Run 16446903

Paper: https://arxiv.org/abs/2008.01464

Formulas:

1-

(J_{1} - J_{2})

2-

r = J_{2} / J_{1}

3-

ℋ = \sum_{b = 1}^{N_{b}} J_{b} 𝑺_{𝒊 (𝒃)} \cdot 𝑺_{𝒋 (𝒃)}

4-

𝑺_{𝒊 (𝒃)}

5-

𝒫 (J_{b}) = \frac{1}{2} [δ (J_{b} - J_{1}) + δ (J_{b} - J_{2})] .

6-

J_{1} + J_{2} = 2

7-

J_{1} > J_{2} > 0

8-

r = J_{2} / J_{1}

9-

k_{B} T_{N} / J

10-

H_{1, b} = (\frac{1}{4} - S_{i (b)}^{z} S_{j (b)}^{z})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1305.3017

Formulas:

1-

0 \overset{''}{.} 5 - 1 \overset{''}{.} 5

2-

1 \overset{''}{.} 0

3-

1 \overset{''}{.} 3

4-

Δ z = {[{(σ_{z})}^{2} + {(Δ λ / \bar{λ_{0}})}^{2}] / N}^{1 / 2},

5-

0 \overset{''}{.} 52

6-

1 \overset{''}{.} 55

7-

0 \overset{''}{.} 6

8-

b = 9 \overset{\circ}{.} 7

9-

0 \overset{''}{.} 06

10-

- 0 \overset{''}{.} 04

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0708.0426

Formulas:

1-

H ({\vec{r_{i}}}) = \sum_{i > j}^{N} ϵ Δ (\vec{r_{i}} - \vec{r_{j}}),

2-

Δ (\vec{r_{i}} - \vec{r_{j}})

3-

0.2 / (N + 6)

4-

0.8 / (N - 3)

5-

T_{o p t}

6-

| i - j | \geq 12

7-

T_{o p t}

8-

T_{o p t}

9-

P_{f o l d}

10-

P_{f o l d} (Γ)

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/hep-ex/0406033

Formulas:

1-

p D^{+} K^{-}

2-

Ξ_{c c}^{+} \to p D^{+} K^{-}

3-

Ξ_{c c}^{+} \to Λ_{c}^{+} K^{-} π^{+}

4-

3518.7 \pm 1.7 MeV / c^{2}

5-

Ξ_{c c}^{+} \to Λ_{c}^{+} K^{-} π^{+}

6-

Ξ_{c c}^{+} \to p D^{+} K^{-}

7-

Ξ_{c c}^{+} \to Λ_{c}^{+} K^{-} π^{+}

8-

Ξ_{c c}^{+} \to p D^{+} K^{-}

9-

σ_{p} / p < 1 %

10-

p ≳ 15 GeV / c

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ex

Result: correct

Paper: https://arxiv.org/abs/0711.4506

Formulas:

1-

(e / a) \approx 7.87

2-

M (μ_{0} H)

3-

μ_{0} H \geq 1

4-

d M_{s} / d H

5-

d M_{s} / d H \approx - 6

6-

d M_{s} / d H \approx - 2

7-

M (μ_{0} H)

8-

M (μ_{0} H)

9-

M (μ_{0} H)

10-

M (μ_{0} H)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0803.1900

Formulas:

1-

N_{H} = 2.05 \times 10^{21}

2-

E / Δ E ≃ 500

3-

≃ 1^{'} - 3^{'}

4-

Γ = {1.29}_{- 0.15}^{+ 0.16}

5-

4.56 \pm 0.48 \times 10^{- 3}

6-

- 2 \ln λ

7-

- 2 \ln λ

8-

- 2 \ln λ \sim 27

9-

Γ = {2.23}_{- 0.27}^{+ 0.36}

10-

N_{H}^{z} \sim 2 \times 10^{22}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1105.5206

Formulas:

1-

H_{0} = 68 \pm 5.5

2-

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

3-

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

4-

i = 1, \dots, N

5-

P_{i} = \frac{2^{- N} N!}{i! (N - i)!},

6-

M_{N + 1} = + \infty

7-

M_{N + 1 - j}

8-

C_{j} = 100 \times (P_{j} + P_{j + 1} + \dots + P_{N - j}) .

9-

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

10-

H_{0} = 100 h km s^{- 1} {Mpc}^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1603.02818

Formulas:

1-

\times 10^{- 14} e r g \cdot s^{- 1} c m^{- 2}

2-

\times 10^{- 14} e r g \cdot s^{- 1} c m^{- 2}

3-

\times 10^{- 14} e r g \cdot s^{- 1} c m^{- 2}

4-

t o p p a n e l

5-

B o t t o m p a n e l

6-

ρ (r, θ) = A r^{β} e x p (- γ c o s (θ))

7-

β = - 1.0, - 0.5, 0.0

8-

f_{AGN} (5 - 40)

9-

L_{[O IV]}

10-

f_{AGN} (5 - 40) / f_{AGN} (8 - 1000)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0011500

Formulas:

1-

300 M_{⊙} {yr}^{- 1}

2-

s_{i} \equiv c_{i} - n_{i} b .

3-

\frac{σ_{(s_{i})}}{s_{i}} = \frac{\sqrt{c_{i} + n_{i} b}}{c_{i} - n_{i} b} .

4-

n i \leq x < n (i + 1)

5-

n j \leq y < n (j + 1)

6-

S_{b} (r) = S_{0} {[1 + {(\frac{r}{r_{c}})}^{2}]}^{0.5 - 3 β},

7-

χ_{i}^{2} = \frac{{(S_{i} - s_{i})}^{2}}{S_{i} + n_{i} b} .

8-

C_{k} = [(s_{k}^{1} / s_{k}^{2}) / s_{k}^{3}] / \dots .

9-

\frac{σ_{(C_{k})}}{C_{k}} = {[\sum_{i = 1}^{N} \frac{c_{k}^{i} + n_{k} b^{i}}{{(c_{k}^{i} - n_{k} b^{i})}^{2}}]}^{1 / 2} .

10-

(s^{0} / s^{1}) / s^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1705.02847

Formulas:

1-

| A ⟩ = \frac{1}{\sqrt{2 T}} T_{-} | π ⟩,

2-

ρ_{n} (r) - ρ_{p} (r)

3-

ρ_{n} (r) - ρ_{p} (r) = ρ_{n (e x c)} (r) + δ ρ (r),

4-

δ ρ (r)

5-

δ ρ (r) = \sum_{i = 1}^{Z} {| φ_{i}^{n} (r) |}^{2} - \sum_{i = 1}^{Z}, {| φ_{i}^{p} (r) |}^{2}

6-

ρ_{n (e x c)}

7-

ρ_{n (e x c)} (r) = \sum_{i = Z + 1}^{N} {| φ_{i}^{n} (r) |}^{2},

8-

φ_{i}^{n (p)} (r)

9-

δ ρ (r)

10-

δ ρ (r)

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/1409.6795

Formulas:

1-

PG (5, q)

2-

PG (5, q)

3-

2 (q^{2} + q + 1)

4-

PG (5, q)

5-

C G (3, q)

6-

PG (5, q)

7-

q^{2} + q + 1

8-

PG (5, q)

9-

C G (3, q)

10-

\frac{1}{2} q^{3} (q - 1) (q^{3} + 1) .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0703103

Formulas:

1-

Q_{l} = 1 / (1 / Q_{0} + 1 / Q_{e})

2-

P_{in} = V_{in rms}^{2} / Z_{0}

3-

P_{out} = V_{out, rms}^{2} / Z_{0}

4-

V = γ \dot{I} / \sqrt{I_{c}^{2} - I^{2}}

5-

γ = Φ_{0} / 2 π = h / 2 e

6-

I = I_{dc} + I_{ac}

7-

| I | << I_{c}

8-

V / \dot{I} = f_{dc} + f_{ac1} I_{a c} + f_{a c 2} I_{ac}^{2} + \dots

9-

f_{dc} = γ / \sqrt{I_{c}^{2} - I_{dc}^{2}}

10-

i_{ac} \sin (ω t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1910.00325

Formulas:

1-

μ_{S} \approx 4.36 μ_{B}

2-

T_{1} = 58 \pm 4 μ

3-

λ_{S O} = - 9.60

4-

\hat{ℋ} = B_{2}^{0} {\hat{O}}_{2}^{0} + B_{4}^{0} {\hat{O}}_{4}^{0} + B_{4}^{4} {\hat{O}}_{4}^{4} + λ_{SO} \hat{𝐋} \cdot \hat{𝐒} + μ_{B} (\hat{𝐋} + 2 \hat{𝐒}) \cdot 𝐁,

