Run 11332332

Paper: https://arxiv.org/abs/1909.00706

Formulas:

1-

M_{⋆} = 1 M_{☉}

2-

R_{0} = 50 au

3-

τ_{dyn, 0} = Ω_{K, 0}^{- 1} = {(R_{0}^{3} / G M_{⋆})}^{1 / 2}

4-

P_{g} = c_{s}^{2} Σ_{g}

5-

c_{s} (R) = c_{s, 0} {(R / R_{0})}^{- 1 / 4}

6-

Σ_{g} = Σ_{0} {(R / R_{0})}^{- 1}

7-

H / R = 0.05 {(R / R_{0})}^{1 / 4}

8-

R_{\min} = 0.2 R_{0} = 10 au

9-

R_{\max} = 4.5 R_{0} = 225 au

10-

α (R) = α_{0} - \frac{α_{0} - α_{DZ}}{2} [1 - \tanh (\frac{R - R_{DZ}}{Δ_{DZ}})],

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1301.4008

Formulas:

1-

F_{1}, F_{2}, \dots, F_{k}

2-

F_{1}, \dots, F_{k}

3-

F_{1}, F_{2}, \dots, F_{k}

4-

G (F_{1}, \dots, F_{k})

5-

⋃_{i = 1}^{k} E (F_{i})

6-

G (F_{1}, \dots, F_{k})

7-

G (F_{1}, \dots, F_{k})

8-

G (F_{1}, \dots, F_{k})

9-

γ_{sd} (F_{1}, F_{2}, \dots, F_{k})

10-

𝒢_{k, δ, n}

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1804.00931

Formulas:

1-

I_{k}^{r} <= I_{c}^{r}

2-

c o n f i d e n c e s c o r e = \frac{\sum_{p \in P} C (O^{r} (p), S^{r} (p))}{P}

3-

O^{r} (p)

4-

S^{r} (p)

5-

C (u, v)

6-

s e g m e n t a t i o n n e t w o r k

7-

f l o w n e t w o r k

8-

k = i - (m + 1), \dots, i

9-

k = i - (l + 1)

10-

d = \frac{\sum_{p \in P} (G {(I_{k})}_{p} - G {(I_{i})}_{p})}{P}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9609202

Formulas:

1-

d_{x^{2} - y^{2}}

2-

d_{x^{2} - y^{2}}

3-

d_{x^{2} - y^{2}}

4-

d_{3 z^{2} - r^{2}}

5-

d_{x^{2} - y^{2}}

6-

d_{3 z^{2} - r^{2}}

7-

N = \sum_{m σ} n_{m σ}

8-

E_{a v} = \frac{1}{/} 2 U N (N - 1) - \frac{1}{/} 4 J N (N - 2) .

9-

E = E_{L D A} - E_{a v} + \frac{1}{/} 2 \sum_{m, m^{'}, σ} U_{m m^{'}} n_{m σ} n_{m^{'} - σ}

10-

+ \frac{1}{/} 2 \sum_{m \neq m^{'}, m^{'}, σ} (U_{m m^{'}} - J_{m m^{'}}) n_{m σ} n_{m^{'} σ} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1809.00597

Formulas:

1-

H_{0} = - \sum_{⟨ i, j ⟩, σ} t c_{i σ}^{+} c_{j σ}

2-

H_{A F}^{x, y, z} = λ_{A F}^{x, y, z} \sum_{i, σ, σ^{'}} κ_{i} {\hat{s}}_{σ, σ^{'}}^{x, y, z} c_{i σ}^{+} c_{i σ^{'}}

3-

{\hat{s}}^{x, y, z}

4-

H_{Δ} = λ_{Δ} \sum_{i, σ} κ_{i} c_{i σ}^{+} c_{i σ}

5-

H_{F M}^{x, y, z} = λ_{F M}^{x, y, z} \sum_{i, σ, σ^{'}} {\hat{s}}_{σ, σ^{'}}^{x, y, z} c_{i σ}^{+} c_{i σ^{'}}

6-

H_{R} = \frac{2 i λ_{R}}{3} \sum_{⟨ i, j ⟩, σ, σ^{'}} {[{(\hat{𝐬} \times 𝐝_{i j})}_{z}]}_{σ, σ^{'}} c_{i σ}^{+} c_{j σ^{'}}

7-

H_{s I} = \frac{i λ_{s I}}{3 \sqrt{3}} \sum_{⟨ ⟨ i, j ⟩ ⟩, σ, σ^{'}} κ_{i} ν_{i j} {\hat{s}}_{σ, σ^{'}}^{z} c_{i σ}^{+} c_{j σ^{'}}

8-

ν_{i j} = (+ 1) - 1

9-

⟨ ⟨ i, j ⟩ ⟩

10-

H_{I} = \frac{i λ_{I}}{3 \sqrt{3}} \sum_{⟨ ⟨ i, j ⟩ ⟩, σ, σ^{'}} ν_{i j} {\hat{s}}_{σ, σ^{'}}^{z} c_{i σ}^{+} c_{j σ^{'}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9802273

Formulas:

1-

m \sin^{3} i

2-

1 \overset{\circ}{.} 2

3-

α (2000) = 20^{h} 43^{m} 13 \overset{s}{.} 52

4-

δ (2000) = + 57 ° 6' 50 \overset{''}{.} 9

5-

21, 000 \pm 3, 000

6-

E (B - V) = 0.58

7-

T_{eff} = 21, 000

8-

\log (g) = 4.0

9-

T_{eff} = 21, 000

10-

\log (g) = 4.0

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9806245

Formulas:

1-

N_{c l} = S

2-

\frac{S R}{N R_{s}}

3-

E = \frac{M^{2} R}{N}

4-

S = M R_{s}

5-

\begin{matrix} T_{+} = \frac{M R}{N R_{s}} \\ = \frac{1}{2} m_{c l} v_{c l}^{2} \end{matrix}

6-

\begin{matrix} Σ_{a, b = 1}^{S} G_{D} \frac{{(m_{c l} v_{c l}^{2})}_{a} {(m_{c l} v_{c l}^{2})}_{b}}{R r^{D - 4}} \end{matrix}

7-

\begin{matrix} p_{-} = \frac{N}{M R} \frac{1}{R_{s}} \\ = \frac{N}{S} \frac{1}{R} \end{matrix}

8-

v_{c l} δ t

9-

\frac{v_{c l}}{R_{s}} \sim T_{+}

10-

\frac{1}{2} m_{c l} v_{c l}^{2} \sim T_{+}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1111.0394

Formulas:

1-

x < L^{*} / 2

2-

δ j (x, t)

3-

δ E (x, t)

4-

δ T (x, t)

5-

δ E (x, t)

6-

δ j (x, t)

7-

ν \frac{d δ T}{d t} = k \frac{d^{2} δ T}{d x^{2}} + j_{c} δ E .

8-

\frac{d^{2} δ E}{d x^{2}} = μ_{0} [\frac{d j}{d E} \frac{d δ E}{d t} - \frac{d j_{c}}{d T} \frac{d δ T}{d t}] .

9-

ν = ν (T)

10-

κ = κ (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9505085

Formulas:

1-

ℋ = 𝒯 + \frac{J}{2} \sum_{< j l >} 𝐒_{j} \cdot 𝐒_{l},

2-

𝒯 = P_{G} [- t \sum_{< j l > σ} c_{j σ}^{†} c_{l σ}] P_{G}

3-

𝒢_{k σ} (ω) = ⟨ Ψ_{0}^{(N)} | c_{k σ}^{†} \frac{1}{ω + E_{0}^{(N)} - ℋ + i η} c_{k σ} | Ψ_{0}^{(N)} ⟩,

4-

| Ψ_{0}^{(N)} ⟩

5-

c_{k σ} = N^{- 1 / 2} \sum_{j}^{N} e^{i 𝐤 \cdot 𝐫_{j}} c_{j σ} .

