="S1.p2.3.m3.7.7.3.2.3.3.cmml">Λ</mi></mrow></msub><mo id="S1.p2.3.m3.7.7.3a" xref="S1.p2.3.m3.7.7.3.cmml">⁡</mo><mrow id="S1.p2.3.m3.7.7.3.1.1" xref="S1.p2.3.m3.7.7.3.1.2.cmml"><mo stretchy="false" id="S1.p2.3.m3.7.7.3.1.1.2" xref="S1.p2.3.m3.7.7.3.1.2.cmml">{</mo><mrow id="S1.p2.3.m3.7.7.3.1.1.1" xref="S1.p2.3.m3.7.7.3.1.1.1.cmml"><msub id="S1.p2.3.m3.7.7.3.1.1.1.1" xref="S1.p2.3.m3.7.7.3.1.1.1.1.cmml"><mrow id="S1.p2.3.m3.4.4.4" xref="S1.p2.3.m3.4.4.3.cmml"><mi id="S1.p2.3.m3.3.3.1" xref="S1.p2.3.m3.3.3.1.cmml">proj</mi><mo id="S1.p2.3.m3.4.4.4.1" xref="S1.p2.3.m3.4.4.3a.cmml">.</mo><mi id="S1.p2.3.m3.4.4.2" xref="S1.p2.3.m3.4.4.2.cmml">dim</mi></mrow><mi id="S1.p2.3.m3.7.7.3.1.1.1.1.2" xref="S1.p2.3.m3.7.7.3.1.1.1.1.2.cmml">R</mi></msub><mo id="S1.p2.3.m3.7.7.3.1.1.1a" xref="S1.p2.3.m3.7.7.3.1.1.1.cmml">⁡</mo><msub id="S1.p2.3.m3.7.7.3.1.1.1.2" xref="S1.p2.3.m3.7.7.3.1.1.1.2.cmml"><mi id="S1.p2.3.m3.7.7.3.1.1.1.2.2" xref="S1.p2.3.m3.7.7.3.1.1.1.2.2.cmml">F</mi><mi id="S1.p2.3.m3.7.7.3.1.1.1.2.3" xref="S1.p2.3.m3.7.7.3.1.1.1.2.3.cmml">λ</mi></msub></mrow><mo stretchy="false" id="S1.p2.3.m3.7.7.3.1.1.3" xref="S1.p2.3.m3.7.7.3.1.2.cmml">}</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml"><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml"><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.5.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml"><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.5.2.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml">(</mo><mi id="S1.Thmtheorem1.Ex1.m1.1.1" xref="S1.Thmtheorem1.Ex1.m1.1.1.cmml">i</mi><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.5.2.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.6" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.6.cmml">⇔</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.2.cmml">i</mi><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.3.cmml">i</mi></mrow><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.7" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.7.cmml">⇔</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.cmml"><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.cmml"><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.cmml">(</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.cmml"><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.2.cmml">i</mi><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.3.cmml">i</mi><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.1a" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.4" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.4.cmml">i</mi></mrow><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.2.1.1.