5-

Δ S_{z} = \pm 1

6-

Δ L_{z} = 0

7-

| S_{z}, L_{z} ⟩ = | - 2, - 2 ⟩ \equiv | 0 ⟩

8-

| - 1, - 2 ⟩ \equiv | 2 ⟩

9-

| + 2, + 2 ⟩ \equiv | 1 ⟩

10-

| + 1, + 2 ⟩ \equiv | 3 ⟩

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1601.04324

Formulas:

1-

O (n^{- 2})

2-

Δ_{qp} = E_{I} - E_{A} = E_{+} + E_{-} - 2 E_{0},

3-

E_{A} = E_{0} - E_{+}

4-

E_{I} = E_{-} - E_{0}

5-

Δ_{exc} = E_{pr} - E_{0},

6-

E (n) = E (\infty) + A n^{- 2},

7-

Δ (n) = Δ (\infty) + B n^{- 1}

8-

ℏ = m_{e} = | e | = 4 π ϵ_{0} = 1

9-

v_{M} = [- 0.2319 - 2 l o g (a n)] / (a n)

10-

R_{\infty}^{*} = μ / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-ex/0603035

Formulas:

1-

M_{P l}^{2} \propto M_{S}^{n + 2} R^{n},

2-

\frac{d σ}{d M} = f_{S M} (M) + η_{L E D} f_{I N T} (M) + η_{L E D}^{2} f_{K K} (M),

3-

f_{I N T}

4-

η_{L E D}

5-

f_{I N T}

6-

η_{L E D}

7-

η_{L E D}^{2}

8-

η_{L E D}

9-

η_{L E D} = λ / M_{S}^{4}

10-

η_{L E D}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1610.06533

Formulas:

1-

M_{m s} \approx 12

2-

ω_{c i}^{- 1}

3-

n_{e, a 0} \approx (0.2 - 0.6) \times 10^{19}

4-

T_{e, a 0} \approx 15

5-

M_{m s} > 4

6-

n / n_{0} > 2

7-

Δ x / d_{i} > 1

8-

v_{s} = 700 \pm 30

9-

M_{m s} = v_{s} / c_{m s} = 12 \pm 5

10-

c_{m s}^{2} = v_{A}^{2} + c_{s}^{2}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9603118

Formulas:

1-

G_{H} = e \frac{\partial J}{\partial μ} .

2-

K_{H} = \frac{\partial J_{Q}}{\partial T},

3-

K_{H} = \frac{π^{2}}{3} \frac{k_{B}^{2}}{h} T .

4-

ℋ_{0} = π \sum_{i j} N_{i} V_{i j} N_{j} .

5-

[N_{i} (x), N_{j} (x^{'})] = \frac{i}{2 π} 𝖪_{i j}^{- 1} \partial_{x} δ (x - x^{'}),

6-

n_{ρ} = \sum_{i} T_{i} N_{i}

7-

ν = \sum_{i j} T_{i} 𝖪_{i j}^{- 1} T_{j}

8-

\partial_{t} n_{ρ} + \partial_{x} J_{ρ}

9-

J_{ρ} = \sum_{i} T_{i} {(𝖪^{- 1} V)}_{i j} N_{j}

10-

ℋ_{Φ} = - n_{ρ} Φ

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0706.4156

Formulas:

1-

Δ \dot{ν} / \dot{ν} = 0.1

2-

> 10^{14} - 10^{15}

3-

ϕ (t) = ϕ_{0} (t_{0}) + ν_{0} (t - t_{0}) + \frac{1}{2} \dot{ν_{0}} {(t - t_{0})}^{2} + \frac{1}{6} \ddot{ν_{0}} {(t - t_{0})}^{3} + \dots,

4-

Δ ν / ν = 2.7 \times 10^{- 6}

5-

ν = ν_{0} (t) + Δ ν + Δ ν_{d} e^{- (t - t_{g}) / τ_{d}} + Δ \dot{ν} (t - t_{g}),

6-

ν_{0} (t)

7-

Δ ν / ν = 1.6 \times 10^{- 5}

8-

Q \equiv Δ ν_{d} / (Δ ν_{d} + Δ ν) = 0.64

9-

Δ \dot{ν} / \dot{ν} = 0.0959 \pm 0.0007

10-

ν = 0.0849041677 (17)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1204.6223

Formulas:

1-

0 \leq z_{i} \leq 127

2-

z_{i_{1}} \equiv (z_{i - 1} + z_{i}) / 2

3-

z_{i_{2}} \equiv (z_{i + 1} + z_{i}) / 2

4-

𝜷_{N} = 𝜷_{1} - \sum_{i = 1}^{N - 1} \frac{d_{i N}}{d_{N}} 𝜶_{i} (𝜷_{i})

5-

𝜶_{i} (𝜷_{i})

6-

D_{i} \equiv D (z_{i})

7-

d_{i N} / d_{N} = (D_{N} - D_{i}) / D_{N}

8-

D_{i_{2}} - D_{i_{1}}

9-

𝜶_{i}^{bcg} (𝜷_{i}) = \frac{2}{(1 + z_{i}) c^{2}} \int_{D_{i_{1}}}^{D_{i_{2}}} 𝒈_{⟂} (D, 𝜷_{i}) 𝑑 D

10-

(D β_{x}, D β_{y})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1511.07741

Formulas:

1-

G = (V, A)

2-

T = (V, A_{T})

3-

G^{'} = (V, A_{T} \cup A^{'})

4-

O (m \log n)

5-

t (v) = d (v)

6-

t (v) \neq d (v)

7-

(d (v), v) \in A

8-

(d (v), v)

9-

(d (v), v) \notin A

10-

t (v) > v

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0312057

Formulas:

1-

𝐫_{ω} = e 𝐄_{ac} / (m ω^{2})

2-

ω = 2 π f_{ac}

3-

ε_{s} (𝐤_{∥})

4-

𝐄 (t) = 𝐄_{0} + 𝐄_{ac} \sin (2 π f_{ac} t),

5-

\frac{d 𝐯}{d t} = e 𝐄_{0} \cdot 𝒦 + 𝐀_{ei} + 𝐀_{ep} + 𝐀_{II} - g 𝐯,

6-

\frac{d h_{e}}{d t} = e 𝐄_{0} \cdot 𝐯 - W_{ep} - W_{II} - g h_{e} + S_{i} + S_{p} + S_{II},

7-

\frac{d N}{d t} = g N,

8-

N = 2 \sum_{s, 𝐤_{∥}} f [(ε_{s} (𝐤_{∥}) - μ) / T_{e}]

9-

f (x) = 1 / [\exp (x) + 1]

10-

n = \pm 1, \pm 2, \dots

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1508.00645

Formulas:

1-

m_{1}, m_{2}, \dots, m_{t}

2-

3 \leq m_{i} \leq n

3-

i = 1, 2, \dots, t

4-

m_{1} + m_{2} + \dots + m_{t} = (\binom{n}{2})

5-

m_{1}, m_{2}, \dots, m_{t}

6-

3 \leq m_{i} \leq n

7-

i = 1, 2, \dots, t

8-

m_{1} + m_{2} + \dots + m_{t} = (\binom{n}{2}) - \frac{n}{2}

9-

{G_{1}, G_{2}, \dots, G_{t}}

10-

i = 1, 2, \dots, t

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0909.3968

Formulas:

1-

e_{ℂ} : π_{2 k + 1}^{s t} \to ℚ / ℤ,

2-

π_{8 k + 1}^{s t}

3-

π_{8 k + 3}^{s t}

4-

\frac{B_{4 k + 2}}{4 k + 2}

5-

π_{8 k + 5}^{s t}

6-

π_{8 k + 7}^{s t}

7-

\frac{B_{4 k + 4}}{8 k + 8}

8-

J_{8 k + 3}

9-

B_{4 k + 2} / (4 k + 2)

10-

μ_{8 k + 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1410.6482

Formulas:

1-

{\tilde{Λ}}_{⊙} = 204^{\circ}

2-

| b | ≲ 20^{\circ}

3-

0.9 < J - K < 1.3

4-

0.22 < J - H - 0.561 (J - K) < 0.46

5-

0.9 < J - K < 1.3

6-

- 0.2 < W1 - W2 < - 0.05

7-

| b | \sim 2^{\circ}

8-

| b | \sim 20^{\circ}

9-

| b | ≲ 15^{\circ}

10-

260^{\circ} < {\tilde{Λ}}_{⊙} < 310^{\circ}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9904039

Formulas:

1-

V = \sum_{i, j, μ, ν} x_{μ} (i) D_{μ ν} (i, j) x_{ν} (j),

2-

x_{μ} (i)

3-

x_{μ}^{'} (i)

4-

x_{μ}^{'} (i) = x_{μ} (i) + Δ x_{μ} (i) .