6-

A_{k} (ω) = - \frac{1}{π} I m 𝒢_{k σ} (ω)

7-

(π / 2, π / 2)

8-

k = (π, π)

9-

(π / 2, π / 2)

10-

Z_{k} = \frac{{| ⟨ Ψ_{k}^{(N - 1)} | c_{k σ} | Ψ_{0}^{(N)} ⟩ |}^{2}}{⟨ Ψ_{0}^{(N)} | c_{k σ}^{†} c_{k σ} | Ψ_{0}^{(N)} ⟩},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0707.0943

Formulas:

1-

{}^{1}S_{0} \to^{3} S_{1}

2-

{}^{1}D_{2} \to^{3} S_{1}

3-

{}^{1}D_{2} \to^{3} S_{1}

4-

p p \to d π^{+}

5-

π^{+} d \to p p

6-

π^{+} d \to p p

7-

π^{+} d \to p p

8-

(σ (π^{-} d) - σ (π - p)) / σ (π^{-} p) \approx 3

9-

π^{-} {}^{12}C \to π^{-} n^{11} C

10-

{}^{12}C π^{-} \to π^{-} n^{11} C

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-ex

Result: incorrect

Paper: https://arxiv.org/abs/1111.0997

Formulas:

1-

T_{N} = T_{T O}

2-

T_{N} \leq T_{T O}

3-

ρ_{b} / ρ_{a} = 1.2, 1.4, 1.5

4-

ϵ = \frac{(a_{O} - b_{O})}{(a_{O} + b_{O})}

5-

ϵ \equiv \frac{(a_{O} - b_{O})}{(a_{O} + b_{O})}

6-

ρ_{b}^{*} \equiv 2 ρ_{t} - ρ_{a}

7-

ρ_{b}^{*} = 2 * ρ_{t} - ρ_{a}

8-

T_{N} \leq T_{T O}

9-

ϵ = \frac{(a_{O} - b_{O})}{(a_{O} + b_{O})}

10-

ϵ = \frac{a_{O} - b_{O}}{a_{O} + b_{O}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/nucl-ex/9608003

Formulas:

1-

40 \leq A_{CN} \leq 60

2-

σ_{FF} = 137 \pm 5 mb

3-

σ_{ER} = 722 \pm 197 mb

4-

L_{crit} = 44 ℏ

5-

E_{lab} = 278 MeV

6-

< Z_{1} + Z_{2} >

7-

280 MeV [5] .

8-

(Z_{1} = 5 to 12)

9-

Z_{1, 2} \geq 5 .

10-

< Z_{1} + Z_{2} >

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/0704.0882

Formulas:

1-

Λ C D M

2-

L = \frac{M_{P l}^{2}}{2 M_{5}^{3}},

3-

H^{2} - \frac{k}{a^{2}} \pm \frac{1}{L} \sqrt{H^{2} - \frac{k}{a^{2}}} = κ ρ,

4-

κ = 8 π G / 3

5-

\frac{2 \ddot{a}}{a} + H^{2} = - 3 κ p,

6-

Λ C D M

7-

p_{Λ} = - ρ_{Λ}

8-

p_{φ} = - A / ρ_{φ}

9-

{\dot{ρ}}_{φ} (a) - 3 H (ρ_{φ} (a) - \frac{A}{ρ_{φ} (a)}) = 0 .

10-

\dot{ρ} (a) = - 3 \frac{\dot{a}}{a} (ρ (a) + p (a) + \frac{1}{2 κ L^{2}}) + \frac{3}{2 L} [ρ (a) - p (a)] .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1501.00519

Formulas:

1-

n (x) = n_{r} (x) + i n_{i} (x)

2-

n_{r} (x) = n_{r} (- x)

3-

n_{i} (x) = - n_{i} (- x)

4-

d_{1} u_{x x} + V_{1} (x) u + i W_{1} (x) u = u^{*} v

5-

d_{2} v_{x x} + κ v + V_{2} (x) v + i W_{2} (x) v = u^{2}

6-

V_{1, 2} (x)

7-

W_{1, 2} (x)

8-

u (x, z) = U (x) e^{- i ω z}

9-

v (x, z) = V (x) e^{- 2 i ω z}

10-

d_{1} U_{x x} + V_{1} (x) U + i W_{1} (x) U = U^{*} V

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/1011.6550

Formulas:

1-

0.1053 < Ω_{CDM} h^{2} < 0.1193

2-

\frac{d n}{d t} = - 3 H n - ⟨ σ_{ann} v ⟩ (n^{2} - n_{eq}^{2}),

3-

⟨ σ_{ann} v ⟩

4-

Ω_{CDM} h^{2} = m_{χ} n_{0} / ρ_{c}

5-

{(g - 2)}_{μ}

6-

𝒪 (α_{s})

7-

H_{k}^{0} = h^{0}, H^{0}, A^{0}

8-

{\tilde{q}}_{i} = {\tilde{q}}_{1}, {\tilde{q}}_{2}

9-

{\tilde{χ}}_{1}^{0} {\tilde{χ}}_{1}^{0} \to A^{0} \to b \bar{b}

10-

𝒪 (α_{s}^{4})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1806.11419

Formulas:

1-

𝒫_{L} = 60 μ

2-

θ = 30 \pm 2^{\circ}

3-

θ = 73 \pm 5^{\circ}

4-

θ = 126 \pm 2^{\circ}

5-

θ = 127 \pm 2^{\circ}

6-

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

7-

10 μ m / h

8-

ν_{\pm} = D \pm g μ_{B} | 𝐁_{𝟎} \cdot 𝐮_{NV} (θ, ϕ) |

9-

g μ_{B} = 28

10-

𝐮_{NV} (θ, ϕ)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1712.02107

Formulas:

1-

W (n) = \frac{(2 n - 1)!!}{(2 n)!!} = \frac{1}{\sqrt{π}} \frac{Γ (n + \frac{1}{2})}{Γ (n + 1)},

2-

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} 𝑑 t, Re (x) > 0 .

3-

\frac{1}{\sqrt{π (n + \frac{4}{π} - 1)}} \leq \frac{(2 n - 1)!!}{(2 n)!!} \leq \frac{1}{\sqrt{π (n + \frac{1}{4})}},

4-

\sqrt{\frac{e}{π}} {(1 - \frac{1}{2 n})}^{n} \frac{\sqrt{n - 1}}{n} < W (n) \leq \frac{4}{3} {(1 - \frac{1}{2 n})}^{n} \frac{\sqrt{n - 1}}{n},

5-

W (n) \sim χ (n) := \sqrt{\frac{e}{π}} {(1 - \frac{1}{2 n})}^{n} \frac{\sqrt{n - 1}}{n}, n \to \infty .

6-

W (n) \sim \sqrt{\frac{e}{π}} {(1 - \frac{1}{2 (n + \frac{1}{3})})}^{n + \frac{1}{3}} \frac{1}{\sqrt{n}} .

7-

W (n) = \frac{(2 n - 1)!!}{(2 n)!!}

8-

W (n) \sim \sqrt{\frac{e}{π}} {(1 - \frac{1}{2 (n + \frac{1}{3})})}^{n + \frac{1}{3}} \frac{1}{\sqrt{n}} \exp (\frac{1}{n^{2}} \frac{a_{1}}{n + b_{1} + \frac{a_{2}}{n + b_{2} + \frac{a_{3}}{n + b_{3}}}}),

9-

a_{1} = \frac{1}{144}, b_{1} = \frac{1}{60}; a_{2} = \frac{781}{3600}, b_{2} = - \frac{4309}{109340}; a_{3} = \frac{51396085}{89664267}, b_{3} = \frac{25682346121}{449571834712} .

