Run 11274858

Paper: https://arxiv.org/abs/1203.0049

Formulas:

1-

J = | J | \cdot e^{i θ}

2-

\hat{H} (t) = - \sum_{⟨ i j ⟩} J_{i j} {\hat{a}}_{i}^{†} {\hat{a}}_{j} + \sum_{i} v_{i} (t) {\hat{n}}_{i} + {\hat{H}}_{on-site} .

3-

J_{i j} > 0

4-

v_{i} (t) = v_{i} (t + T)

5-

{⟨ v_{i} ⟩}_{T} \equiv \frac{1}{T} \int_{0}^{T} d t v_{i} (t)

6-

𝑭 = - m \ddot{𝒙}

7-

v_{i} = - 𝒓_{i} \cdot 𝑭

8-

{\hat{H}}_{eff} = - \sum_{i j} | J_{i j}^{eff} | e^{i θ_{i j}} {\hat{a}}_{i}^{†} {\hat{a}}_{j} + {\hat{H}}_{on-site}

9-

| J_{i j}^{eff} | e^{i θ_{i j}} = J_{i j} {⟨ e^{i (χ_{j} - χ_{i}) / ℏ} ⟩}_{T}

10-

χ_{i} (t) = - \int_{t_{0}}^{t} d t^{'} v_{i} (t^{'}) + {⟨ \int_{t_{0}}^{t} d t^{'} v_{i} (t^{'}) ⟩}_{T}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1310.0369

Formulas:

1-

d_{x z} / d_{y z}

2-

d_{x z} / d_{y z}

3-

d_{x z} / d_{y z}

4-

(A = 2 π e F / ℏ)

5-

τ (H, T)

6-

\frac{m^{⋆}}{m_{b}} - 1

7-

λ = \frac{m^{⋆}}{m_{b}} - 1

8-

F = F_{0} / \cos (θ)

9-

F = F_{0} / \cos (θ)

10-

x = 14.69 m^{⋆} T / H

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1304.0008

Formulas:

1-

S_{n} (f)

2-

ρ (t) = \frac{| z (t) |}{σ}

3-

z (t) = 4 \int_{f_{L}}^{f_{H}} \frac{\tilde{s} (f) {\tilde{h}}^{*} (f)}{S_{n} (f)} ⅆ f

4-

σ^{2} = 4 \int_{f_{L}}^{f_{H}} \frac{{| \tilde{h} (f) |}^{2}}{S_{n} (f)} ⅆ f .

5-

\tilde{s} (f)

6-

\tilde{h} (f) = h_{0} f^{- 7 / 6} e^{- i ψ (f)}

7-

f_{ISCO} = c^{3} / (6 \sqrt{6} π G M)

8-

s (t) = A \exp (- \frac{{(t - t_{0})}^{2}}{τ^{2}}) \cos (2 π f_{0} t + ϕ_{0})

9-

τ = Q / 2 π f_{0}

10-

\tilde{s} (f) = \frac{\sqrt{π}}{2} τ \exp (- π^{2} τ^{2} {(f - f_{0})}^{2} + i ϕ_{0}) [1 + \exp (- Q^{2} \frac{f}{f_{0}} - 2 i ϕ_{0})] .

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/2005.11058

Formulas:

1-

λ_{I}^{A2} = - λ_{I}^{B2}

2-

λ_{ex}^{A1} = λ_{ex}^{B1}

3-

ℋ = ℋ_{orb} + ℋ_{soc} + ℋ_{R} + ℋ_{ex} + E_{D} .

4-

- \frac{\sqrt{3} γ_{0} a}{2} μ_{0} \otimes (τ k_{x} σ_{x} + k_{y} σ_{y}) \otimes s_{0}

5-

+ \frac{γ_{1}}{2} (μ_{x} \otimes σ_{x} - μ_{y} \otimes σ_{y}) \otimes s_{0}

6-

- \frac{\sqrt{3} γ_{3} a}{4} μ_{x} \otimes (τ k_{x} σ_{x} - k_{y} σ_{y}) \otimes s_{0}

7-

- \frac{\sqrt{3} γ_{3} a}{4} μ_{y} \otimes (τ k_{x} σ_{y} + k_{y} σ_{x}) \otimes s_{0}

8-

- \frac{\sqrt{3} γ_{4} a}{2} (τ k_{x} μ_{x} - k_{y} μ_{y}) \otimes σ_{0} \otimes s_{0}

9-

+ V μ_{z} \otimes σ_{0} \otimes s_{0}

10-

+ Δ (μ_{+} \otimes σ_{+} + μ_{-} \otimes σ_{-}) \otimes s_{0},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1605.06825

Formulas:

1-

{1, 2, \dots, s t}

2-

σ_{1}, \dots, σ_{k}

3-

σ_{1}, \dots, σ_{k}

4-

π (i) ⇑ π (j)

5-

([n], ⇑)

6-

σ_{1}, \dots, σ_{k}

7-

N E_{4, 3}

8-

E N_{4, 3}

9-

{1, 2, \dots, n}

10-

P = (X, ⇑)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1508.04169

Formulas:

1-

ε (t) = 0

2-

J (ε) = Tr [U_{T}^{ε} ρ_{0} U_{T}^{† ε} O]

3-

ρ_{0} = | i ⟩ ⟨ i |

4-

\frac{d U_{t}^{ε}}{d t} = - i (H_{0} + V ε (t)) U_{t}^{ε}, U_{0}^{ε} = 𝕀

5-

H_{0} = \sum_{j = 1}^{3} ε_{j} | j ⟩ ⟨ j |

6-

\nabla J_{ε} = 0

7-

H_{ε} = δ^{2} J / δ ε^{2}

8-

δ J_{ε} := J (ε + δ ε) - J (ε) = R (δ ε) + O (δ ε^{n + 1})

9-

R (δ ε) = O (δ ε^{n})

10-

R (δ ε) \leq 0

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1402.4932

Formulas:

1-

ℋ = - t \sum_{⟨ i, j ⟩, σ} (c_{i σ}^{†} c_{j σ} + H.c.) + V \sum_{⟨ i, j ⟩} n_{i} n_{j} + U \sum_{i} n_{i ↑} n_{i ↓} .

2-

n_{i} = n_{i ↑} + n_{i ↓}

3-

n_{i σ} = c_{i σ}^{†} c_{i σ}

4-

| t | ≪ V ≪ U

5-

N_{e} = 2 N / 3

6-

n = N_{e} / N = 2 / 3

7-

2^{2 N / 3}

8-

\sim 2^{2 N / 3} \times {1.1137}^{N}

9-

t / (U - V)

10-

ℋ_{eff} = ℋ_{ring} + ℋ_{spin} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1603.04559

Formulas:

1-

\frac{2 m - n + 2}{7}

2-

S \subseteq V (G)

3-

ϕ (G) \leq k

4-

ϕ (G) \leq \frac{3 n}{10}

5-

ϕ (G) \leq \frac{5 m}{23}

6-

\frac{5}{3} (n - 2)

7-

\frac{25 n - 50}{69}

8-

ϕ (G) \leq \frac{m}{g}

9-

ϕ (G) \leq \frac{2 m - n + 2}{7}

10-

\frac{5}{3} (n - 2)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0501494

Formulas:

1-

Δ T (θ, φ) = \sum_{ℓ = 2}^{\infty} \sum_{m = - ℓ}^{ℓ} | a_{ℓ m} | e^{i ϕ_{ℓ m}} Y_{ℓ m} (θ, φ),

2-

| a_{ℓ m} |

3-

[0, 2 π]

4-

C_{ℓ} = \frac{1}{2 ℓ + 1} \sum_{m = - ℓ}^{ℓ} {| a_{ℓ m} |}^{2}

5-

D_{ℓ} = \frac{ℓ (ℓ + 1)}{2 π} \frac{C_{ℓ}}{T_{0}} .

6-

a_{ℓ m}^{^{'}} = B_{ℓ} a_{ℓ m},

7-

B_{ℓ} = \exp [- σ^{2} ℓ (ℓ + 1) / 2]

8-

σ = FWHM / \sqrt{8 \ln 2}

9-

a_{ℓ m}^{^{''}} = \frac{a_{ℓ m}}{B_{ℓ} + α},

10-

| a_{ℓ m} |

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0605577

Formulas:

1-

q \equiv M_{2} / M_{1} \approx 1

2-

\sim 1 \overset{''}{.} 5

3-

M_{V e g a}

4-

S p T = 7.26 - 3.44 (F 110 W - F 170 M)

5-

J_{M K O} - F 110 W

6-

- 1.15 + 0.025 (F 110 W - F 170 M)

7-

+ 0.035 {(F 110 W - F 170 M)}^{2}

8-

H_{M K O} - F 170 M

9-

- 0.75 + 0.89 (F 110 W - F 170 M)

10-

- 0.27 {(F 110 W - F 170 M)}^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1708.02158

Formulas:

1-

x + i y = z \mapsto f (z)

2-

f^{'} (z) = \frac{\partial f}{\partial z} = \frac{1}{2} (\frac{\partial f (x + i y)}{\partial x} - i \frac{\partial f (x + i y)}{\partial y}) \neq 0

3-

\frac{\partial f}{\partial \bar{z}} = \frac{1}{2} (\frac{\partial f (x + i y)}{\partial x} + i \frac{\partial f (x + i y)}{\partial y}) = 0 .

4-

R_{τ} = {w \in ℂ : 0 < Re (w) < 1, 0 < Im (w) < τ}

5-

p_{1}, \dots, p_{4} \in \partial G

6-

f : G \to R_{τ}

7-

p_{1} \mapsto 0, p_{2} \mapsto 1, p_{3} \mapsto 1 + i τ and p_{4} \mapsto i τ .