5-

D_{μ ν}^{'} (i, j)

6-

x_{μ} (i)

7-

V^{'} = \sum_{i, j, μ, ν} x_{μ} (i) D_{μ ν}^{'} (i, j) x_{ν} (j) + \sum_{i, μ} F_{μ} (i) x_{μ} (i) + V_{0}^{'},

8-

F_{μ} (i)

9-

D_{μ ν} (i, j)

10-

\frac{1}{2} \sum_{i} \sum_{j = 1}^{6} \frac{1}{2} K_{1} {[{\hat{𝐫}}_{i j} \cdot (𝐫_{i} - 𝐫_{j})]}^{2} + \frac{1}{2} \sum_{i} K_{2} 𝐫_{i}^{2},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0002055

Formulas:

1-

d J / d ω

2-

𝒥 = 85 {MeV}^{- 1}

3-

h^{'} = h_{def} (ε_{2}, ε_{4}) - ω (j_{1} \sin ϑ + j_{3} \cos ϑ),

4-

E^{'} (ω, ϑ, ε_{2}, ε_{4}) = E_{L D} (ε_{2}, ε_{4}) - \tilde{E} + ⟨ ω | h^{'} | ω ⟩ .

5-

h_{def} (ε_{2}, ε_{4})

6-

\vec{J} = ⟨ ω | \vec{j} | ω ⟩

7-

ϑ \neq 90^{\circ} or 0^{\circ}

8-

J (ω) = \sqrt{J_{1}^{2} + J_{3}^{2}}

9-

ω (J) = E (I) - E (I - 1)

10-

J = (I - \frac{1}{2}) + \frac{1}{2} = I

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1403.7932

Formulas:

1-

v_{1}, e_{1}, v_{2}, \dots, v_{n}, e_{n}

2-

v_{1}, e_{1}, v_{2}, \dots, v_{n}, e_{n}

3-

{v_{1}, \dots, v_{n}}

4-

K_{n}^{(3)} - M

5-

| E (K_{n}^{(k)}) ∖ M |

6-

K_{n}^{(k)} - M

7-

K_{n}^{(3)} - M

8-

n / (k - ℓ)

9-

n / (k - ℓ)

10-

N := \frac{k - ℓ}{n} (\binom{n}{k})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0802.2765

Formulas:

1-

i n i t i o

2-

P \bar{6} m 2

3-

P \bar{6} m 2

4-

P \bar{6} m 2

5-

k_{4} ⟨ u^{2} ⟩

6-

i n i t i o

7-

i n i t i o

8-

λ_{𝐪 λ} = \frac{2}{π N (0)} \frac{γ_{𝐪 λ}}{ω_{𝐪 λ}^{2}}

9-

i n i t i o

10-

P h o n o n s

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0612013

Formulas:

1-

{| ψ_{i} ⟩}

2-

ℋ_{Q} = \otimes_{j = 1}^{n} ℋ_{j}

3-

𝒮 = {| ψ_{i} ⟩}

4-

| ψ_{i} ⟩ \in 𝒮

5-

𝒮 = {| ψ_{i} ⟩}

6-

| ψ_{x} ⟩ \in 𝒮

7-

\forall i \neq x ⟨ ψ_{i} | ϕ ⟩ = 0

8-

⟨ ψ_{x} | ϕ ⟩ \neq 0

9-

{| ⟨ ψ_{x} | ϕ ⟩ |}^{2}

10-

M^{†} M = A^{†} A \otimes B^{†} B \otimes \dots

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0512422

Formulas:

1-

G_{α β} = d I_{α} / d V_{β}

2-

G_{α β} (Φ) = G_{β α} (- Φ)

3-

(𝒢_{a}) 𝒢_{s}

4-

Φ_{c} ≪ Φ_{0} = e h / c

5-

𝒢_{s} (Φ)

6-

𝒢_{a} (Φ)

7-

Φ_{0} = e h / c

8-

Φ {(τ_{d} / τ_{erg})}^{1 / 2}

9-

Φ_{c} \sim Φ_{0} {(τ_{erg} / τ_{d})}^{1 / 2} ≪ Φ_{0}

10-

E_{T} = ℏ / τ_{erg}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/2007.02881

Formulas:

1-

Y_{hom} > 10^{3} Pa

2-

Y_{hie} \sim 0.03 p + Y_{hie, 0}

3-

Y_{hie} \sim 1 Pa

4-

a_{c} = {[\frac{9 π γ (1 - ν^{2})}{2 E R}]}^{1 / 3} R,

5-

R_{sil} = 0.5 μ m

6-

R_{sil} = 0.5 μ m

7-

χ_{org} = \frac{f_{org} / ρ_{org}}{f_{org} / ρ_{org} + f_{sil} / ρ_{sil}},

8-

χ_{sil} = \frac{f_{sil} / ρ_{sil}}{f_{org} / ρ_{org} + f_{sil} / ρ_{sil}},

9-

R = χ_{sil}^{- 1 / 3} R_{sil},

10-

Δ = (1 - χ_{sil}^{1 / 3}) R .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0801.0297

Formulas:

1-

p (t_{b}) = 1

2-

p (t) = 0

3-

0 \leq t < t_{b}

4-

t_{b} + d / v_{b}

5-

\frac{d}{v_{w}} < t_{b} + \frac{d}{v_{b}} .

6-

\int_{0}^{t_{w}} (p (t) (\frac{d}{v_{b}} + t)) 𝑑 t + (1 - \int_{0}^{t_{w}} p (t) 𝑑 t) (\frac{d}{v_{w}} + t_{w}),

7-

\int_{0}^{t_{w}} (p (t) (\frac{d}{v_{b}} + t)) 𝑑 t + (1 - \int_{0}^{t_{w}} p (t) 𝑑 t) (\frac{d}{v_{w}} + t_{w}) = \frac{d}{v_{w}} .

8-

p (t) = 1 / t_{b}

9-

= \int_{0}^{t_{w}} (\frac{1}{t_{b}} (\frac{d}{v_{b}} + t)) 𝑑 t + (1 - \int_{0}^{t_{w}} \frac{1}{t_{b}} 𝑑 t) (\frac{d}{v_{w}} + t_{w})

10-

= - \frac{t_{w}^{2}}{2 t_{b}} + (\frac{d (\frac{1}{v_{b}} - \frac{1}{v_{w}})}{t_{b}} + 1) t_{w} + \frac{d}{v_{w}} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0507306

Formulas:

1-

S F R_{U V}

2-

h_{0.7}^{- 2} M_{☉} {yr}^{- 1}

3-

h_{0.7}^{- 2} M_{☉} {yr}^{- 1}

4-

Ω_{t o t}

5-

α_{2000} = 12^{h} 52^{m} 56 \overset{s}{.} 888, δ_{2000} = - 29 ° 25' 55 \overset{''}{.} 50

6-

z_{850} - J s

7-

z_{850} - K s

8-

λ_{l a b}

9-

λ_{o b s}

10-

z_{i o n}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9901287

Formulas:

1-

O (α_{s})

2-

e p \to W + X

3-

O (10 %)

4-

e^{+} e^{-} \to W^{+} W^{-}

5-

ℒ_{CI} = \sum_{\begin{matrix} i, k = L, R \\ q = u, d, \dots \end{matrix}} η_{i k}^{q} \frac{4 π}{{(Λ_{i k}^{q})}^{2}} ({\bar{e}}_{i} γ^{μ} e_{i}) ({\bar{q}}_{k} γ_{μ} q_{k}) .

6-

e p \to e X

7-

e^{+} e^{-} \to hadr

8-

p \bar{p} \to ℓ^{+} ℓ^{-} X

9-

e p \to e X

10-

e^{+} e^{-} \to hadrons

Formulas (html):

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Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0202299

Formulas:

1-

\frac{\partial \bar{B}}{\partial t} = curl (\bar{U} \times \bar{B} + ℰ - η curl \bar{B}),

2-

ℰ = \bar{u \times b}

3-

u = U - \bar{U}

4-

b = B - \bar{B}

5-

ℰ_{i} = α_{i j} {\bar{B}}_{j} + β_{i j k} N e w A_{j} {\bar{B}}_{k} + \dots .

6-

α_{i j}^{S} = (α_{i j} + α_{j i}) / 2; α_{i j}^{A} = (α_{i j} - α_{j i}) / 2 .

7-

α^{A} \cdot \bar{B} = γ \times \bar{B}

8-

γ = - (\begin{matrix} α_{y z}^{A} \\ α_{z x}^{A} \\ α_{x y}^{A} \end{matrix}) .