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.3.cmml">⟹</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.cmml"><mo stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.cmml">(</mo><mrow id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.cmml"><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.2.cmml">i</mi><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.1" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.3.cmml">v</mi></mrow><mo rspace="4.7pt" stretchy="false" id="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.3" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Thmtheorem1.Ex1.m1.2.2.1.2" xref="S1.Thmtheorem1.Ex1.m1.2.2.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S1.p4.2.m2.2.3" xref="S1.p4.2.m2.2.3.cmml"><msub id="S1.p4.2.m2.2.3.1" xref="S1.p4.2.m2.2.3.1.cmml"><mrow id="S1.p4.2.m2.2.2.4" xref="S1.p4.2.m2.2.2.3.cmml"><mi id="S1.p4.2.m2.1.1.1" xref="S1.p4.2.m2.1.1.1.cmml">inj</mi><mo id="S1.p4.2.m2.2.2.4.1" xref="S1.p4.2.m2.2.2.3a.cmml">.</mo><mi id="S1.p4.2.m2.2.2.2" xref="S1.p4.2.m2.2.2.2.cmml">dim</mi></mrow><mi id="S1.p4.2.m2.2.3.1.2" xref="S1.p4.2.m2.2.3.1.2.cmml">R</mi></msub><mo id="S1.p4.2.m2.2.3a" xref="S1.p4.2.m2.2.3.cmml">⁡</mo><mi id="S1.p4.2.m2.2.3.2" xref="S1.p4.2.m2.2.3.2.cmml">R</mi></mrow></math>, <math><mrow id="S1.p6.4.m4.1.1.1" xref="S1.p6.4.m4.1.1.1.1.cmml"><mo stretchy="false" id="S1.p6.4.m4.1.1.1.2" xref="S1.p6.4.m4.1.1.1.1.cmml">(</mo><mrow id="S1.p6.4.m4.1.1.1.1" xref="S1.p6.4.m4.1.1.1.1.cmml"><mi id="S1.p6.4.m4.1.1.1.1.2" xref="S1.p6.4.m4.1.1.1.1.2.cmml">i</mi><mo id="S1.p6.4.m4.1.1.1.1.1" xref="S1.p6.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p6.4.m4.1.1.1.1.3" xref="S1.p6.4.m4.1.1.1.1.3.cmml">i</mi><mo id="S1.p6.4.m4.1.1.1.1.1a" xref="S1.p6.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p6.4.m4.1.1.1.1.4" xref="S1.p6.4.m4.1.1.1.1.4.cmml">i</mi></mrow><mo stretchy="false" id="S1.p6.4.m4.1.1.1.3" xref="S1.p6.4.m4.1.1.1.1.cmml">)</mo></mrow></math>, <math><mrow id="S1.I2.i1.p1.2.m2.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.1.cmml"><mo stretchy="false" id="S1.I2.i1.p1.2.m2.1.1.1.2" xref="S1.I2.i1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.I2.i1.p1.2.m2.1.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.1.cmml"><mi id="S1.I2.i1.p1.2.m2.1.1.1.1.2" xref="S1.I2.i1.p1.2.m2.1.1.1.1.2.cmml">i</mi><mo id="S1.I2.i1.p1.2.m2.1.1.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I2.i1.p1.2.m2.1.1.1.1.3" xref="S1.I2.i1.p1.2.m2.1.1.1.1.3.cmml">i</mi><mo id="S1.I2.i1.p1.2.m2.1.1.1.1.1a" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I2.i1.p1.2.m2.1.1.1.1.4" xref="S1.I2.i1.p1.2.m2.1.1.1.1.4.cmml">i</mi></mrow><mo stretchy="false" id="S1.I2.i1.p1.2.m2.1.1.1.3" xref="S1.I2.i1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow></math>, <math><mrow id="S1.I2.i2.p1.3.m3.1.1" xref="S1.I2.i2.p1.3.m3.1.1.cmml"><mrow id="S1.I2.i2.p1.3.m3.1.1.2" xref="S1.I2.i2.p1.3.m3.1.1.2.cmml"><mi id="S1.I2.i2.p1.3.m3.1.1.2.1" xref="S1.I2.i2.p1.3.m3.1.1.2.1.cmml">splf</mi><mo id="S1.I2.i2.p1.3.m3.1.1.2a" 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Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0011321