10-

\sqrt{\frac{e}{π}} {(1 - \frac{1}{2 (n + \frac{1}{3})})}^{n + \frac{1}{3}} \frac{1}{\sqrt{n}} \exp (\frac{1}{n^{2}} \frac{a_{1}}{n + b_{1} + \frac{a_{2}}{n + b_{2} + \frac{a_{3}}{n + b_{3}}}})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0711.0946

Formulas:

1-

χ \in [0, π / 2]

2-

χ_{v} \in (0, π / 2]

3-

ϵ_{a, b, c}

4-

\exp (- i ω t)

5-

k_{0} = ω \sqrt{ϵ_{0} μ_{0}}

6-

λ_{0} = 2 π / k_{0}

7-

η_{0} = \sqrt{μ_{0} / ϵ_{0}}

8-

0 \leq z \leq L_{m e t}

9-

ϵ_{m e t}

10-

z \geq L_{m e t} + L_{s t f}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/hep-th/9608165

Formulas:

1-

S (ϕ) = \frac{1}{n} g_{n} ϕ^{n}

2-

(g_{n} \frac{\partial}{\partial J^{n - 1}} - J) Z (J) = 0

3-

Z (J) = \int_{Γ} 𝑑 ϕ e^{- S (ϕ) + J ϕ}

4-

{e^{- S (ϕ) + J ϕ} |}_{\partial Γ} = 0

5-

R e S (ϕ) \to + \infty

6-

R e g_{k} ϕ^{k} \to + \infty

7-

S = β c o s ϕ

8-

Γ_{1} = [- i \infty, + i \infty]

9-

Γ_{2} = [- i \infty, 0] + [0, 2 π] + [2 π, 2 π + i \infty]

10-

[β (\partial_{J} - \partial_{\bar{J}}) - 2 (J \partial_{J} - \bar{J} \partial_{\bar{J}})] Z (J, \bar{J}) = 0

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1301.3055

Formulas:

1-

V_{LSR} = + 65.4 \pm 0.5

2-

\log (Q) \geq 50

3-

\log (Q) \sim 48.5

4-

b = - 0 \overset{\circ}{.} 4

5-

l = 19 \overset{\circ}{.} 025, b = - 0 \overset{\circ}{.} 38

6-

V_{LSR} = 65.4 \pm 0.5

7-

l = 18 \overset{\circ}{.} 95, b = - 0 \overset{\circ}{.} 01

8-

0 \leq V_{LSR} \leq + 130

9-

V_{LSR} = 65.4 \pm 0.5

10-

F_{1420} = 40 \pm 0.6

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0509666

Formulas:

1-

i ℏ \frac{\partial}{\partial t} ψ (𝐫, t) = (- \frac{ℏ^{2}}{2 m} \nabla^{2} + V (𝐫, t) + g {| ψ (𝐫, t) |}^{2}) ψ (𝐫, t)

2-

ψ (𝐫, t)

3-

V (𝐫, t)

4-

g = 4 π a_{s} ℏ^{2} / m

5-

V (ρ, z, t) = U (t) (1 - e^{- ρ^{2} / r_{l b}^{2}} \cos^{2} (k z))

6-

+ \frac{1}{2} m (ω_{⟂}^{2} ρ^{2} + ω_{z}^{2} z^{2}) + m g z

7-

ρ = \sqrt{x^{2} + y^{2}}

8-

2 \sqrt{2} ω_{⟂}

9-

E_{R} = ℏ^{2} k^{2} / 2 m

10-

U_{f} = 7.2 E_{R}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/nucl-ex/0411032

Formulas:

1-

N N \to N Y K

2-

J^{P} = 1 / 2^{+}

3-

I (J^{P})

4-

0 ({\frac{1}{2}}^{+})

5-

I (J^{P})

6-

K^{-} + p \to Y π

7-

Y p \to Y p

8-

a_{s} = - 1.8 + (+ 2.3 - 4.2) fm and a_{t} = - 1.6 + (+ 1.1 - 0.8) fm

9-

- 6 f m < a_{s} < + 2.5 f m

10-

- 2 f m < a_{t} < 1.3 f m

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0503498

Formulas:

1-

H_{0} / 2 = ϕ_{0} / 2 w s =

2-

J_{c} (H)

3-

J_{c} (H)

4-

J_{c} (H)

5-

J_{c} (H)

6-

J_{c} (H)

7-

w ≪ λ_{J} \equiv γ s

8-

J_{c} (H)

9-

l_{B} = γ s (π γ s^{2} B / \sqrt{2} ϕ_{0})

10-

w < 2 l_{B}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1811.04523

Formulas:

1-

L_{2}^{(a, b)}

2-

V_{s} (t)

3-

L_{2}^{(a)} = 102.7

4-

L_{2}^{(b)} = 970

5-

N^{(a)} = N^{(b)}

6-

\frac{d i_{n}}{d τ} = \frac{L_{2}^{(a)}}{L_{1}} (v_{n + 1} + v_{n - 1} - 2 v_{n}) - \frac{L_{2}^{(a)}}{L_{2}^{(n)}} v_{n}

7-

\frac{d v_{n}}{d τ} = \frac{1}{c (v_{n})} [i - i_{d} + \frac{\cos (Ω τ)}{ω_{0} R C_{0}} - \frac{v_{n}}{ω_{0} R_{e} C_{0}}],

8-

L_{2}^{(n)} = L_{2}^{(a)}

9-

L_{2}^{(n)} = L_{2}^{(b)}

10-

τ = ω_{0} t

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/1408.1280

Formulas:

1-

c^{2} d τ^{2} = c^{2} d t^{2} - {| d 𝐫 |}^{2}

2-

{| d 𝐫 |}^{2} = d x^{2} + d y^{2} + d z^{2}

3-

d τ, d t, | d 𝐫 |

4-

c^{2} d t^{2} = c^{2} d τ^{2} + {| d 𝐫 |}^{2}

5-

\cos θ = | d 𝐫 | / c d t = v / c

6-

= \int_{P}^{Q} 𝑑 t = \int_{P}^{Q} 𝑑 τ \sqrt{1 + \cot^{2} θ}

7-

= t \sin θ = t \sqrt{1 - u^{2} / c^{2}} < t

8-

\sqrt{1 - u^{2} / c^{2}}

9-

O P O_{2}

10-

\cos θ = 2 x_{0} / c t = u / c

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1606.07975

Formulas:

1-

R_{s} = 2 G M / c^{2} =

2-

t = R_{s} / c =

3-

R_{s} = 2 G M / c^{2}

4-

V (R_{1}, R_{2}, M) = \int_{R_{1}}^{R_{2}} \frac{4 π r^{2}}{\sqrt{1 - R_{s} / r}} 𝑑 r

5-

Δ V (R_{1}, R_{2}, M) = V (R_{1}, R_{2}, M) - V (R_{1}, R_{2}, 0)

6-

Δ V_{t o t a l} = Δ V (R_{1}, R_{2}, 62) - Δ V (R_{1}, R_{2}, 29) - Δ V (R_{1}, R_{2}, 36)

7-

R_{2} = 4 R_{s}

8-

R_{1} = 1.5 R_{s}

9-

\sqrt[3]{Δ V_{t o t a l}} / c \approx 0.0012

10-

d s^{2} = - d t^{2} + a (t) (\frac{d r^{2}}{1 + k r^{2} / R^{2}} + r^{2} d Ω^{2})

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1312.0409

Formulas:

1-

\min_{Ψ} {⟨ Ψ | \hat{T} + {\hat{V}}_{ne} + {\hat{W}}_{ee}^{lr, μ} | Ψ ⟩ + E_{Hxc}^{sr, μ} [n_{Ψ}]}

2-

⟨ Ψ^{μ} | \hat{T} + {\hat{V}}_{ne} + {\hat{W}}_{ee}^{lr, μ} | Ψ^{μ} ⟩ + E_{Hxc}^{sr, μ} [n_{Ψ^{μ}}],

3-

{\hat{W}}_{ee}^{lr, μ}

4-

w_{ee}^{lr, μ} (r_{12}) = \erf (μ r_{12}) / r_{12}

5-

E_{Hxc}^{sr, μ} [n]

6-

E_{Hxc}^{sr, μ} [n] = E_{H}^{sr, μ} [n] + E_{x}^{sr, μ} [n] + E_{c}^{sr, μ} [n],

7-

E_{H}^{sr, μ} [n]

8-

E_{x}^{sr, μ} [n] = ⟨ Φ^{KS} [n] | {\hat{W}}_{ee}^{sr, μ} | Φ^{KS} [n] ⟩ - E_{H}^{sr, μ} [n]

9-

Φ^{KS} [n]

10-

E_{Hxc}^{sr, μ} [n] = ⟨ Ψ^{μ} [n] | (β - 1) {\hat{W}}_{ee}^{lr, μ} + α {\hat{W}}_{ee} | Ψ^{μ} [n] ⟩ + {\bar{E}}_{Hxc}^{μ, α, β} [n],

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1512.00399

Formulas:

1-

𝐱_{k + 1} = {𝐅𝐱}_{k} + 𝚪 𝐯_{k}

2-

E (𝐯_{k}) = 𝟎

3-

E (𝐯_{k} 𝐯_{k}^{T}) = 𝐐_{k}

4-

k = 0, 1, 2, \dots

5-

𝒩 = {1, 2, \dots, N}

6-

𝐳_{k}^{i} = 𝐇^{i} 𝐱_{k} + 𝐰_{k}^{i}

7-

E (𝐰_{k}^{i}) = 𝟎

8-

E (𝐰_{k}^{i} 𝐰_{k}^{i}^{T}) = 𝐑_{w}^{i}

9-

k = 1, 2, \dots

10-

i = 1, 2, \dots, N

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/math/0411301

Formulas:

1-

r = 1 - s^{2}

2-

μ = ν h^{- 1} .