9-

w_{tor} = \bar{U} + γ^{(tor)}

10-

\bar{B} = [{\bar{B}}_{x} (z), {\bar{B}}_{y} (z), {\bar{B}}_{z}]

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1305.0269

Formulas:

1-

{\dot{M}}_{e - \lim} \approx \frac{ϵ π F_{XUV} R_{XUV}^{3}}{G M_{p} K_{tide}}

2-

K_{tide} = (1 - \frac{3}{2 ξ} + \frac{1}{2 ξ^{3}})

3-

ξ = \frac{R_{Hill}}{R_{XUV}}

4-

J K^{- 1} g^{- 1}

5-

F_{th} \propto M_{core}^{2.4}

6-

t_{loss} = \frac{G M_{p}^{2}}{π ϵ R_{p}^{3} F_{XUV, E100}} \frac{F_{\oplus}}{F_{p}} \approx 12 G y r

7-

F_{XUV, E100} = 504

8-

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

9-

F_{th} \propto M_{core}^{2.4}

10-

F_{th} \sim \exp (- (t - 140 Myr) / 80 Myr) F_{\oplus} + 3.4 F_{\oplus}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9707172

Formulas:

1-

\frac{ℏ e B}{m_{e} c}

2-

\sim m_{e} c^{2}

3-

B_{c} = \frac{m_{e}^{2} c^{3}}{e ℏ} = 4.4 \times 10^{13} Gs

4-

W_{n} = W_{0} [1 + 0.17 {(B / B_{c})}^{2} + \dots] at B ≪ B_{c}

5-

W_{n} = 0.77 W_{0} (B / B_{c}) at B ≫ B_{c}

6-

α_{T} = k T / m_{e} c^{2}

7-

α_{ρ} = ϵ_{F e} / m_{e} c^{2}

8-

B_{ϕ} = B_{ϕ 0} + B_{p} (t / P)

9-

2 \times 10^{4} B_{p}

10-

B_{p} = 10^{11} \div 10^{13}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9202042

Formulas:

1-

A = \frac{1}{g} \int d^{2} x {(\partial_{ν} \vec{η})}^{2}

2-

\vec{η} = (n_{x}, n_{y}, n_{z})

3-

S_{i j}^{k l} (β)

4-

S_{i j}^{k l} (β) = \frac{π + i β}{(2 π + i β) (π - i β)} (\frac{2 π i β}{π + i β} δ_{i, j} δ_{k, l} - 2 π δ_{k, j} δ_{i, l} - i β δ_{k, i} δ_{l, j})

5-

e^{i L m s h (β_{i})} = \prod_{j = 1, i \neq j}^{N} \frac{β_{i} - β_{j} + i π}{β_{i} - β_{j} - i π} \prod_{j = 1}^{M} \frac{β_{i} - λ_{j} + i π}{β_{i} - λ_{j} - i π}, i = 1, 2, \dots, N

6-

\prod_{j = 1, i \neq j}^{M} \frac{λ_{i} - λ_{j} + i π}{λ_{i} - λ_{j} - i π} \prod_{j = 1}^{N} \frac{β_{j} - λ_{i} + i π}{β_{j} - λ_{i} - i π}, i = 1, 2, \dots, M

7-

A_{i}, i = 1, 2, 3

8-

E = \sum_{i = 1}^{N} m c h (β_{i})

9-

e t a l l

10-

R (λ - ν) T (λ) \otimes T (ν) = T (ν) \otimes T (λ) R (λ - ν)

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9809082

Formulas:

1-

\frac{d f}{d t} = \frac{1}{3} p \frac{\partial f}{\partial p} (\nabla \cdot \vec{v}) + \nabla \cdot (κ \nabla f) + Q,

2-

f (\vec{x}, p, t)

3-

p_{i} \leq p \leq p_{i + 1}

4-

f (p) = f_{i} {(p / p_{i})}^{- q_{i}}

5-

\frac{d n_{i}}{d t} = - \frac{1}{3} (\nabla \cdot v) q_{i} n_{i} + \nabla \cdot (< κ > \nabla n) + Q^{'}

6-

n_{i} = 4 π \int_{p_{i}}^{p_{i + 1}} p^{2} f 𝑑 p

7-

< κ > = \frac{\int p^{2} κ \nabla f d p}{\int p^{2} \nabla f d p}

8-

Q^{'} = 4 π \int_{p_{i}}^{p_{i + 1}} p^{2} Q 𝑑 p

9-

n_{i} = 4 π \frac{f_{i} p_{i}^{3}}{q_{i} - 3} [1 - {(\frac{p_{i}}{p_{i + 1}})}^{q_{i} - 3}] .

10-

f_{i + 1} = f_{i} {(p_{i} / p_{i + 1})}^{q_{i}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1809.00180

Formulas:

1-

M_{m a x}

2-

150 / \sin (l)

3-

σ_{ϖ} / ϖ < 30 %

4-

σ_{ϖ} / ϖ > 30 %

5-

V_{m a x}

6-

V_{m a x}

7-

V_{m a x}

8-

V_{m a x}

9-

E (B - V)

10-

3.1 \times E (B - V)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1603.01428

Formulas:

1-

(i i i)

2-

- 1 / r + C

3-

- 1 / z + C

4-

IP = - ε_{vbm}^{p . cell} + {\bar{v}}_{Coul}^{p . cell} - {\bar{v}}_{Coul}^{s . cell},

5-

{\bar{v}}_{Coul}^{p . cell}

6-

{\bar{v}}_{Coul}^{s . cell}

7-

{\bar{v}}_{Coul}^{s . cell}

8-

Δ ε_{vbm}^{p . cell}

9-

Δ {\bar{v}}_{Coul}^{p . cell}

10-

Δ {\bar{v}}_{Coul}^{s . cell}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9705219

Formulas:

1-

a_{𝐤} (𝐪)

2-

a_{𝐤}^{†} (𝐪)

3-

c_{𝐤 + 𝐪 / 𝟐}^{†} c_{𝐤 - 𝐪 / 𝟐} = n_{F} (𝐤) \frac{N}{⟨ N ⟩} δ_{𝐪, 𝟎} + (\sqrt{\frac{N}{⟨ N ⟩}}) [Λ_{𝐤} (𝐪) a_{𝐤} (- 𝐪) + Λ_{𝐤} (- 𝐪) a_{𝐤}^{†} (𝐪)]

4-

+ \sum_{𝐪_{1}} Λ_{𝐤 + 𝐪 / 𝟐 - 𝐪_{𝟏} / 𝟐} (- 𝐪_{𝟏}) Λ_{𝐤 - 𝐪_{𝟏} / 𝟐} (𝐪 - 𝐪_{𝟏}) a_{𝐤 + 𝐪 / 2 - 𝐪_{1} / 2}^{†} (𝐪_{1}) a_{𝐤 - 𝐪_{1} / 2} (- 𝐪 + 𝐪_{1})

5-

- \sum_{𝐪_{1}} Λ_{𝐤 - 𝐪 / 𝟐 + 𝐪_{𝟏} / 𝟐} (- 𝐪_{𝟏}) Λ_{𝐤 + 𝐪_{𝟏} / 𝟐} (𝐪 - 𝐪_{𝟏}) a_{𝐤 - 𝐪 / 2 + 𝐪_{1} / 2}^{†} (𝐪_{1}) a_{𝐤 + 𝐪_{1} / 2} (- 𝐪 + 𝐪_{1})

6-

a_{𝐤} (𝐪)

7-

a_{𝐤}^{†} (𝐪)

8-

{c_{𝐤}, c_{𝐤^{^{'}}}} = 0; {c_{𝐤}, c_{𝐤^{^{'}}}^{†}} = δ_{𝐤, 𝐤^{^{'}}}

9-

[a_{𝐤} (𝐪), a_{𝐤^{^{'}}} (𝐪^{^{'}})] = 0; [a_{𝐤} (𝐪), a_{𝐤^{^{'}}}^{†} (𝐪^{^{'}})] = δ_{𝐤, 𝐤^{^{'}}} δ_{𝐪, 𝐪^{^{'}}}

10-

Λ_{𝐤} (𝐪) = \sqrt{n_{F} (𝐤 + 𝐪 / 𝟐) (1 - n_{F} (𝐤 - 𝐪 / 𝟐))}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1101.4796

Formulas:

1-

n_{u} (S, z)

2-

μ (\hat{ϕ}, z)

3-

n_{l} (S, z) = \frac{n_{u} (S / μ (\hat{ϕ}, z), z)}{μ (\hat{ϕ}, z)} .