8-

τ = τ (G)

9-

\hat{ℂ} = ℂ \cup {\infty}

10-

σ (z, d z) = Q (z) d z^{2}

Formulas (html):

Correct Categorie: stat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1608.01571

Formulas:

1-

H = ℏ Δ_{a} {\hat{σ}}^{†} \hat{σ} + ℏ Δ_{c} {\hat{a}}^{†} \hat{a} + ℏ g ({\hat{a}}^{†} \hat{σ} + {\hat{σ}}^{†} \hat{a}) + H_{d}

2-

Δ_{a} = ω_{a} - ω_{d}

3-

Δ_{c} = ω_{c} - ω_{d}

4-

\hat{n} = {\hat{a}}^{†} \hat{a}

5-

H_{d} = ℏ η_{c} (\hat{a} + {\hat{a}}^{†})

6-

H_{d} = ℏ η_{a} (\hat{σ} + {\hat{σ}}^{†})

7-

| n, \pm ⟩ = (| n, g ⟩ \pm | n - 1, e ⟩) / \sqrt{2}

8-

| 0, g ⟩ \to | 1, \pm ⟩

9-

{\tilde{η}}_{𝑎} = \pm \sqrt{2} η_{a}

10-

{\tilde{η}}_{𝑐} = \sqrt{2} η_{c}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0307504

Formulas:

1-

T = MJD 40 609.564056 (11) + 0.521183398 (26) E

2-

- {0.23}_{- 0.06}^{+ 0.12}

3-

- {0.06}_{- 0.21}^{+ 0.29}

4-

- {0.23}_{- 0.06}^{+ 0.12}

5-

- {0.17}_{- 0.12}^{+ 0.29}

6-

- {0.19}_{- 0.13}^{+ 0.17}

7-

- {0.17}_{- 0.25}^{+ 0.33}

8-

- {0.23}_{- 0.06}^{+ 0.12}

9-

- {0.12}_{- 0.23}^{+ 0.24}

10-

- {0.23}_{- 0.06}^{+ 0.12}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1408.1243

Formulas:

1-

S (t) = \int [k δ (t - t^{'}) + R_{n} (t - t^{'})] G (t^{'}) 𝑑 t^{'};

2-

R_{n} (t)

3-

R_{n} (t) \propto - \frac{\partial}{\partial t} ⟨ χ_{x y} (t) χ_{x y} (0) ⟩

4-

R_{n} (t)

5-

I m [{\tilde{R}}_{n} (ν)] \propto I m {F T [S (t)] / F T [G (t)]}

6-

I m [{\tilde{R}}_{n} (ν)]

7-

R_{n} (t) = A R_{d r y} (t) + B \frac{d}{d t} e x p [- {(\frac{t}{τ})}^{β}] + \sum_{i} C_{i} e x p (- γ_{i}^{2} t^{2}) \sin (ω_{i} t)

8-

R_{n} (t)

9-

R_{d r y} (t)

10-

τ \propto {(T - T_{c})}^{- γ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1005.0123

Formulas:

1-

δ t = β t_{p}^{2 / 3} t^{1 / 3}

2-

\dot{G} / G \sim 2 \pm 4 \times 10^{- 12} {yr}^{- 1}

3-

- 1.6 \times 10^{- 12} {yr}^{- 1} < \dot{G} / G < 0

4-

- 10^{- 11} {yr}^{- 1} \leq \frac{\dot{G}}{G} < 0

5-

| \frac{\dot{G}}{G} | \leq 4.10 \times 10^{- 11} {yr}^{- 1}

6-

S = \int d^{5} x \sqrt{- g} (M_{5}^{3} R - ρ_{Λ 5}) + \int d^{4} x \sqrt{- γ} (L_{b r} - V + r_{c} M_{5}^{3} R_{4}) .

7-

H^{2} + \frac{k}{a^{2}} = \frac{8 π}{3 M_{p}^{2}} ρ + \frac{8 π}{3 M_{p}^{2}} ρ_{Λ} .

8-

ρ = ρ_{m} + ρ_{Λ}

9-

M_{p}^{2} = {(8 π G)}^{- 1}

10-

ρ_{Λ} \equiv ρ_{Λ 4} = \frac{M_{p}^{2}}{32 π M_{5}^{3}} ρ_{Λ 5} + \frac{3 M_{p}^{2}}{2 π {(\frac{L_{5}}{8 π} - 2 r_{c})}^{2}} .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1410.3134

Formulas:

1-

\ddot{θ} + \frac{σ^{2}}{2} {(\frac{a}{r})}^{3} \sin 2 (θ - f) = 0, with σ = n \sqrt{\frac{3 (B - A)}{C}},

2-

\ddot{θ} + \frac{σ^{2}}{2} \sum_{k = - \infty}^{+ \infty} X_{k}^{- 3, 2} (e) \sin (2 θ - k n t) = 0,

3-

X_{k}^{- 3, 2} (e)

4-

e^{| k - 2 |}

5-

\dot{θ} = k n / 2

6-

σ {\sqrt{e}}^{| k - 2 |}

7-

X_{k}^{- 3, 2} (0)