3-

μ (E) = ν (h^{- 1} (E))

4-

{[0, 1]}^{n}

5-

𝒞 = {0, 1}^{N}

6-

μ (1 / 2)

7-

{[0, 1]}^{n}

8-

r \sim_{t o p} s

9-

0 < r, s < 1

10-

s = \sum_{i = 0}^{n} a_{i} r^{i} {(1 - r)}^{n - i},

Formulas (html):

Correct Categorie: math

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1805.10332

Formulas:

1-

K_{z z} = \frac{H}{3} {(\frac{L}{H})}^{4 / 3} {(\frac{R σ T^{4}}{μ ρ c_{p}})}^{1 / 3},

2-

\frac{R T}{μ g}

3-

L = H m a x (0.1, Γ / Γ_{a}),

4-

10^{5} {cm}^{- 2} s^{- 1}

5-

ϕ_{i} \approx \frac{b_{i} f_{i}}{H}

6-

v_{i} \approx \frac{b_{i}}{n H} = \frac{D_{i}}{H} .

7-

(1.9 - 7.1) \cdot 10^{9}

8-

{cm}^{- 2} s^{- 1}

9-

c_{p, O_{2}} = 1118.1 - 0.722 T + 2.07 \cdot 10^{- 3} T^{2} - 1.30 \cdot 10^{- 6} T^{3} - 2.63 \cdot 10^{5} T^{- 2} J k g^{- 1} K^{- 1} .

10-

4 \cdot 10^{25} molec . {cm}^{- 2}

Formulas (html):