4-

N_{u} (S) \propto S^{- β}

5-

\frac{N_{l} (S)}{N_{u} (S)} = μ^{β - 1}

6-

β = β (S)

7-

w_{cross} (θ) = ⟨ δ n_{1} (\hat{ϕ}) δ n_{2} (\hat{ϕ^{'}}) ⟩,

8-

δ n_{i} \equiv n_{i} (\hat{ϕ}) / {\bar{n}}_{i} - 1

9-

δ n_{i} (\hat{ϕ}) = δ n_{i}^{c} (\hat{ϕ}) + δ n_{i}^{μ} (\hat{ϕ}) .

10-

δ n_{i}^{c} (\hat{ϕ}) = b_{i} \int_{0}^{χ_{H}} 𝑑 χ W_{i} (χ) δ (r (χ) \hat{ϕ}, a),

Formulas (html):

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xref="S2.E1.m1.4.4.4.4.2.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.4.4.1.2" xref="S2.E1.m1.4.4.4.4.2.cmml">(</mo><mrow id="S2.E1.m1.4.4.4.4.1.1" xref="S2.E1.m1.4.4.4.4.1.1.cmml"><mrow id="S2.E1.m1.4.4.4.4.1.1.2" xref="S2.E1.m1.4.4.4.4.1.1.2.cmml"><mi id="S2.E1.m1.4.4.4.4.1.1.2.2" xref="S2.E1.m1.4.4.4.4.1.1.2.2.cmml">S</mi><mo id="S2.E1.m1.4.4.4.4.1.1.2.1" xref="S2.E1.m1.4.4.4.4.1.1.2.1.cmml">/</mo><mi id="S2.E1.m1.4.4.4.4.1.1.2.3" xref="S2.E1.m1.4.4.4.4.1.1.2.3.cmml">μ</mi></mrow><mo id="S2.E1.m1.4.4.4.4.1.1.1" xref="S2.E1.m1.4.4.4.4.1.1.1.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.4.4.1.1.3.2" xref="S2.E1.m1.4.4.4.4.1.1.3.1.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.4.4.1.1.3.2.1" xref="S2.E1.m1.4.4.4.4.1.1.3.1.cmml">(</mo><mover accent="true" id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.cmml">^</mo></mover><mo id="S2.E1.m1.4.4.4.4.1.1.3.2.2" xref="S2.E1.m1.4.4.4.4.1.1.3.1.cmml">,</mo><mi id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml">z</mi><mo stretchy="false" id="S2.E1.m1.4.4.4.4.1.1.3.2.3" xref="S2.E1.m1.4.4.4.4.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.4.4.4.1.3" xref="S2.E1.m1.4.4.4.4.2.cmml">,</mo><mi id="S2.E1.m1.3.3.3.3" xref="S2.E1.m1.3.3.3.3.cmml">z</mi><mo stretchy="false" id="S2.E1.m1.4.4.4.4.1.4" xref="S2.E1.m1.4.4.4.4.2.cmml">)</mo></mrow></mrow><mrow id="S2.E1.m1.6.6.6" xref="S2.E1.m1.6.6.6.cmml"><mi id="S2.E1.m1.6.6.6.4" xref="S2.E1.m1.6.6.6.4.cmml">μ</mi><mo id="S2.E1.m1.6.6.6.3" xref="S2.E1.m1.6.6.6.3.cmml">⁢</mo><mrow id="S2.E1.m1.6.6.6.5.2" xref="S2.E1.m1.6.6.6.5.1.cmml"><mo stretchy="false" id="S2.E1.m1.6.6.6.5.2.1" xref="S2.E1.m1.6.6.6.5.1.cmml">(</mo><mover accent="true" id="S2.E1.m1.5.5.5.1" xref="S2.E1.m1.5.5.5.1.cmml"><mi id="S2.E1.m1.5.5.5.1.2" xref="S2.E1.m1.5.5.5.1.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E1.m1.5.5.5.1.1" xref="S2.E1.m1.5.5.5.1.1.cmml">^</mo></mover><mo id="S2.E1.m1.6.6.6.5.2.2" xref="S2.E1.m1.6.6.6.5.1.cmml">,</mo><mi id="S2.E1.m1.6.6.6.2" xref="S2.E1.m1.6.6.6.2.cmml">z</mi><mo stretchy="false" id="S2.E1.m1.6.6.6.5.2.3" xref="S2.E1.m1.6.6.6.5.1.cmml">)</mo></mrow></mrow></mfrac></mrow><mo id="S2.E1.m1.9.9.1.2" xref="S2.E1.m1.9.9.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.p1.8.m2.1.2" xref="S2.p1.8.m2.1.2.cmml"><mrow id="S2.p1.8.m2.1.2.2" xref="S2.p1.8.m2.1.2.2.cmml"><msub id="S2.p1.8.m2.1.2.2.2" xref="S2.p1.8.m2.1.2.2.2.cmml"><mi id="S2.p1.8.m2.1.2.2.2.2" xref="S2.p1.8.m2.1.2.2.2.2.cmml">N</mi><mi mathvariant="normal" id="S2.p1.8.m2.1.2.2.2.3" xref="S2.p1.8.m2.1.2.2.2.3.cmml">u</mi></msub><mo id="S2.p1.8.m2.1.2.2.1" xref="S2.p1.8.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.p1.8.m2.1.2.2.3.2" xref="S2.p1.8.m2.1.2.2.cmml"><mo stretchy="false" id="S2.p1.8.m2.1.2.2.3.2.1" xref="S2.p1.8.m2.1.2.2.cmml">(</mo><mi id="S2.p1.8.m2.1.1" xref="S2.p1.8.m2.1.1.cmml">S</mi><mo stretchy="false" id="S2.p1.8.m2.1.2.2.3.2.2" xref="S2.p1.8.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.8.m2.1.2.1" xref="S2.p1.8.m2.1.2.1.cmml">∝</mo><msup id="S2.p1.8.m2.1.2.3" xref="S2.p1.8.m2.1.2.3.cmml"><mi id="S2.p1.8.m2.1.2.3.2" xref="S2.p1.8.m2.1.2.3.2.cmml">S</mi><mrow id="S2.p1.8.m2.1.2.3.3" xref="S2.p1.8.m2.1.2.3.3.cmml"><mo id="S2.p1.8.m2.1.2.3.3.1" xref="S2.p1.8.m2.1.2.3.3.1.cmml">-</mo><mi id="S2.p1.8.m2.1.2.3.3.2" xref="S2.p1.8.m2.1.2.3.3.2.cmml">β</mi></mrow></msup></mrow></math>, <math><mrow id="S2.E2.m1.2.3" xref="S2.E2.m1.2.3.cmml"><mfrac id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><msub id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml"><mi id="S2.E2.m1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.3.2.cmml">N</mi><mi mathvariant="normal" id="S2.E2.m1.1.1.1.3.3" xref="S2.E2.m1.1.1.1.3.3.cmml">l</mi></msub><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.4.2" xref="S2.E2.m1.1.1.1.cmml"><mo stretchy="false" id="S2.E2.m1.1.1.1.4.2.1" xref="S2.E2.m1.1.1.1.cmml">(</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">S</mi><mo stretchy="false" id="S2.E2.m1.1.1.1.4.2.2" xref="S2.E2.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E2.m1.2.2.2" xref="S2.E2.m1.2.2.2.cmml"><msub id="S2.E2.m1.2.2.2.3" xref="S2.E2.m1.2.2.2.3.cmml"><mi id="S2.E2.m1.2.2.2.3.2" xref="S2.E2.m1.2.2.2.3.2.cmml">N</mi><mi mathvariant="normal" id="S2.E2.m1.2.2.2.3.3" xref="S2.E2.m1.2.2.2.3.3.cmml">u</mi></msub><mo id="S2.E2.m1.2.2.2.2" xref="S2.E2.m1.2.2.2.2.cmml">⁢</mo><mrow id="S2.E2.m1.2.2.2.4.2" xref="S2.E2.m1.2.2.2.cmml"><mo stretchy="false" id="S2.E2.m1.2.2.2.4.2.1" xref="S2.E2.m1.2.2.2.cmml">(</mo><mi id="S2.E2.m1.2.2.2.1" xref="S2.E2.m1.2.2.2.1.cmml">S</mi><mo stretchy="false" id="S2.E2.m1.2.2.2.4.2.2" xref="S2.E2.m1.2.2.2.cmml">)</mo></mrow></mrow></mfrac><mo id="S2.E2.m1.2.3.1" xref="S2.E2.m1.2.3.1.cmml">=</mo><msup id="S2.E2.m1.2.3.2" xref="S2.E2.m1.2.3.2.cmml"><mi id="S2.E2.m1.2.3.2.2" xref="S2.E2.m1.2.3.2.2.cmml">μ</mi><mrow id="S2.E2.m1.2.3.2.3" xref="S2.E2.m1.2.3.2.3.cmml"><mi id="S2.E2.m1.2.3.2.3.2" xref="S2.E2.m1.2.3.2.3.2.cmml">β</mi><mo id="S2.E2.m1.2.3.2.3.1" xref="S2.E2.m1.2.3.2.3.1.cmml">-</mo><mn id="S2.E2.m1.2.3.2.3.3" xref="S2.E2.m1.2.3.2.3.3.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.p1.9.m1.1.2" xref="S2.p1.9.m1.1.2.cmml"><mi id="S2.p1.9.m1.1.2.2" xref="S2.p1.9.m1.1.2.2.cmml">β</mi><mo id="S2.p1.9.m1.1.2.1" xref="S2.p1.9.m1.1.2.1.cmml">=</mo><mrow id="S2.p1.9.m1.1.2.3" xref="S2.p1.9.m1.1.2.3.cmml"><mi id="S2.p1.9.m1.1.2.3.2" xref="S2.p1.9.m1.1.2.3.2.cmml">β</mi><mo id="S2.p1.9.m1.1.2.3.1" xref="S2.p1.9.