8-

(γ, \dot{θ} / n)

9-

μ = (m_{1} + m_{2}) / (m_{0} + m_{1} + m_{2})

10-

\begin{matrix} λ_{1} (t) \approx n t + δ ζ (t) + λ^{0}, & a_{1} (t) \approx \bar{a} (1 - δ \frac{2}{3} \frac{\dot{ζ} (t)}{n}) \\ λ_{2} (t) \approx n t - (1 - δ) ζ (t) + λ^{0}, & a_{2} (t) \approx \bar{a} (1 + (1 - δ) \frac{2}{3} \frac{\dot{ζ} (t)}{n}) \end{matrix}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/2004.02370

Formulas:

1-

𝒢 = (𝒩, ℒ)

2-

𝒩 := {0, \dots, N}

3-

p_{n} + j q_{n}

4-

[𝐚_{0} 𝐀]

5-

𝐀 \in {0, \pm 1}^{N \times N}

6-

𝐯_{t} ≃ 𝐆^{- 1} 𝐩_{t} + 𝐁^{- 1} 𝐪_{t} + 𝟏

7-

𝐆 := 𝐀^{⊤} {dg}^{- 1} (𝐫) 𝐀

8-

𝐁 := 𝐀^{⊤} {dg}^{- 1} (𝐱) 𝐀

9-

{\tilde{𝐯}}_{t} ≃ 𝐆^{- 1} {\tilde{𝐩}}_{t} + 𝐁^{- 1} {\tilde{𝐪}}_{t}

10-

{\tilde{𝐯}}_{t} := 𝐯_{t} - 𝐯_{t - 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0603200

Formulas:

1-

\sim 0.6 n m

2-

v_{y} = 200 m / s

3-

v_{y} = 200 m / s

4-

λ_{d b} = 2.8

5-

y = v_{y} t

6-

z = - \frac{1}{2} g t^{2}

7-

ψ (x, t)

8-

ψ_{0} (x) = {(2 π σ_{0}^{2})}^{- \frac{1}{4}} \exp^{- \frac{x^{2}}{4 σ_{0}^{2}}}

9-

σ_{0} = 2 μ m

10-

ψ (x, t) = {(2 π s {(t)}^{2})}^{- \frac{1}{4}} \exp^{- \frac{x^{2}}{4 σ_{0} s (t)}}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0612151

Formulas:

1-

ρ ≐ {| ψ |}^{2}

2-

ψ \in L^{2} (R)

3-

ρ (x, t)

4-

ψ \in L^{2} (R^{n})

5-

ρ ≐ {| ψ |}^{2}

6-

L^{2} (R^{n})

7-

L^{2} (R)

8-

(ℱ ψ) (p)

9-

ρ (x) = {| ψ (x) |}^{2}

10-

\tilde{ρ} (p) = {| (ℱ ψ) (p) |}^{2}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0005407

Formulas:

1-

05^{h} 37^{m} 50^{s}

2-

05^{h} 38^{m} 45^{s}

3-

05^{h} 37^{m} 50^{s}

4-

- 69 ° 10' 00 ″

5-

- 69 ° 06' 00 ″

6-

- 69 ° 10' 00 ″

7-

120 ″ \times 70 ″

8-

- 69 ° 10' 50 ″

9-

05^{h} 37^{m} 44 \overset{s}{.} 5

10-

- 69 ° 10' 07 \overset{''}{.} 5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1005.1390

Formulas:

1-

L_{γ} \propto ψ M_{gas}

2-

(1.03 \pm 0.17) \times 10^{- 5} photons {cm}^{- 2} s^{- 1} {sr}^{- 1}

3-

H_{0} = 70 {Mpc}^{- 1} km s^{- 1}

4-

≳ 10^{6} M_{⊙}

5-

ϕ (L_{γ}, z)

6-

L_{γ} \propto ψ M_{gas}

7-

{(\frac{ψ}{M_{⊙} {yr}^{- 1}} \times \frac{M_{gas}}{10^{9} M_{⊙}})}^{0.86 \pm 0.06} [10^{39} erg / s]

8-

{(\frac{ψ}{M_{⊙} {yr}^{- 1}})}^{1.2 \pm 0.09} [10^{39} erg / s],

9-

L_{γ} \propto {(ψ M_{gas})}^{1}

10-

L_{γ} \propto {(ψ)}^{1}

Formulas (html):