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xref="S2.SS1.p6.3.m3.1.1.4.cmml"><mi mathvariant="normal" id="S2.SS1.p6.3.m3.1.1.4.2" xref="S2.SS1.p6.3.m3.1.1.4.2.cmml">s</mi><mrow id="S2.SS1.p6.3.m3.1.1.4.3" xref="S2.SS1.p6.3.m3.1.1.4.3.cmml"><mo id="S2.SS1.p6.3.m3.1.1.4.3.1" xref="S2.SS1.p6.3.m3.1.1.4.3.1.cmml">-</mo><mn id="S2.SS1.p6.3.m3.1.1.4.3.2" xref="S2.SS1.p6.3.m3.1.1.4.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><msub id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.2.2" xref="S2.E3.m1.1.1.2.2.cmml">ϕ</mi><mi id="S2.E3.m1.1.1.2.3" xref="S2.E3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.cmml">≈</mo><mfrac id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml"><mrow id="S2.E3.m1.1.1.3.2" xref="S2.E3.m1.1.1.3.2.cmml"><msub id="S2.E3.m1.1.1.3.2.2" xref="S2.E3.m1.1.1.3.2.2.cmml"><mi id="S2.E3.m1.1.1.3.2.2.2" xref="S2.E3.m1.1.1.3.2.2.2.cmml">b</mi><mi id="S2.E3.m1.1.1.3.2.2.3" xref="S2.E3.m1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S2.E3.m1.1.1.3.2.1" xref="S2.E3.m1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E3.m1.1.1.3.2.3" xref="S2.E3.m1.1.1.3.2.3.cmml"><mi id="S2.E3.m1.1.1.3.2.3.2" xref="S2.E3.m1.1.1.3.2.3.2.cmml">f</mi><mi id="S2.E3.m1.1.1.3.2.3.3" xref="S2.E3.m1.1.1.3.2.3.3.cmml">i</mi></msub></mrow><mi id="S2.E3.m1.1.1.3.3" xref="S2.E3.m1.1.1.3.3.cmml">H</mi></mfrac></mrow></math>, <math><mrow id="S2.E4.m1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml"><mrow id="S2.E4.m1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml"><msub id="S2.E4.m1.1.1.1.1.2" xref="S2.E4.m1.1.1.1.1.2.cmml"><mi id="S2.E4.m1.1.1.1.1.2.2" xref="S2.E4.m1.1.1.1.1.2.2.cmml">v</mi><mi id="S2.E4.m1.1.1.1.1.2.3" xref="S2.E4.m1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E4.m1.1.1.1.1.3" xref="S2.E4.m1.1.1.1.1.3.cmml">≈</mo><mfrac id="S2.E4.m1.1.1.1.1.4" xref="S2.E4.m1.1.1.1.1.4.cmml"><msub id="S2.E4.m1.1.1.1.1.4.2" xref="S2.E4.m1.1.1.1.1.4.2.cmml"><mi id="S2.E4.m1.1.1.1.1.4.2.2" xref="S2.E4.m1.1.1.1.1.4.2.2.cmml">b</mi><mi id="S2.E4.m1.1.1.1.1.4.2.3" xref="S2.E4.m1.1.1.1.1.4.2.3.cmml">i</mi></msub><mrow id="S2.E4.m1.1.1.1.1.4.3" xref="S2.E4.m1.1.1.1.1.4.3.cmml"><mi id="S2.E4.m1.1.1.1.1.4.3.2" xref="S2.E4.m1.1.1.1.1.4.3.2.cmml">n</mi><mo id="S2.E4.m1.1.1.1.1.4.3.1" xref="S2.E4.m1.1.1.1.1.4.3.1.cmml">⁢</mo><mi id="S2.E4.m1.1.1.1.1.4.3.3" xref="S2.E4.m1.1.1.1.1.4.3.3.cmml">H</mi></mrow></mfrac><mo id="S2.E4.m1.1.1.1.1.5" xref="S2.E4.m1.1.1.1.1.5.cmml">=</mo><mfrac id="S2.E4.m1.1.1.1.1.6" xref="S2.E4.m1.1.1.1.1.6.cmml"><msub id="S2.E4.m1.1.1.1.1.6.2" xref="S2.E4.m1.1.1.1.1.6.2.cmml"><mi id="S2.E4.m1.1.1.1.1.6.2.2" xref="S2.E4.m1.1.1.1.1.6.2.2.cmml">D</mi><mi id="S2.E4.m1.1.1.1.1.6.2.3" xref="S2.E4.m1.1.1.1.1.6.2.3.cmml">i</mi></msub><mi id="S2.E4.m1.1.1.1.1.6.3" xref="S2.E4.m1.1.1.1.1.6.3.cmml">H</mi></mfrac></mrow><mo id="S2.E4.m1.1.1.1.2" xref="S2.E4.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.SS1.p9.4.m4.1.1" xref="S2.SS1.p9.4.m4.1.1.cmml"><mrow id="S2.SS1.p9.4.m4.1.1.1.1" xref="S2.SS1.p9.4.m4.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p9.4.m4.1.1.1.1.2" xref="S2.SS1.p9.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p9.4.m4.1.1.1.1.1" xref="S2.SS1.p9.4.m4.1.1.1.1.1.cmml"><mn id="S2.SS1.p9.4.m4.1.1.1.1.1.2" xref="S2.SS1.p9.4.m4.1.1.1.1.1.2.cmml">1.9</mn><mo id="S2.SS1.p9.4.m4.1.1.1.1.1.1" xref="S2.SS1.p9.4.m4.1.1.1.1.1.1.cmml">-</mo><mn id="S2.SS1.p9.4.m4.1.1.1.1.1.3" xref="S2.SS1.p9.4.m4.1.1.1.1.1.3.cmml">7.1</mn></mrow><mo stretchy="false" id="S2.SS1.p9.4.m4.1.1.1.1.3" xref="S2.SS1.p9.4.m4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p9.4.m4.1.1.2" xref="S2.SS1.p9.4.m4.1.1.2.cmml">⋅</mo><msup id="S2.SS1.p9.4.m4.1.1.3" xref="S2.SS1.p9.4.m4.1.1.3.cmml"><mn id="S2.SS1.p9.4.m4.1.1.3.2" xref="S2.SS1.p9.4.m4.1.1.3.2.cmml">10</mn><mn id="S2.SS1.p9.4.m4.1.1.3.3" xref="S2.SS1.p9.4.m4.1.1.3.3.cmml">9</mn></msup></mrow></math>, <math><mrow id="S2.SS1.p9.6.m6.1.1" xref="S2.SS1.p9.6.m6.1.1.cmml"><msup id="S2.SS1.p9.6.m6.1.1.2" xref="S2.SS1.p9.6.m6.1.1.2.cmml"><mi id="S2.SS1.p9.6.m6.1.1.2.2" xref="S2.SS1.p9.6.m6.1.1.2.2.cmml">cm</mi><mrow id="S2.SS1.p9.6.m6.1.1.2.3" xref="S2.SS1.p9.6.m6.1.1.2.3.cmml"><mo id="S2.SS1.p9.6.m6.1.1.2.3.1" xref="S2.SS1.p9.6.m6.1.1.2.3.1.cmml">-</mo><mn id="S2.SS1.p9.6.m6.1.1.2.3.2" xref="S2.SS1.p9.6.m6.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S2.SS1.p9.6.m6.1.1.1" xref="S2.SS1.p9.6.m6.1.1.1.cmml">⁢</mo><msup id="S2.SS1.p9.6.m6.1.1.3" xref="S2.SS1.p9.6.m6.1.1.3.cmml"><mi mathvariant="normal" id="S2.SS1.p9.6.m6.1.1.3.2" xref="S2.SS1.p9.6.m6.1.1.3.2.cmml">s</mi><mrow id="S2.SS1.p9.6.m6.1.1.3.3" xref="S2.SS1.p9.6.m6.1.1.3.3.cmml"><mo id="S2.SS1.p9.6.m6.1.1.3.3.1" xref="S2.SS1.p9.6.m6.1.1.3.3.1.cmml">-</mo><mn id="S2.SS1.p9.6.m6.1.1.3.3.2" xref="S2.SS1.p9.6.m6.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.E5.m1.3.3.1" xref="S2.E5.m1.3.3.1.1.cmml"><mrow id="S2.E5.m1.3.3.1.1" xref="S2.E5.m1.3.3.1.1.cmml"><msub id="S2.E5.m1.3.3.1.1.2" xref="S2.E5.m1.3.3.1.1.2.cmml"><mi id="S2.E5.m1.3.3.1.1.2.2" xref="S2.E5.m1.3.3.1.1.2.2.cmml">c</mi><mrow id="S2.E5.m1.2.2.2.2" xref="S2.E5.m1.2.2.2.3.cmml"><mi id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.cmml">p</mi><mo id="S2.E5.m1.2.2.2.2.2" xref="S2.E5.m1.2.2.2.3.cmml">,</mo><msub id="S2.E5.m1.2.2.2.2.1" xref="S2.E5.m1.2.2.2.2.1.cmml"><mi id="S2.E5.m1.2.2.2.2.1.2" xref="S2.E5.m1.2.2.2.2.1.2.cmml">O</mi><mn id="S2.E5.m1.2.2.2.2.1.3" xref="S2.E5.m1.2.2.2.2.1.3.cmml">2</mn></msub></mrow></msub><mo id="S2.E5.m1.3.3.1.1.1" xref="S2.E5.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.3.3.1.1.3" xref="S2.E5.m1.3.3.1.1.3.cmml"><mrow id="S2.E5.m1.3.3.1.1.3.2" xref="S2.E5.m1.3.3.1.1.3.2.cmml"><mrow id="S2.E5.m1.3.3.1.1.3.2.2" xref="S2.E5.m1.3.3.1.1.3.2.2.cmml"><mn id="S2.E5.m1.3.3.1.1.3.2.2.2" xref="S2.E5.m1.3.3.1.1.3.2.2.2.cmml">1118.1</mn><mo id="S2.E5.m1.3.3.1.1.3.2.2.1" xref="S2.E5.m1.3.3.1.1.3.2.2.1.cmml">-</mo><mrow id="S2.E5.m1.3.3.1.1.3.2.2.3" xref="S2.E5.m1.3.3.1.1.3.2.2.3.cmml"><mn id="S2.E5.m1.3.3.1.1.3.2.2.3.2" xref="S2.E5.m1.3.3.1.1.3.2.2.3.2.cmml">0.722</mn><mo id="S2.E5.m1.3.3.1.1.3.2.2.3.1" xref="S2.E5.m1.3.3.