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.9.m1.1.2.3.3.2" xref="S2.p1.9.m1.1.2.3.cmml"><mo stretchy="false" id="S2.p1.9.m1.1.2.3.3.2.1" xref="S2.p1.9.m1.1.2.3.cmml">(</mo><mi id="S2.p1.9.m1.1.1" xref="S2.p1.9.m1.1.1.cmml">S</mi><mo stretchy="false" id="S2.p1.9.m1.1.2.3.3.2.2" xref="S2.p1.9.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.E3.m1.4.4.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1.3" xref="S2.E3.m1.4.4.1.1.3.cmml"><msub id="S2.E3.m1.4.4.1.1.3.2" xref="S2.E3.m1.4.4.1.1.3.2.cmml"><mi id="S2.E3.m1.4.4.1.1.3.2.2" xref="S2.E3.m1.4.4.1.1.3.2.2.cmml">w</mi><mtext id="S2.E3.m1.4.4.1.1.3.2.3" xref="S2.E3.m1.4.4.1.1.3.2.3a.cmml">cross</mtext></msub><mo id="S2.E3.m1.4.4.1.1.3.1" xref="S2.E3.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.3.3.2" xref="S2.E3.m1.4.4.1.1.3.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.3.3.2.1" xref="S2.E3.m1.4.4.1.1.3.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">θ</mi><mo stretchy="false" id="S2.E3.m1.4.4.1.1.3.3.2.2" xref="S2.E3.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.4.1.1.2" xref="S2.E3.m1.4.4.1.1.2.cmml">=</mo><mrow id="S2.E3.m1.4.4.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.2.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.2.1.cmml">⟨</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.1.1.2.cmml">δ</mi><mo id="S2.E3.m1.4.4.1.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.E3.m1.4.4.1.1.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.1.1.3.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.3.2" xref="S2.E3.m1.4.4.1.1.1.1.1.3.2.cmml">n</mi><mn id="S2.E3.m1.4.4.1.1.1.1.1.3.3" xref="S2.E3.m1.4.4.1.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S2.E3.m1.4.4.1.1.1.1.1.1a" xref="S2.E3.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.4.2" xref="S2.E3.m1.2.2.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.4.2.1" xref="S2.E3.m1.2.2.cmml">(</mo><mover accent="true" id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml"><mi id="S2.E3.m1.2.2.2" xref="S2.E3.m1.2.2.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E3.m1.2.2.1" xref="S2.E3.m1.2.2.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.4.2.2" xref="S2.E3.m1.2.2.cmml">)</mo></mrow><mo id="S2.E3.m1.4.4.1.1.1.1.1.1b" xref="S2.E3.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E3.m1.4.4.1.1.1.1.1.5" xref="S2.E3.m1.4.4.1.1.1.1.1.5.cmml">δ</mi><mo id="S2.E3.m1.4.4.1.1.1.1.1.1c" xref="S2.E3.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.E3.m1.4.4.1.1.1.1.1.6" xref="S2.E3.m1.4.4.1.1.1.1.1.6.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.6.2" xref="S2.E3.m1.4.4.1.1.1.1.1.6.2.cmml">n</mi><mn id="S2.E3.m1.4.4.1.1.1.1.1.6.3" xref="S2.E3.m1.4.4.1.1.1.1.1.6.3.cmml">2</mn></msub><mo id="S2.E3.m1.4.4.1.1.1.1.1.1d" xref="S2.E3.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.7.2" xref="S2.E3.m1.3.3.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.7.2.1" xref="S2.E3.m1.3.3.cmml">(</mo><mover accent="true" id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml"><msup id="S2.E3.m1.3.3.2" xref="S2.E3.m1.3.3.2.cmml"><mi id="S2.E3.m1.3.3.2.2" xref="S2.E3.m1.3.3.2.2.cmml">ϕ</mi><mo id="S2.E3.m1.3.3.2.3" xref="S2.E3.m1.3.3.2.3.cmml">′</mo></msup><mo stretchy="false" id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.7.2.2" xref="S2.E3.m1.3.3.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S2.E3.m1.4.4.1.2" xref="S2.E3.m1.4.4.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.p2.1.m1.1.2" xref="S2.p2.1.m1.1.2.cmml"><mrow id="S2.p2.1.m1.1.2.2" xref="S2.p2.1.m1.1.2.2.cmml"><mi id="S2.p2.1.m1.1.2.2.2" xref="S2.p2.1.m1.1.2.2.2.cmml">δ</mi><mo id="S2.p2.1.m1.1.2.2.1" xref="S2.p2.1.m1.1.2.2.1.cmml">⁢</mo><msub id="S2.p2.1.m1.1.2.2.3" xref="S2.p2.1.m1.1.2.2.3.cmml"><mi id="S2.p2.1.m1.1.2.2.3.2" xref="S2.p2.1.m1.1.2.2.3.2.cmml">n</mi><mi id="S2.p2.1.m1.1.2.2.3.3" xref="S2.p2.1.m1.1.2.2.3.3.cmml">i</mi></msub></mrow><mo id="S2.p2.1.m1.1.2.1" xref="S2.p2.1.m1.1.2.1.cmml">≡</mo><mrow id="S2.p2.1.m1.1.2.3" xref="S2.p2.1.m1.1.2.3.cmml"><mrow id="S2.p2.1.m1.1.2.3.2" xref="S2.p2.1.m1.1.2.3.2.cmml"><mrow id="S2.p2.1.m1.1.2.3.2.2" xref="S2.p2.1.m1.1.2.3.2.2.cmml"><msub id="S2.p2.1.m1.1.2.3.2.2.2" xref="S2.p2.1.m1.1.2.3.2.2.2.cmml"><mi id="S2.p2.1.m1.1.2.3.2.2.2.2" xref="S2.p2.1.m1.1.2.3.2.2.2.2.cmml">n</mi><mi id="S2.p2.1.m1.1.2.3.2.2.2.3" xref="S2.p2.1.m1.1.2.3.2.2.2.3.cmml">i</mi></msub><mo id="S2.p2.1.m1.1.2.3.2.2.1" xref="S2.p2.1.m1.1.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.p2.1.m1.1.2.3.2.2.3.2" xref="S2.p2.1.m1.1.1.cmml"><mo stretchy="false" id="S2.p2.1.m1.1.2.3.2.2.3.2.1" xref="S2.p2.1.m1.1.1.cmml">(</mo><mover accent="true" 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">ϕ</mi><mo stretchy="false" id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">^</mo></mover><mo stretchy="false" id="S2.p2.1.m1.1.2.3.2.2.3.2.2" xref="S2.p2.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.1.m1.1.2.3.2.1" xref="S2.p2.1.m1.1.2.3.2.1.cmml">/</mo><msub id="S2.p2.1.m1.1.2.3.2.3" xref="S2.p2.1.m1.1.2.3.2.3.cmml"><mover accent="true" id="S2.p2.1.m1.1.2.3.2.3.2" xref="S2.p2.1.m1.1.2.3.2.3.2.cmml"><mi id="S2.p2.1.m1.1.2.3.2.3.2.2" xref="S2.p2.1.m1.1.2.3.2.3.2.2.cmml">n</mi><mo stretchy="false" id="S2.p2.1.m1.1.2.3.2.3.2.1" xref="S2.p2.1.m1.1.2.3.2.3.2.1.cmml">¯</mo></mover><mi id="S2.p2.1.m1.1.2.3.2.3.3" xref="S2.p2.1.m1.1.2.3.2.3.3.cmml">i</mi></msub></mrow><mo id="S2.p2.1.m1.1.2.3.1" xref="S2.p2.1.m1.1.2.3.1.cmml">-</mo><mn id="S2.p2.1.m1.1.2.3.3" xref="S2.p2.1.m1.1.2.3.3.cmml">1</mn></mrow></mrow></math>, <math><mrow id="S2.E4.m1.4.4.1" xref="S2.E4.m1.4.4.1.1.cmml"><mrow id="S2.E4.m1.4.4.1.1" xref="S2.E4.m1.4.4.1.1.cmml"><mrow id="S2.E4.m1.4.4.1.1.2" xref="S2.E4.m1.4.4.1.1.2.cmml"><mi id="S2.E4.m1.4.4.1.1.2.2" xref="S2.E4.m1.4.4.1.1.2.2.cmml">δ</mi><mo id="S2.E4.m1.4.4.1.1.2.1" xref="S2.E4.m1.4.4.1.1.2.1.cmml">⁢</mo><msub id="S2.E4.m1.4.4.1.1.2.3" xref="S2.E4.m1.4.4.1.1.2.3.cmml"><mi id="S2.E4.m1.4.4.1.1.2.3.2" xref="S2.E4.m1.4.4.1.1.2.3.2.cmml">n</mi><mi id="S2.E4.m1.4.4.1.1.2.3.3" xref="S2.E4.m1.4.4.1.1.2.3.3.cmml">i</mi></msub><mo id="S2.E4.m1.4.4.1.1.2.1a" xref="S2.E4.