<math><mrow id="id6.4.m4.1.1" xref="id6.4.m4.1.1.cmml"><msub id="id6.4.m4.1.1.2" xref="id6.4.m4.1.1.2.cmml"><mi id="id6.4.m4.1.1.2.2" xref="id6.4.m4.1.1.2.2.cmml">L</mi><mi id="id6.4.m4.1.1.2.3" xref="id6.4.m4.1.1.2.3.cmml">γ</mi></msub><mo id="id6.4.m4.1.1.1" xref="id6.4.m4.1.1.1.cmml">∝</mo><mrow id="id6.4.m4.1.1.3" xref="id6.4.m4.1.1.3.cmml"><mi id="id6.4.m4.1.1.3.2" xref="id6.4.m4.1.1.3.2.cmml">ψ</mi><mo id="id6.4.m4.1.1.3.1" xref="id6.4.m4.1.1.3.1.cmml">⁢</mo><msub id="id6.4.m4.1.1.3.3" xref="id6.4.m4.1.1.3.3.cmml"><mi id="id6.4.m4.1.1.3.3.2" xref="id6.4.m4.1.1.3.3.2.cmml">M</mi><mi id="id6.4.m4.1.1.3.3.3" xref="id6.4.m4.1.1.3.3.3.cmml">gas</mi></msub></mrow></mrow></math>, <math><mrow id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mrow id="S1.p1.2.m2.1.1.1" xref="S1.p1.2.m2.1.1.1.cmml"><mrow id="S1.p1.2.m2.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p1.2.m2.1.1.1.1.1.2" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.2.m2.1.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml"><mn id="S1.p1.2.m2.1.1.1.1.1.1.2" xref="S1.p1.2.m2.1.1.1.1.1.1.2.cmml">1.03</mn><mo id="S1.p1.2.m2.1.1.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.1.cmml">±</mo><mn id="S1.p1.2.m2.1.1.1.1.1.1.3" xref="S1.p1.2.m2.1.1.1.1.1.1.3.cmml">0.17</mn></mrow><mo stretchy="false" id="S1.p1.2.m2.1.1.1.1.1.3" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p1.2.m2.1.1.1.2" xref="S1.p1.2.m2.1.1.1.2.cmml">×</mo><mpadded width="+2.8pt" id="S1.p1.2.m2.1.1.1.3" xref="S1.p1.2.m2.1.1.1.3.cmml"><msup id="S1.p1.2.m2.1.1.1.3a" xref="S1.p1.2.m2.1.1.1.3.cmml"><mn id="S1.p1.2.m2.1.1.1.3.2" xref="S1.p1.2.m2.1.1.1.3.2.cmml">10</mn><mrow id="S1.p1.2.m2.1.1.1.3.3" xref="S1.p1.2.m2.1.1.1.3.3.cmml"><mo id="S1.p1.2.m2.1.1.1.3.3.1" xref="S1.p1.2.m2.1.1.1.3.3.1.cmml">-</mo><mn id="S1.p1.2.m2.1.1.1.3.3.2" xref="S1.p1.2.m2.1.1.1.3.3.2.cmml">5</mn></mrow></msup></mpadded></mrow><mo id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">⁢</mo><mpadded width="+2.8pt" id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml"><mi id="S1.p1.2.m2.1.1.3a" xref="S1.p1.2.m2.1.1.3.cmml">photons</mi></mpadded><mo id="S1.p1.2.m2.1.1.2a" xref="S1.p1.2.m2.1.1.2.cmml">⁢</mo><mpadded width="+2.8pt" id="S1.p1.2.m2.1.1.4" xref="S1.p1.2.m2.1.1.4.cmml"><msup id="S1.p1.2.m2.1.1.4a" xref="S1.p1.2.m2.1.1.4.cmml"><mi id="S1.p1.2.m2.1.1.4.2" xref="S1.p1.2.m2.1.1.4.2.cmml">cm</mi><mrow id="S1.p1.2.m2.1.1.4.3" xref="S1.p1.2.m2.1.1.4.3.cmml"><mo id="S1.p1.2.m2.1.1.4.3.1" xref="S1.p1.2.m2.1.1.4.3.1.cmml">-</mo><mn id="S1.p1.2.m2.1.1.4.3.2" xref="S1.p1.2.m2.1.1.4.3.2.cmml">2</mn></mrow></msup></mpadded><mo id="S1.p1.2.m2.1.1.2b" xref="S1.p1.2.m2.1.1.2.cmml">⁢</mo><mpadded width="+2.8pt" id="S1.p1.2.m2.1.1.5" xref="S1.p1.2.m2.1.1.5.cmml"><msup id="S1.p1.2.m2.1.1.5a" xref="S1.p1.2.m2.1.1.5.cmml"><mi mathvariant="normal" id="S1.p1.2.m2.1.1.5.2" xref="S1.p1.2.m2.1.1.5.2.cmml">s</mi><mrow