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E5.m1.3.3.1.1.3.2.2.3.3" xref="S2.E5.m1.3.3.1.1.3.2.2.3.3.cmml">T</mi></mrow></mrow><mo id="S2.E5.m1.3.3.1.1.3.2.1" xref="S2.E5.m1.3.3.1.1.3.2.1.cmml">+</mo><mrow id="S2.E5.m1.3.3.1.1.3.2.3" xref="S2.E5.m1.3.3.1.1.3.2.3.cmml"><mrow id="S2.E5.m1.3.3.1.1.3.2.3.2" xref="S2.E5.m1.3.3.1.1.3.2.3.2.cmml"><mn id="S2.E5.m1.3.3.1.1.3.2.3.2.2" xref="S2.E5.m1.3.3.1.1.3.2.3.2.2.cmml">2.07</mn><mo id="S2.E5.m1.3.3.1.1.3.2.3.2.1" xref="S2.E5.m1.3.3.1.1.3.2.3.2.1.cmml">⋅</mo><msup id="S2.E5.m1.3.3.1.1.3.2.3.2.3" xref="S2.E5.m1.3.3.1.1.3.2.3.2.3.cmml"><mn id="S2.E5.m1.3.3.1.1.3.2.3.2.3.2" xref="S2.E5.m1.3.3.1.1.3.2.3.2.3.2.cmml">10</mn><mrow id="S2.E5.m1.3.3.1.1.3.2.3.2.3.3" xref="S2.E5.m1.3.3.1.1.3.2.3.2.3.3.cmml"><mo id="S2.E5.m1.3.3.1.1.3.2.3.2.3.3.1" xref="S2.E5.m1.3.3.1.1.3.2.3.2.3.3.1.cmml">-</mo><mn id="S2.E5.m1.3.3.1.1.3.2.3.2.3.3.2" xref="S2.E5.m1.3.3.1.1.3.2.3.2.3.3.2.cmml">3</mn></mrow></msup></mrow><mo id="S2.E5.m1.3.3.1.1.3.2.3.1" xref="S2.E5.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><msup id="S2.E5.m1.3.3.1.1.3.2.3.3" xref="S2.E5.m1.3.3.1.1.3.2.3.3.cmml"><mi id="S2.E5.m1.3.3.1.1.3.2.3.3.2" xref="S2.E5.m1.3.3.1.1.3.2.3.3.2.cmml">T</mi><mn id="S2.E5.m1.3.3.1.1.3.2.3.3.3" xref="S2.E5.m1.3.3.1.1.3.2.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S2.E5.m1.3.3.1.1.3.1" xref="S2.E5.m1.3.3.1.1.3.1.cmml">-</mo><mrow id="S2.E5.m1.3.3.1.1.3.3" xref="S2.E5.m1.3.3.1.1.3.3.cmml"><mrow id="S2.E5.m1.3.3.1.1.3.3.2" xref="S2.E5.m1.3.3.1.1.3.3.2.cmml"><mn id="S2.E5.m1.3.3.1.1.3.3.2.2" xref="S2.E5.m1.3.3.1.1.3.3.2.2.cmml">1.30</mn><mo id="S2.E5.m1.3.3.1.1.3.3.2.1" xref="S2.E5.m1.3.3.1.1.3.3.2.1.cmml">⋅</mo><msup id="S2.E5.m1.3.3.1.1.3.3.2.3" xref="S2.E5.m1.3.3.1.1.3.3.2.3.cmml"><mn id="S2.E5.m1.3.3.1.1.3.3.2.3.2" xref="S2.E5.m1.3.3.1.1.3.3.2.3.2.cmml">10</mn><mrow id="S2.E5.m1.3.3.1.1.3.3.2.3.3" xref="S2.E5.m1.3.3.1.1.3.3.2.3.3.cmml"><mo id="S2.E5.m1.3.3.1.1.3.3.2.3.3.1" xref="S2.E5.m1.3.3.1.1.3.3.2.3.3.1.cmml">-</mo><mn id="S2.E5.m1.3.3.1.1.3.3.2.3.3.2" xref="S2.E5.m1.3.3.1.1.3.3.2.3.3.2.cmml">6</mn></mrow></msup></mrow><mo id="S2.E5.m1.3.3.1.1.3.3.1" xref="S2.E5.m1.3.3.1.1.3.3.1.cmml">⁢</mo><msup id="S2.E5.m1.3.3.1.1.3.3.3" xref="S2.E5.m1.3.3.1.1.3.3.3.cmml"><mi id="S2.E5.m1.3.3.1.1.3.3.3.2" xref="S2.E5.m1.3.3.1.1.3.3.3.2.cmml">T</mi><mn id="S2.E5.m1.3.3.1.1.3.3.3.3" xref="S2.E5.m1.3.3.1.1.3.3.3.3.cmml">3</mn></msup></mrow><mo id="S2.E5.m1.3.3.1.1.3.1a" xref="S2.E5.m1.3.3.1.1.3.1.cmml">-</mo><mrow id="S2.E5.m1.3.3.1.1.3.4" xref="S2.E5.m1.3.3.1.1.3.4.cmml"><mrow id="S2.E5.m1.3.3.1.1.3.4.2" xref="S2.E5.m1.3.3.1.1.3.4.2.cmml"><mn id="S2.E5.m1.3.3.1.1.3.4.2.2" xref="S2.E5.m1.3.3.1.1.3.4.2.2.cmml">2.63</mn><mo id="S2.E5.m1.3.3.1.1.3.4.2.1" xref="S2.E5.m1.3.3.1.1.3.4.2.1.cmml">⋅</mo><msup id="S2.E5.m1.3.3.1.1.3.4.2.3" xref="S2.E5.m1.3.3.1.1.3.4.2.3.cmml"><mn id="S2.E5.m1.3.3.1.1.3.4.2.3.2" xref="S2.E5.m1.3.3.1.1.3.4.2.3.2.cmml">10</mn><mn id="S2.E5.m1.3.3.1.1.3.4.2.3.3" xref="S2.E5.m1.3.3.1.1.3.4.2.3.3.cmml">5</mn></msup></mrow><mo id="S2.E5.m1.3.3.1.1.3.4.1" xref="S2.E5.m1.3.3.1.1.3.4.1.cmml">⁢</mo><msup id="S2.E5.m1.3.3.1.1.3.4.3" xref="S2.E5.m1.3.3.1.1.3.4.3.cmml"><mi id="S2.E5.m1.3.3.1.1.3.4.3.2" xref="S2.E5.m1.3.3.1.1.3.4.3.2.cmml">T</mi><mrow id="S2.E5.m1.3.3.1.1.3.4.3.3" xref="S2.E5.m1.3.3.1.1.3.4.3.3.cmml"><mo id="S2.E5.m1.3.3.1.1.3.4.3.3.1" xref="S2.E5.m1.3.3.1.1.3.4.3.3.1.cmml">-</mo><mn id="S2.E5.m1.3.3.1.1.3.4.3.3.2" xref="S2.E5.m1.3.3.1.1.3.4.3.3.2.cmml">2</mn></mrow></msup><mo id="S2.E5.m1.3.3.1.1.3.4.1a" xref="S2.E5.m1.3.3.1.1.3.4.1.cmml">⁢</mo><mi id="S2.E5.m1.3.3.1.1.3.4.4" xref="S2.E5.m1.3.3.1.1.3.4.4.cmml">J</mi><mo id="S2.E5.m1.3.3.1.1.3.4.1b" xref="S2.E5.m1.3.3.1.1.3.4.1.cmml">⁢</mo><mi id="S2.E5.m1.3.3.1.1.3.4.5" xref="S2.E5.m1.3.3.1.1.3.4.5.cmml">k</mi><mo id="S2.E5.m1.3.3.1.1.3.4.1c" xref="S2.E5.m1.3.3.1.1.3.4.1.cmml">⁢</mo><msup id="S2.E5.m1.3.3.1.1.3.4.6" xref="S2.E5.m1.3.3.1.1.3.4.6.cmml"><mi id="S2.E5.m1.3.3.1.1.3.4.6.2" xref="S2.E5.m1.3.3.1.1.3.4.6.2.cmml">g</mi><mrow id="S2.E5.m1.3.3.1.1.3.4.6.3" xref="S2.E5.m1.3.3.1.1.3.4.6.3.cmml"><mo id="S2.E5.m1.3.3.1.1.3.4.6.3.1" xref="S2.E5.m1.3.3.1.1.3.4.6.3.1.cmml">-</mo><mn id="S2.E5.m1.3.3.1.1.3.4.6.3.2" xref="S2.E5.m1.3.3.1.1.3.4.6.3.2.cmml">1</mn></mrow></msup><mo id="S2.E5.m1.3.3.1.1.3.4.1d" xref="S2.E5.m1.3.3.1.1.3.4.1.cmml">⁢</mo><msup id="S2.E5.m1.3.3.1.1.3.4.7" xref="S2.E5.m1.3.3.1.1.3.4.7.cmml"><mi id="S2.E5.m1.3.3.1.1.3.4.7.2" xref="S2.E5.m1.3.3.1.1.3.4.7.2.cmml">K</mi><mrow id="S2.E5.m1.3.3.1.1.3.4.7.3" xref="S2.E5.m1.3.3.1.1.3.4.7.3.cmml"><mo id="S2.E5.m1.3.3.1.1.3.4.7.3.1" xref="S2.E5.m1.3.3.1.1.3.4.7.3.1.cmml">-</mo><mn id="S2.E5.m1.3.3.1.1.3.4.7.3.2" xref="S2.E5.m1.3.3.1.1.3.4.7.3.2.cmml">1</mn></mrow></msup></mrow></mrow></mrow><mo id="S2.E5.m1.3.3.1.2" xref="S2.E5.m1.3.3.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S3.p6.2.m2.2.2.2" xref="S3.p6.2.m2.2.2.3.cmml"><mrow id="S3.p6.2.m2.1.1.1.1" xref="S3.p6.2.m2.1.1.1.1.cmml"><mrow id="S3.p6.2.m2.1.1.1.1.2" xref="S3.p6.2.m2.1.1.1.1.2.cmml"><mn id="S3.p6.2.m2.1.1.1.1.2.2" xref="S3.p6.2.m2.1.1.1.1.2.2.cmml">4</mn><mo id="S3.p6.2.m2.1.1.1.1.2.1" xref="S3.p6.2.m2.1.1.1.1.2.1.cmml">⋅</mo><msup id="S3.p6.2.m2.1.1.1.1.2.3" xref="S3.p6.2.m2.1.1.1.1.2.3.cmml"><mn id="S3.p6.2.m2.1.1.1.1.2.3.2" xref="S3.p6.2.m2.1.1.1.1.2.3.2.cmml">10</mn><mn id="S3.p6.2.m2.1.1.1.1.2.3.3" xref="S3.p6.2.m2.1.1.1.1.2.3.3.cmml">25</mn></msup></mrow><mo id="S3.p6.2.m2.1.1.1.1.1" xref="S3.p6.2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S3.p6.2.m2.1.1.1.1.3" xref="S3.p6.2.m2.1.1.1.1.3.cmml">molec</mi></mrow><mo id="S3.p6.2.m2.2.2.2.3" xref="S3.p6.2.m2.2.2.3a.cmml">.</mo><msup id="S3.p6.2.m2.2.2.2.2" xref="S3.p6.2.m2.2.2.2.2.cmml"><mi id="S3.p6.2.m2.2.2.2.2.2" xref="S3.p6.2.m2.2.2.2.2.2.cmml">cm</mi><mrow id="S3.p6.2.m2.2.2.2.2.3" xref="S3.p6.2.m2.2.2.2.2.3.cmml"><mo id="S3.p6.2.m2.2.2.2.2.3.1" xref="S3.p6.2.m2.2.2.2.2.3.1.cmml">-</mo><mn id="S3.p6.2.m2.2.2.2.2.3.2" xref="S3.p6.2.m2.2.2.2.2.3.2.cmml">2</mn></mrow></msup></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2004.02601