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.4.4.1.1.2.4.2" xref="S2.E4.m1.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.4.4.1.1.2.4.2.1" xref="S2.E4.m1.1.1.cmml">(</mo><mover accent="true" id="S2.E4.m1.1.1" xref="S2.E4.m1.1.1.cmml"><mi id="S2.E4.m1.1.1.2" xref="S2.E4.m1.1.1.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E4.m1.1.1.1" xref="S2.E4.m1.1.1.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E4.m1.4.4.1.1.2.4.2.2" xref="S2.E4.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E4.m1.4.4.1.1.1" xref="S2.E4.m1.4.4.1.1.1.cmml">=</mo><mrow id="S2.E4.m1.4.4.1.1.3" xref="S2.E4.m1.4.4.1.1.3.cmml"><mrow id="S2.E4.m1.4.4.1.1.3.2" xref="S2.E4.m1.4.4.1.1.3.2.cmml"><mi id="S2.E4.m1.4.4.1.1.3.2.2" xref="S2.E4.m1.4.4.1.1.3.2.2.cmml">δ</mi><mo id="S2.E4.m1.4.4.1.1.3.2.1" xref="S2.E4.m1.4.4.1.1.3.2.1.cmml">⁢</mo><msubsup id="S2.E4.m1.4.4.1.1.3.2.3" xref="S2.E4.m1.4.4.1.1.3.2.3.cmml"><mi id="S2.E4.m1.4.4.1.1.3.2.3.2.2" xref="S2.E4.m1.4.4.1.1.3.2.3.2.2.cmml">n</mi><mi id="S2.E4.m1.4.4.1.1.3.2.3.2.3" xref="S2.E4.m1.4.4.1.1.3.2.3.2.3.cmml">i</mi><mi mathvariant="normal" id="S2.E4.m1.4.4.1.1.3.2.3.3" xref="S2.E4.m1.4.4.1.1.3.2.3.3.cmml">c</mi></msubsup><mo id="S2.E4.m1.4.4.1.1.3.2.1a" xref="S2.E4.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.4.4.1.1.3.2.4.2" xref="S2.E4.m1.2.2.cmml"><mo stretchy="false" id="S2.E4.m1.4.4.1.1.3.2.4.2.1" xref="S2.E4.m1.2.2.cmml">(</mo><mover accent="true" id="S2.E4.m1.2.2" xref="S2.E4.m1.2.2.cmml"><mi id="S2.E4.m1.2.2.2" xref="S2.E4.m1.2.2.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E4.m1.2.2.1" xref="S2.E4.m1.2.2.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E4.m1.4.4.1.1.3.2.4.2.2" xref="S2.E4.m1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E4.m1.4.4.1.1.3.1" xref="S2.E4.m1.4.4.1.1.3.1.cmml">+</mo><mrow id="S2.E4.m1.4.4.1.1.3.3" xref="S2.E4.m1.4.4.1.1.3.3.cmml"><mi id="S2.E4.m1.4.4.1.1.3.3.2" xref="S2.E4.m1.4.4.1.1.3.3.2.cmml">δ</mi><mo id="S2.E4.m1.4.4.1.1.3.3.1" xref="S2.E4.m1.4.4.1.1.3.3.1.cmml">⁢</mo><msubsup id="S2.E4.m1.4.4.1.1.3.3.3" xref="S2.E4.m1.4.4.1.1.3.3.3.cmml"><mi id="S2.E4.m1.4.4.1.1.3.3.3.2.2" xref="S2.E4.m1.4.4.1.1.3.3.3.2.2.cmml">n</mi><mi id="S2.E4.m1.4.4.1.1.3.3.3.2.3" xref="S2.E4.m1.4.4.1.1.3.3.3.2.3.cmml">i</mi><mi id="S2.E4.m1.4.4.1.1.3.3.3.3" xref="S2.E4.m1.4.4.1.1.3.3.3.3.cmml">μ</mi></msubsup><mo id="S2.E4.m1.4.4.1.1.3.3.1a" xref="S2.E4.m1.4.4.1.1.3.3.1.cmml">⁢</mo><mrow id="S2.E4.m1.4.4.1.1.3.3.4.2" xref="S2.E4.m1.3.3.cmml"><mo stretchy="false" id="S2.E4.m1.4.4.1.1.3.3.4.2.1" xref="S2.E4.m1.3.3.cmml">(</mo><mover accent="true" id="S2.E4.m1.3.3" xref="S2.E4.m1.3.3.cmml"><mi id="S2.E4.m1.3.3.2" xref="S2.E4.m1.3.3.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E4.m1.3.3.1" xref="S2.E4.m1.3.3.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E4.m1.4.4.1.1.3.3.4.2.2" xref="S2.E4.m1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E4.m1.4.4.1.2" xref="S2.E4.m1.4.4.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.E5.m1.5.5.1" xref="S2.E5.m1.5.5.1.1.cmml"><mrow id="S2.E5.m1.5.5.1.1" xref="S2.E5.m1.5.5.1.1.cmml"><mrow id="S2.E5.m1.5.5.1.1.3" xref="S2.E5.m1.5.5.1.1.3.cmml"><mi id="S2.E5.m1.5.5.1.1.3.2" xref="S2.E5.m1.5.5.1.1.3.2.cmml">δ</mi><mo id="S2.E5.m1.5.5.1.1.3.1" xref="S2.E5.m1.5.5.1.1.3.1.cmml">⁢</mo><msubsup id="S2.E5.m1.5.5.1.1.3.3" xref="S2.E5.m1.5.5.1.1.3.3.cmml"><mi id="S2.E5.m1.5.5.1.1.3.3.2.2" xref="S2.E5.m1.5.5.1.1.3.3.2.2.cmml">n</mi><mi id="S2.E5.m1.5.5.1.1.3.3.2.3" xref="S2.E5.m1.5.5.1.1.3.3.2.3.cmml">i</mi><mi mathvariant="normal" id="S2.E5.m1.5.5.1.1.3.3.3" xref="S2.E5.m1.5.5.1.1.3.3.3.cmml">c</mi></msubsup><mo id="S2.E5.m1.5.5.1.1.3.1a" xref="S2.E5.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.E5.m1.5.5.1.1.3.4.2" xref="S2.E5.m1.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.5.5.1.1.3.4.2.1" xref="S2.E5.m1.1.1.cmml">(</mo><mover accent="true" id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mi id="S2.E5.m1.1.1.2" xref="S2.E5.m1.1.1.2.cmml">ϕ</mi><mo stretchy="false" id="S2.E5.m1.1.1.1" xref="S2.E5.m1.1.1.1.cmml">^</mo></mover><mo stretchy="false" id="S2.E5.m1.5.5.1.1.3.4.2.2" xref="S2.E5.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.5.5.1.1.2" xref="S2.E5.m1.5.5.1.1.2.cmml">=</mo><mrow id="S2.E5.m1.5.5.1.1.1" xref="S2.E5.m1.5.5.1.1.1.cmml"><msub id="S2.E5.m1.5.5.1.1.1.3" xref="S2.E5.m1.5.5.1.1.1.3.cmml"><mi id="S2.E5.m1.5.5.1.1.1.3.2" xref="S2.E5.m1.5.5.1.1.1.3.2.cmml">b</mi><mi id="S2.E5.m1.5.5.1.1.1.3.3" xref="S2.E5.m1.5.5.1.1.1.3.3.cmml">i</mi></msub><mo id="S2.E5.m1.5.5.1.1.1.2" xref="S2.E5.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.5.5.1.1.1.1" xref="S2.E5.m1.5.5.1.1.1.1.cmml"><msubsup id="S2.E5.m1.5.5.1.1.1.1.2" xref="S2.E5.m1.5.5.1.1.1.1.2.cmml"><mo largeop="true" symmetric="true" id="S2.E5.m1.5.5.1.1.1.1.2.2.2" xref="S2.E5.m1.5.5.1.1.1.1.2.2.2.cmml">∫</mo><mn id="S2.E5.m1.5.5.1.1.1.1.2.2.3" xref="S2.E5.m1.5.5.1.1.1.1.2.2.3.cmml">0</mn><msub id="S2.E5.m1.5.5.1.1.1.1.2.3" xref="S2.E5.m1.5.5.1.1.1.1.2.3.cmml"><mi id="S2.E5.m1.5.5.1.1.1.1.2.3.2" xref="S2.E5.m1.5.5.1.1.1.1.2.3.2.cmml">χ</mi><mi 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xref="S2.E5.m1.5.5.1.1.cmml">,</mo></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1712.00967