id="S1.p1.2.m2.1.1.5.3" xref="S1.p1.2.m2.1.1.5.3.cmml"><mo id="S1.p1.2.m2.1.1.5.3.1" xref="S1.p1.2.m2.1.1.5.3.1.cmml">-</mo><mn id="S1.p1.2.m2.1.1.5.3.2" xref="S1.p1.2.m2.1.1.5.3.2.cmml">1</mn></mrow></msup></mpadded><mo id="S1.p1.2.m2.1.1.2c" xref="S1.p1.2.m2.1.1.2.cmml">⁢</mo><msup id="S1.p1.2.m2.1.1.6" xref="S1.p1.2.m2.1.1.6.cmml"><mi id="S1.p1.2.m2.1.1.6.2" xref="S1.p1.2.m2.1.1.6.2.cmml">sr</mi><mrow id="S1.p1.2.m2.1.1.6.3" xref="S1.p1.2.m2.1.1.6.3.cmml"><mo id="S1.p1.2.m2.1.1.6.3.1" xref="S1.p1.2.m2.1.1.6.3.1.cmml">-</mo><mn id="S1.p1.2.m2.1.1.6.3.2" xref="S1.p1.2.m2.1.1.6.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S1.p7.3.m3.1.1" xref="S1.p7.3.m3.1.1.cmml"><msub id="S1.p7.3.m3.1.1.2" xref="S1.p7.3.m3.1.1.2.cmml"><mi id="S1.p7.3.m3.1.1.2.2" xref="S1.p7.3.m3.1.1.2.2.cmml">H</mi><mn id="S1.p7.3.m3.1.1.2.3" xref="S1.p7.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S1.p7.3.m3.1.1.1" xref="S1.p7.3.m3.1.1.1.cmml">=</mo><mrow id="S1.p7.3.m3.1.1.3" xref="S1.p7.3.m3.1.1.3.cmml"><mpadded width="+3.3pt" id="S1.p7.3.m3.1.1.3.2" xref="S1.p7.3.m3.1.1.3.2.cmml"><mn id="S1.p7.3.m3.1.1.3.2a" xref="S1.p7.3.m3.1.1.3.2.cmml">70</mn></mpadded><mo id="S1.p7.3.m3.1.1.3.1" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mpadded width="+3.3pt" id="S1.p7.3.m3.1.1.3.3" xref="S1.p7.3.m3.1.1.3.3.cmml"><msup id="S1.p7.3.m3.1.1.3.3a" xref="S1.p7.3.m3.1.1.3.3.cmml"><mi id="S1.p7.3.m3.1.1.3.3.2" xref="S1.p7.3.m3.1.1.3.3.2.cmml">Mpc</mi><mrow id="S1.p7.3.m3.1.1.3.3.3" xref="S1.p7.3.m3.1.1.3.3.3.cmml"><mo id="S1.p7.3.m3.1.1.3.3.3.1" xref="S1.p7.3.m3.1.1.3.3.3.1.cmml">-</mo><mn id="S1.p7.3.m3.1.1.3.3.3.2" xref="S1.p7.3.m3.1.1.3.3.3.2.cmml">1</mn></mrow></msup></mpadded><mo id="S1.p7.3.m3.1.1.3.1a" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mpadded width="+3.3pt" id="S1.p7.3.m3.1.1.3.4" xref="S1.p7.3.m3.1.1.3.4.cmml"><mi id="S1.p7.3.m3.1.1.3.4a" xref="S1.p7.3.m3.1.1.3.4.cmml">km</mi></mpadded><mo id="S1.p7.3.m3.1.1.3.1b" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><msup id="S1.p7.3.m3.1.1.3.5" xref="S1.p7.3.m3.1.1.3.5.cmml"><mi mathvariant="normal" id="S1.p7.3.m3.1.1.3.5.2" xref="S1.p7.3.m3.1.1.3.5.2.cmml">s</mi><mrow id="S1.p7.3.m3.1.1.3.5.3" xref="S1.p7.3.m3.1.1.3.5.3.cmml"><mo id="S1.p7.3.m3.1.1.3.5.3.1" xref="S1.p7.3.m3.1.1.3.5.3.1.cmml">-</mo><mn id="S1.p7.3.m3.1.1.3.5.3.2" xref="S1.p7.3.m3.1.1.3.5.3.2.cmml">1</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml"/><mo id="S2.SS1.p3.2.m2.1.1.1" xref="S2.SS1.p3.2.m2.1.1.1.cmml">≳</mo><mrow id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml"><msup id="S2.SS1.p3.2.m2.1.1.3.2" xref="S2.SS1.p3.2.m2.1.1.3.2.cmml"><mn id="S2.SS1.p3.2.m2.1.1.3.2.2" xref="S2.SS1.p3.2.m2.1.1.3.2.2.cmml">10</mn><mn id="S2.SS1.p3.2.m2.1.1.3.2.3" xref="S2.SS1.p3.2.m2.1.1.3.2.3.cmml">6</mn></msup><mo id="S2.SS1.p3.2.m2.1.1.3.1" xref="S2.SS1.p3.