Formulas:

1-

- β S (I + A),

2-

β S (I + A) - γ E,

3-

γ p E - μ_{1} I,

4-

γ (1 - p) E - μ_{2} A,

5-

μ_{1} I + μ_{2} A,

6-

\frac{1}{μ_{2}} > \frac{1}{μ_{1}}

7-

S + I + E + A + R = 1

8-

(1, 0, 0, 0)

9-

J = (\begin{matrix} - β (I + A) & 0 & - β S & - β S \\ β (I + A) & - γ & β S & β S \\ 0 & γ p & - μ_{1} & 0 \\ 0 & γ (1 - p) & 0 & - μ_{2} \end{matrix})

10-

J (1, 0, 0, 0) = (\begin{matrix} 0 & 0 & - β & - β \\ 0 & - γ & β & β \\ 0 & γ p & - μ_{1} & 0 \\ 0 & γ (1 - p) & 0 & - μ_{2} \end{matrix}) .

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9706098

Formulas:

1-

t = e^{- 2 / T}

2-

ℋ = J \sum_{⟨ i, j ⟩} s_{i} s_{j},

3-

s_{0} = \frac{2}{π} k_{B} \int_{0}^{π / 3} \ln (2 \cos ω) d ω ≃ 0.323 k_{B} .

4-

S (𝐩) = \sum_{𝐫} e^{i 𝐩 \cdot 𝐫} ⟨ s_{𝟎} s_{𝐫} ⟩,

5-

t = e^{- 2 / T}

6-

⟨ s_{𝟎} s_{𝐫} ⟩ \sim \frac{\cos (\frac{2 π r}{3})}{\sqrt{r}} for r ≫ 1,

7-

\cos (\frac{2 π r}{3})

8-

X_{6}^{(s)} = \frac{1}{4}

9-

\sim r^{- 2 X_{6}^{(s)}}

10-

𝐪_{1, 2} = (\pm \frac{4 π}{3}, 0)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1605.09182

Formulas:

1-

(3 d_{ν})

2-

\sum_{m, σ} (ε_{m} - μ) c_{m σ}^{†} c_{m σ} + \sum_{ν, σ} (ε_{d ν} - μ) d_{ν σ}^{†} d_{ν σ}

3-

\sum_{m, ν, σ} (V_{m ν} c_{m σ}^{†} d_{ν σ} + V_{m ν}^{*} d_{ν σ}^{†} c_{m σ})

4-

\sum_{ν} U_{ν} n_{ν ↑} n_{ν ↓}

5-

n_{ν σ} = d_{ν σ}^{†} d_{ν σ}

6-

ψ_{n} (𝐫)

7-

F (𝐫) ψ_{n} (𝐫) = E_{n} ψ_{n} (𝐫),

8-

F (𝐫) = (- \frac{ℏ^{2}}{2 m_{e}} \nabla_{𝐫}^{2} + V_{e x t} (𝐫) + \int d^{3} r^{'} \frac{ρ (𝐫^{'})}{| 𝐫 - 𝐫^{'} |} + V_{x c} (𝐫))

9-

| ψ_{n} ⟩ = \sum_{i}^{N} C_{n i} | ϕ_{i} ⟩,

10-

F_{i j} = \int d^{3} r ϕ_{i}^{*} (𝐫) F (𝐫) ϕ_{j} (𝐫)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1702.03086

Formulas:

1-

s_{\max} = 3000 µm

2-

ρ_{s} = 2.7 g / {cm}^{3}

3-

p : (p + 1)

4-

m ≳ 10^{- 6} g

5-

m ≲ 10^{- 9} g

6-

m \sim 3 \times 10^{- 7} g

7-

ρ_{s} = 2 g / {cm}^{3}

8-

m \sim 1.5 \times 10^{- 5} g

9-

ρ_{s} = 2 g / {cm}^{3}

10-

n_{d} (s) \propto s^{- 2.72}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2011.07298

Formulas:

1-

T = T_{J} + T_{b}

2-

H = {\begin{matrix} b \sum_{j} σ_{j}^{z}, 0 \leq t < T_{b} \\ \sum_{j} J_{j} σ_{j}^{x} σ_{j + 1}^{x}, T_{b} \leq t < T_{b} + T_{J} . \end{matrix}

3-

σ_{j}^{x, y, z}

4-

J_{j} = J + A_{J} \sin (2 π Q j a + φ_{J}) .

5-

Q = (\sqrt{5} + 1) / 2

6-

q_{n + 1} / q_{n}

7-

q_{n + 1} = q_{n} + q_{n - 1}

8-

q_{1} = 1, q_{2} = 1

9-

U_{F} = \exp [- i T_{J} \sum_{j} J_{j} σ_{j}^{x} σ_{j + 1}^{x}] \exp [- i T_{b} b \sum_{j} σ_{j}^{z}] .

10-

P = \prod_{j} σ_{j}^{z}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1409.4045

Formulas:

1-

5 \times 5 μ m^{2}

2-

V_{3} (B) \neq V_{3} (- B)

3-

\partial_{V} G_{3} / G_{3}

4-

N_{3} \leq 1 ≪ N_{1}, N_{2}

5-

V_{3} (B)

6-

l_{e} = 4.8 μ m

7-

l_{S O} = 36 nm

8-

\partial_{B} V_{3} / I

9-

\partial_{V} G_{3} / G_{3}

10-

R_{C} = V_{C} / I

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1110.2495

Formulas:

1-

ℋ = \sum_{⟨ i, j ⟩} J_{1} 𝐒_{i} \cdot 𝐒_{j} + J_{2} {(𝐒_{i} \cdot 𝐒_{j})}^{2},

2-

0 < J_{1} \leq J_{2}

3-

| Ψ ⟩ = \prod_{i = 1}^{N} | ψ_{i} ⟩,

4-

= i \frac{| 1 ⟩ - | \bar{1} ⟩}{\sqrt{2}},

5-

= \frac{| 1 ⟩ + | \bar{1} ⟩}{\sqrt{2}},

6-

= - i | 0 ⟩ .

7-

| 𝐝 ⟩ = \sum_{α} d_{α} | α ⟩

8-

𝐝 = 𝐮 + i 𝐯

9-

u^{2} + v^{2} = 1

10-

⟨ 𝐒 ⟩ = 2 𝐮 \times 𝐯

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1709.03345

Formulas:

1-

3 \times 10^{11} M_{⊙}

2-

\log M^{*} > 11.0

3-

n_{S è rsic} > 2.0

4-

I_{i, j} = \sum_{k \in 𝒱} \frac{x_{i}^{(k)} x_{j}^{(k)}}{r_{k}^{2}},

5-

r_{k} = \sqrt{x_{k}^{2} + y_{k}^{2} / p^{2} + z_{k}^{2} / q^{2}}

6-

b / a - c / a < 0.2

7-

\log M^{*} > 11.6

8-

\sim 80 km s^{- 1}

9-

\sim 200 km / s

10-

\sim 70 km / s

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0508212

Formulas:

1-

f (ϵ_{0}) \propto P_{0} ϵ_{0}^{- (2 α + 1)},

2-

s (ν) \propto P D^{α + 2} {(B_{⟂}^{'})}^{α + 1} ν^{- α}

3-

{[1 - ϵ (ν) / ϵ_{\infty}]}^{2 α - 1},

4-

ϵ_{\infty} \leq ϵ (ν)

5-

d Q = \frac{α + 1}{α + 5 / 3} d I \cos 2 χ,

6-

d U = \frac{α + 1}{α + 5 / 3} d I \sin 2 χ .