Formulas:

1-

N_{T E} (C) / N_{T R} (C)

2-

i \in N (c)

3-

2^{[- 0.1, 0.1]}

4-

2^{[- 1, 1]}

5-

[- 20, 20]

6-

1 / 2 \times 1 / 2

7-

1 / 2 \times 1 / 2

8-

1 / 2 \times 1 / 2

9-

1 / 2 \times 1 / 2

10-

𝒪 (1 / \sqrt{k})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9608050

Formulas:

1-

d s^{2} = - (1 - \frac{2 M}{r}) d t^{2} + {(1 - \frac{2 M}{r})}^{- 1} d r^{2} + r^{2} d Ω^{2},

2-

\bar{r} {(1 + \frac{M}{2 \bar{r}})}^{2},

3-

- α {(\bar{r})}^{2} d t^{2} + ψ {(\bar{r})}^{4} (d {\bar{r}}^{2} + {\bar{r}}^{2} d Ω^{2}),

4-

α (\bar{r})

5-

(1 - \frac{M}{2 \bar{r}}) / (1 + \frac{M}{2 \bar{r}}),

6-

ψ (\bar{r})

7-

1 + \frac{M}{2 \bar{r}} .

8-

d x^{2} + d y^{2} + d z^{2} = d {\bar{r}}^{2} + {\bar{r}}^{2} d Ω^{2}, \bar{r} = {(x^{2} + y^{2} + z^{2})}^{1 / 2} .

9-

\bar{r} = M / 2

10-

\bar{r} \leftrightarrow {\bar{r}}^{'} = M^{2} / (4 \bar{r})

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/0705.3444

Formulas:

1-

μ \equiv M_{C} / (M_{⋆} + M_{C})

2-

Q_{B} \equiv a_{B} (1 + e_{B})

3-

m_{p} < 1 M_{\oplus}

4-

i 0_3

5-

i 0_1

6-

i 0_2

7-

i 0_4

8-

i 0_1

9-

i 0_2

10-

i 15_2

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1308.4591

Formulas:

1-

E (k^{'}) = ℏ v_{F} k^{'} = β k^{'}

2-

{⟨ g_{q}^{2} ⟩}_{𝙵}

3-

{⟨ g_{Γ}^{2} ⟩}_{𝙵}

4-

{⟨ g_{K}^{2} ⟩}_{𝙵}

5-

\frac{{⟨ g_{K}^{2} ⟩}_{𝙵} ω_{K}}{{⟨ g_{Γ}^{2} ⟩}_{𝙵} ω_{Γ}}

6-

g_{(𝐤 + 𝐪) m, 𝐤 n}^{ν} = \sqrt{\frac{ℏ}{2 M ω_{𝐪}^{ν}}} ⟨ m 𝐤 + 𝐪 | \frac{δ V_{s c f}}{δ u_{𝐪}^{ν}} | n 𝐤 ⟩,

7-

δ V_{s c f} \equiv V_{s c f} (u_{𝐪}^{ν}) - V_{s c f} (0)

8-

{⟨ g_{q}^{2} ⟩}_{F} = \sum_{i, j}^{π} {| g_{(K + q) i, K j} |}^{2} / 4

9-

{⟨ g_{K}^{2} ⟩}_{F} = \sum_{i, j}^{π} {| g_{(2 K) i, K j} |}^{2} / 4

10-

{⟨ g_{Γ}^{2} ⟩}_{F} = \sum_{i, j}^{π} {| g_{K i, K j} |}^{2} / 4

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/2002.02174

Formulas:

1-

T_{c} ≃ \frac{ω_{\log}}{1.2} \exp (- \frac{1.04 (1 + λ)}{λ - μ^{*} (1 + 0.62 λ)}),

2-

T_{c} = \frac{f_{1} f_{2} ω_{\log}}{1.2} \exp (- \frac{1.04 (1 + λ)}{λ - μ^{*} (1 + 0.62 λ)}),

3-

f_{1} = {[1 + \frac{λ}{2.46 (1 + 3.8 μ^{*})}]}^{1 / 3},

4-

f_{2} = 1 + \frac{(\sqrt{ω_{2}} / ω_{\log} - 1) λ^{2}}{λ^{2} + [1.82 (1 + 6.3 μ^{*})] {(\sqrt{ω_{2}} / ω_{\log})}^{2}} .

5-

ω_{2} = \frac{1}{λ} \int_{0}^{\infty} Ω α^{2} F (Ω) 𝑑 Ω

6-

α^{2} F (Ω)

7-

α^{2} F (Ω)

8-

d y n a m i c a l l y

9-

α^{2} F (ω)

10-

λ (ω) = \int_{0}^{ω} α^{2} F (ω^{'}) 𝑑 ω^{'}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1309.1283

Formulas:

1-

\frac{d σ_{i}}{d t} = - ψ (t) X_{i} (t) + \int_{M_{L}}^{M_{B m}} ψ (t - τ_{m}) Q_{m i} (t - τ_{m}) ϕ (m) 𝑑 m

2-

+ A_{B} \int_{M_{B m}}^{M_{B M}} ϕ (m) [\int_{μ_{B m i n}}^{0.5} f (μ_{B}) ψ (t - τ_{m_{2}}) Q_{m i} (t - τ_{m_{2}}) 𝑑 μ_{B}] 𝑑 m

3-

+ (1 - A_{B}) \int_{M_{B m}}^{M_{B M}} ψ (t - τ_{m}) Q_{m i} (t - τ_{m}) ϕ (m) 𝑑 m

4-

+ \int_{M_{B M}}^{M_{U}} ψ (t - τ_{m}) Q_{m i} (t - τ_{m}) ϕ (m) 𝑑 m

5-

+ \frac{d σ_{i, i n f}}{d t}

6-

μ = \frac{M_{1}}{(M_{1} + M_{2})}

7-

M_{B} = M_{1} + M_{2}

8-

\frac{d σ_{i} {(r, t)}_{i n f}}{d t} = A (r) {(X_{i})}_{i n f} e^{- \frac{t}{τ_{H}}} + B (r) {(X_{i})}_{i n f} e^{- \frac{t - t_{m a x}}{τ_{D}}}

9-

σ_{i} {(r, t)}_{i n f}

10-

{(X_{i})}_{i n f}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1703.00398

Formulas:

1-

μ_{x} (t)

2-

\ln μ_{x} (t) = α_{x} + β_{x} k_{t} + ϵ_{x, t} .

3-

D_{x} (t)

4-

D_{x} (t) \sim P o i s s o n (E T R_{x} (t), μ_{x} (t))

5-

E T R_{x} (t)

6-

μ_{x} (t)

7-

\ln μ_{x} (t) = α_{x} + β_{x} k_{t} .

8-

\ln μ_{x} (t) = m + α_{x} + β_{t} + γ_{t - x} .

9-

\ln μ_{x} (t) = α_{x} + β_{x}^{(0)} i_{t - x} + β_{x}^{(1)} k_{t} .

10-

β_{x}^{(0)} i_{t - x}

Formulas (html):

Correct Categorie: stat

Guessed Categorie: stat

Result: correct

Run 16446903 (Agent672)