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p3.2.m2.1.1.3.3" xref="S2.SS1.p3.2.m2.1.1.3.3.cmml"><mi id="S2.SS1.p3.2.m2.1.1.3.3.2" xref="S2.SS1.p3.2.m2.1.1.3.3.2.cmml">M</mi><mo id="S2.SS1.p3.2.m2.1.1.3.3.3" xref="S2.SS1.p3.2.m2.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S2.SS2.p1.1.m1.2.2" xref="S2.SS2.p1.1.m1.2.2.cmml"><mi id="S2.SS2.p1.1.m1.2.2.3" xref="S2.SS2.p1.1.m1.2.2.3.cmml">ϕ</mi><mo id="S2.SS2.p1.1.m1.2.2.2" xref="S2.SS2.p1.1.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p1.1.m1.2.2.1.1" xref="S2.SS2.p1.1.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.SS2.p1.1.m1.2.2.1.1.2" xref="S2.SS2.p1.1.m1.2.2.1.2.cmml">(</mo><msub id="S2.SS2.p1.1.m1.2.2.1.1.1" xref="S2.SS2.p1.1.m1.2.2.1.1.1.cmml"><mi id="S2.SS2.p1.1.m1.2.2.1.1.1.2" xref="S2.SS2.p1.1.m1.2.2.1.1.1.2.cmml">L</mi><mi id="S2.SS2.p1.1.m1.2.2.1.1.1.3" xref="S2.SS2.p1.1.m1.2.2.1.1.1.3.cmml">γ</mi></msub><mo id="S2.SS2.p1.1.m1.2.2.1.1.3" xref="S2.SS2.p1.1.m1.2.2.1.2.cmml">,</mo><mi id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">z</mi><mo stretchy="false" id="S2.SS2.p1.1.m1.2.2.1.1.4" xref="S2.SS2.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml"><msub id="S2.SS2.p2.3.m3.1.1.2" xref="S2.SS2.p2.3.m3.1.1.2.cmml"><mi id="S2.SS2.p2.3.m3.1.1.2.2" xref="S2.SS2.p2.3.m3.1.1.2.2.cmml">L</mi><mi id="S2.SS2.p2.3.m3.1.1.2.3" xref="S2.SS2.p2.3.m3.1.1.2.3.cmml">γ</mi></msub><mo id="S2.SS2.p2.3.m3.1.1.1" xref="S2.SS2.p2.3.m3.1.1.1.cmml">∝</mo><mrow id="S2.SS2.p2.3.m3.1.1.3" xref="S2.SS2.p2.3.m3.1.1.3.cmml"><mi id="S2.SS2.p2.3.m3.1.1.3.2" xref="S2.SS2.p2.3.m3.1.1.3.2.cmml">ψ</mi><mo id="S2.SS2.p2.3.m3.1.1.3.1" xref="S2.SS2.p2.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS2.p2.3.m3.1.1.3.3" xref="S2.SS2.p2.3.m3.1.1.3.3.cmml"><mi id="S2.SS2.p2.3.m3.1.1.3.3.2" xref="S2.SS2.p2.3.m3.1.1.3.3.2.cmml">M</mi><mi id="S2.SS2.p2.3.m3.1.1.3.3.3" xref="S2.SS2.p2.3.m3.1.1.3.3.3.cmml">gas</mi></msub></mrow></mrow></math>, <math><mrow id="S2.E1.m3.2.2" xref="S2.E1.m3.2.2.cmml"><msup id="S2.E1.m3.1.1.1" xref="S2.E1.m3.1.1.1.cmml"><mrow id="S2.E1.m3.1.1.1.1.1" xref="S2.E1.m3.1.1.1.1.1.1.cmml"><mo id="S2.E1.m3.1.1.1.1.1.2" xref="S2.E1.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m3.1.1.1.1.1.1" xref="S2.E1.m3.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.E1.m3.1.1.1.1.1.1.2" xref="S2.E1.m3.1.1.1.1.1.1.2.cmml"><mfrac id="S2.E1.m3.1.1.1.1.1.1.2a" xref="S2.E1.m3.1.1.1.1.1.1.2.cmml"><mi id="S2.E1.m3.1.1.1.1.1.1.2.2" xref="S2.E1.m3.1.1.1.1.1.1.2.2.cmml">ψ</mi><mrow id="S2.E1.m3.1.1.1.1.1.1.2.3" xref="S2.E1.m3.1.1.1.1.1.1.2.3.cmml"><mpadded width="+2.8pt" id="S2.E1.m3.1.1.1.1.1.1.2.3.2" xref="S2.E1.m3.1.1.1.1.1.1.2.3.2.cmml"><msub id="S2.E1.m3.1.1.1.1.1.1.2.3.2a" xref="S2.E1.m3.1.1.1.1.1.1.2.3.2.cmml"><mi id="S2.E1.m3.1.1.1.1.1.1.2.3.2.2" xref="S2.E1.m3.1.1.1.1.1.1.2.3.2.2.cmml">M</mi><mo id="S2.E1.m3.1.1.1.1.1.1.2.3.2.3" xref="S2.E1.m3.1.1.1.1.1.1.2.3.2.3.cmml">⊙</mo></msub></mpadded><mo id="S2.E1.m3.1.1.1.1.1.1.2.3.1" xref="S2.E1.m3.