7-

\tan ψ = \frac{B_{z}}{B_{y}} \frac{| 𝜷 | - 𝜷 \cdot 𝐧 / | 𝜷 |}{1 - 𝜷 \cdot 𝐧},

8-

𝜷 \cdot 𝐁 = 0 \Rightarrow B_{x} = 0

9-

{(r \sin θ - R_{1})}^{2} + {(r \cos θ)}^{2} - R_{2}^{2} \leq 0,

10-

L = 5 \times 10^{38}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9804194

Formulas:

1-

3 \times 10^{8} M_{⊙}

2-

V = 155 \pm 17 km s^{- 1}

3-

σ \approx 70 km s^{- 1}

4-

σ \approx 400 km s^{- 1}

5-

500 km s^{- 1}

6-

V_{pred} = 92 km s^{- 1}

7-

M_{∙} = {3.3}_{- 1.3}^{+ 2.3} \times 10^{8} M_{⊙}

8-

1000 km s^{- 1}

9-

300 km s^{- 1}

10-

1^{''} = 284.6 pc

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9701221

Formulas:

1-

ν = 1 / (2 m + 1)

2-

σ_{x x} - σ_{x y}

3-

s l (2, 𝐙)

4-

σ_{x y} < \frac{1}{2}

5-

ℰ = \int_{0}^{\infty} 𝑑 ϵ ϵ n_{b e} (T) ρ_{q p} (ϵ) .

6-

n_{b e} = \frac{1}{e^{ϵ / T} - 1}

7-

C_{V} = \partial ℰ / \partial T

8-

ρ_{q p} (ϵ)

9-

σ_{x y} = 1 / 2, 1 / 4

10-

ν = 1 / (2 m + 1)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1812.01103

Formulas:

1-

R_{i} (τ) = \log P r i c e (τ) - \log P r i c e (τ - 1)

2-

P r i c e (τ)

3-

t = 1, 2, \dots, M

4-

ρ_{i, j} (t) = \sum_{t^{'} < τ} s g n (V_{i} (t^{'}) - V_{i} (τ)) s g n (V_{j} (t^{'}) - V_{j} (τ)),

5-

V_{i} \in {R_{i}, O_{i}}

6-

- 1 \leq ρ_{i, j} \leq 1

7-

d_{i, j} (t) = \sqrt{2 (1 - ρ_{i, j} (t))}

8-

2 \geq d_{i, j} \geq 0

9-

N (N - 1) / 2

10-

d_{1} (t), d_{2} (t), \dots, d_{N (N - 1) / 2} (t)

Formulas (html):

Correct Categorie: q-fin

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0403047

Formulas:

1-

ϵ_{e f f}

2-

μ_{e f f}

3-

Re (ϵ_{e f f}) < 0

4-

Re (μ_{e f f}) < 0

5-

a_{m m} = \frac{m}{H^{loc}} .

6-

R_{c} = Re (\frac{2 π r}{σ^{'} π r_{0}^{2}}) .

7-

σ^{'} = \frac{2 σ J_{1} (κ r_{0})}{κ r_{0} J_{0} (κ r_{0})}, κ = \frac{(1 - j)}{δ},

8-

δ = 1 / \sqrt{σ ω μ_{0}}

9-

I^{(1, 2)} = 0

10-

a_{m m} = - \frac{ω^{2} μ_{0}^{2} S^{2}}{L + M} \frac{1}{(ω^{2} - ω_{0}^{2}) + j ω Γ} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1510.00210

Formulas:

1-

T / e^{2} ρ

2-

r_{0} \approx 10^{- 5}

3-

ω << {(4 π^{2} e^{2} ρ / m L_{c})}^{1 / 2}

4-

L_{c} \approx 10^{- 3}

5-

Δ ρ_{σ} = δ ρ_{σ} + \frac{Π_{σ}}{Π} ρ_{p}, \sum_{σ} δ ρ_{σ} = 0 .

6-

Π_{σ} ρ_{p} / Π

7-

Π = \sum_{σ} Π_{σ}

8-

i ω Δ ρ_{σ} + j_{σ}^{'} = - ν_{s} Π^{*} (μ_{σ} - μ_{- σ})

9-

(i ω + ν) m u_{σ} + {(δ μ_{σ} + e δ φ)}^{'} = - m ν_{e e} \frac{ρ_{- σ}}{ρ_{σ}} (u_{σ} - u_{- σ})

10-

Π_{σ} δ μ_{σ} = δ ρ_{σ},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1502.04709

Formulas:

1-

R_{2.2} = 2.2 R_{d}

2-

ℱ (R) \equiv V_{bar} / V_{tot}

3-

(M_{*} / L) (R)

4-

f_{DM} (R) = M_{DM} (R) / M_{tot} (R) \approx 1 - ℱ^{2}

5-

M_{DM} (R)

6-

M_{tot} (R)

7-

f_{DM} (R)

8-

f_{DM} (R)

9-

ℱ \equiv V_{disk} (R_{\max}) / V_{tot} (R_{\max}) > 0.85,

10-

Δ \log V (L)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1706.04121

Formulas:

1-

R_{a d j}^{2}

2-

F (1, 71)

3-

F (1, 71)

4-

F (1, 71)

5-

F (1, 71)

6-

R_{a d j}^{2}

7-

R_{a d j}^{2}

8-

R_{a d j}^{2}

9-

R_{a d j}^{2}

10-

F (1, 71)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9609111

Formulas:

1-

Δ (V - I) \propto + 0.04 [F e / H]

2-

V - I = 0.053 P^{- 2.1}

3-

u v b y

4-

n (a) = \int K (q, a) N (q) 𝑑 q

5-

K (q, a)

6-

I_{m a x}

7-

{(V - I)}_{m a x}

8-

l (θ) = \sum_{i = 0}^{4} a_{i} \cos 2 π i θ

9-

a_{4} = a_{2} (0.125 - a_{2})

10-

Δ (V - I) = 0.02

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0702147

Formulas:

1-

v (r, t)

2-

v = \frac{R^{2} \dot{R}}{r^{2}},

3-

{\dot{R}}^{2} = \frac{2 p_{\infty}}{3 ρ_{\infty}} (\frac{a^{3}}{R^{3}} - 1),

4-

τ_{R} = a \sqrt{\frac{3 π ρ_{\infty}}{2 p_{\infty}}} \frac{Γ (5 / 6)}{Γ (1 / 3)},

5-

Γ (m / n)

6-

R \ddot{R} + \frac{3}{2} {\dot{R}}^{2} + \frac{p_{\infty}}{ρ_{\infty}} = 0,

7-

ρ (r, t)

8-

\frac{\partial ρ}{\partial t} + \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} ρ v) = 0 .

9-

\frac{1}{R^{2}} \frac{\partial ρ}{\partial R} + \frac{1}{r^{2}} \frac{\partial ρ}{\partial r} = 0 .

10-

ρ (r, R) = \frac{ρ_{b} - ρ_{a} e^{- λ}}{1 - e^{- λ}} + \frac{ρ_{a} - ρ_{b}}{1 - e^{- λ}} \exp [- (\frac{r^{3}}{a^{3}} - \frac{R^{3}}{a^{3}})],

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0005067

Formulas:

1-

A + B ⟷ C .

2-

N_{C} (t)

3-

N_{A, e q}

4-

N_{B, e q}

5-

\sum_{i} α_{i}^{(r)} N_{i} ⟷ 0

6-

x^{(r)} = \sum_{i} α_{i}^{(r)} \frac{μ_{i}}{T} .

7-

{\dot{N}}_{i} = \sum_{(r)} α_{i}^{(r)} R^{(r)} (1 - e^{x^{(r)}}) .

8-

V T^{3 / 2} =

9-

N_{i} = N_{i, e q} e^{μ_{i} / T}

10-

N_{i, e q} = V d_{i} {(\frac{m_{i} T}{2 π})}^{3 / 2} e^{- m_{i} / T} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0704.3091

Formulas:

1-

{3, 3, 5}

2-

{3, 3, 5}

3-

{5, 3, 3}

4-

3 {3} 3 {3} 3 {3} 3

5-

3 {3} 3 {3} 3 {3} 3

6-

ω = \exp (\frac{i π}{30})

7-

a > b > c > d

8-

45 x^{8} - 90 x^{6} + 60 x^{4} - 15 x^{2} + 1

9-

\begin{matrix} A_{n} = (a ω^{2 n}, d ω^{22 n}), & B_{n} = (b ω^{2 n + 1}, c ω^{22 n + 11}), \\ C_{n} = (c ω^{2 n + 1}, - b ω^{22 n + 11}), & D_{n} = (d ω^{2 n}, - a ω^{22 n}), \end{matrix}

10-

n \in {0, 1, 2, \dots, 28, 29}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

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