Formulas:

1-

\frac{\partial 𝐮}{\partial t} = - 𝐮 \cdot \nabla 𝐮 - c_{s}^{2} \nabla \ln ρ + \frac{𝐉 \times 𝐁}{ρ}

2-

+ \frac{μ}{ρ} (\nabla^{2} 𝐮 + \frac{1}{3} \nabla \nabla \cdot 𝐮),

3-

\frac{\partial \ln ρ}{\partial t} = - 𝐮 \cdot \nabla \ln ρ - \nabla \cdot 𝐮,

4-

\frac{\partial 𝐀}{\partial t} = 𝐮 \times 𝐁 + η \nabla^{2} 𝐀,

5-

⟨ ρ_{0} ⟩ = ρ_{0}

6-

L_{BOX} = 2 π

7-

\sqrt{μ_{0} ρ_{0}} c

8-

c = k_{1} = ρ_{0} = μ_{0} = 1

9-

ν \equiv μ / ρ_{0}

10-

c_{s} = 1 / \sqrt{3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1902.04898

Formulas:

1-

E_{g} (φ)

2-

E_{g} (0) \approx 3.1 E_{Th} = 3.1 ℏ D / L^{2}

3-

j_{s} (E, φ)

4-

f (E, φ, T)

5-

I (φ, T)

6-

\frac{1}{e R_{N}} \int [1 - 2 f (E, φ, T)] j_{s} (E, φ) 𝑑 E,

7-

\sum_{n = 1}^{\infty} I_{c, n} (T) \sin (n φ)

8-

j_{s} (E, φ)

9-

I (φ, T)

10-

I_{c, n} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1912.11806

Formulas:

1-

ρ [\frac{\partial 𝒗}{\partial t} + (𝒗 \cdot \nabla) 𝒗] = - \nabla p + μ \nabla^{2} 𝒗 + ρ 𝒈,

2-

\nabla \cdot 𝒗 = 0,

3-

ρ (𝒓, t)

4-

𝒗 (𝒓, t) = (v_{x} (𝒓, t), v_{y} (𝒓, t))

5-

p (𝒓, t)

6-

𝒓 = (x, y)

7-

c (𝒓, t)

8-

(0 \leq c \leq 1)

9-

ρ (𝒓, t) = ρ_{high} + (ρ_{low} - ρ_{high}) (1 - c (𝒓, t)),

10-

\frac{\partial c}{\partial t} + \nabla \cdot (c 𝒗) = D \nabla^{2} c,

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/2008.00490

Formulas:

1-

𝑨 \in ℝ^{H W \times H W}

2-

𝒗_{c i} \in ℝ^{C \times 1 \times 1}

3-

𝒗_{h i} \in ℝ^{1 \times H \times 1}

4-

𝒗_{w i} \in ℝ^{1 \times 1 \times W}

5-

𝑨 \in ℝ^{C \times H \times W}

6-

𝑨 = \sum_{i = 1}^{r} λ_{i} 𝒗_{c i} \otimes 𝒗_{h i} \otimes 𝒗_{w i},

7-

𝒗_{c i} \in ℝ^{C \times 1 \times 1}

8-

𝒗_{h i} \in ℝ^{1 \times H \times 1}

9-

𝒗_{w i} \in ℝ^{1 \times 1 \times W}

10-

𝑨 \in ℝ^{C \times H \times W}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1711.07021

Formulas:

1-

T (G) = T (H)

2-

u, v \in V (G)

3-

| V (G) |

4-

| E (G) |

5-

N_{G} (v)

6-

d_{G} (v)

7-

| N_{G} (v) |

8-

d_{G} (v) = k

9-

v \in V (G)

10-

(v_{1}, v_{n})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0506253

Formulas:

1-

l = - \infty \dots \infty

2-

p = 0 \dots \infty

3-

ℓ = h_{s} (n - n_{0}) / λ

4-

Δ ϕ \equiv 2 π ℓ

5-

p = \min (n, m)

6-

P_{c} (α_{s}, α_{i}) = C {(1 - \frac{| α_{s} - α_{i} |}{π})}^{2},

7-

ℓ = \pm 3.48 \pm 0.02

8-

w_{0} = 0.78 \pm 0.01

9-

α_{i} = 0, π / 2, π, 3 π / 2

10-

K > \frac{1}{4} {(\sqrt{\frac{L}{w_{0}^{2} k_{p}}} + \sqrt{\frac{w_{0}^{2} k_{p}}{L}})}^{2} .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1106.0131

Formulas:

1-

𝖫^{2} (a, b), 0 \leq a < b \leq \infty

2-

(Γ_{a, b} (k) u) (x) = \int_{a}^{b} k (x + y) u (y) 𝑑 y,

3-

k = k (t), t > 0

4-

Γ_{0, \infty} (k)

5-

𝖫^{2} (0, \infty)

6-

Γ_{a, b} (k)

7-

0 \leq a < b < \infty

8-

ϰ = ϰ (ξ), ξ \in ℝ,

9-

Γ_{0, \infty} (k)

10-

\hat{ϰ} (t)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Run 11332329 (Agent397)