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><msup id="S2.E1.m3.1.1.1.1.1.1.2.3.3" xref="S2.E1.m3.1.1.1.1.1.1.2.3.3.cmml"><mi id="S2.E1.m3.1.1.1.1.1.1.2.3.3.2" xref="S2.E1.m3.1.1.1.1.1.1.2.3.3.2.cmml">yr</mi><mrow id="S2.E1.m3.1.1.1.1.1.1.2.3.3.3" xref="S2.E1.m3.1.1.1.1.1.1.2.3.3.3.cmml"><mo id="S2.E1.m3.1.1.1.1.1.1.2.3.3.3.1" xref="S2.E1.m3.1.1.1.1.1.1.2.3.3.3.1.cmml">-</mo><mn id="S2.E1.m3.1.1.1.1.1.1.2.3.3.3.2" xref="S2.E1.m3.1.1.1.1.1.1.2.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mfrac></mstyle><mo id="S2.E1.m3.1.1.1.1.1.1.1" xref="S2.E1.m3.1.1.1.1.1.1.1.cmml">×</mo><mstyle displaystyle="true" id="S2.E1.m3.1.1.1.1.1.1.3" xref="S2.E1.m3.1.1.1.1.1.1.3.cmml"><mfrac id="S2.E1.m3.1.1.1.1.1.1.3a" xref="S2.E1.m3.1.1.1.1.1.1.3.cmml"><msub id="S2.E1.m3.1.1.1.1.1.1.3.2" xref="S2.E1.m3.1.1.1.1.1.1.3.2.cmml"><mi id="S2.E1.m3.1.1.1.1.1.1.3.2.2" xref="S2.E1.m3.1.1.1.1.1.1.3.2.2.cmml">M</mi><mi id="S2.E1.m3.1.1.1.1.1.1.3.2.3" xref="S2.E1.m3.1.1.1.1.1.1.3.2.3.cmml">gas</mi></msub><mrow id="S2.E1.m3.1.1.1.1.1.1.3.3" xref="S2.E1.m3.1.1.1.1.1.1.3.3.cmml"><msup id="S2.E1.m3.1.1.1.1.1.1.3.3.2" xref="S2.E1.m3.1.1.1.1.1.1.3.3.2.cmml"><mn id="S2.E1.m3.1.1.1.1.1.1.3.3.2.2" xref="S2.E1.m3.1.1.1.1.1.1.3.3.2.2.cmml">10</mn><mn id="S2.E1.m3.1.1.1.1.1.1.3.3.2.3" xref="S2.E1.m3.1.1.1.1.1.1.3.3.2.3.cmml">9</mn></msup><mo id="S2.E1.m3.1.1.1.1.1.1.3.3.1" xref="S2.E1.m3.1.1.1.1.1.1.3.3.1.cmml">⁢</mo><msub id="S2.E1.m3.1.1.1.1.1.1.3.3.3" xref="S2.E1.m3.1.1.1.1.1.1.3.3.3.cmml"><mi id="S2.E1.m3.1.1.1.1.1.1.3.3.3.2" xref="S2.E1.m3.1.1.1.1.1.1.3.3.3.2.cmml">M</mi><mo id="S2.E1.m3.1.1.1.1.1.1.3.3.3.3" xref="S2.E1.m3.1.1.1.1.1.1.3.3.3.3.cmml">⊙</mo></msub></mrow></mfrac></mstyle></mrow><mo id="S2.E1.m3.1.1.1.1.1.3" xref="S2.E1.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.E1.m3.1.1.1.3" xref="S2.E1.m3.1.1.1.3.cmml"><mn id="S2.E1.m3.1.1.1.3.2" xref="S2.E1.m3.1.1.1.3.2.cmml">0.86</mn><mo id="S2.E1.m3.1.1.1.3.1" xref="S2.E1.m3.1.1.1.3.1.cmml">±</mo><mn id="S2.E1.m3.1.1.1.3.3" xref="S2.E1.m3.1.1.1.3.3.cmml">0.06</mn></mrow></msup><mo id="S2.E1.m3.2.2.3" xref="S2.E1.m3.2.2.3.cmml">⁢</mo><mrow id="S2.E1.m3.2.2.2.1" xref="S2.E1.m3.2.2.2.2.cmml"><mo stretchy="false" id="S2.E1.m3.2.2.2.1.2" xref="S2.E1.m3.2.2.2.2.1.cmml">[</mo><mrow id="S2.E1.m3.2.2.2.1.1" xref="S2.E1.m3.2.2.2.1.1.cmml"><mrow id="S2.E1.m3.2.2.2.1.1.2" xref="S2.E1.m3.2.2.2.1.1.2.cmml"><msup id="S2.E1.m3.2.2.2.1.1.2.2" xref="S2.E1.m3.2.2.2.1.1.2.2.cmml"><mn id="S2.E1.m3.2.2.2.1.1.2.2.2" xref="S2.E1.m3.2.2.2.1.1.2.2.2.cmml">10</mn><mn id="S2.E1.m3.2.2.2.1.1.2.2.3" xref="S2.E1.m3.2.2.2.1.1.2.2.3.cmml">39</mn></msup><mo id="S2.E1.m3.2.2.2.1.1.2.1" xref="S2.E1.m3.2.2.2.1.1.2.1.cmml">⁢</mo><mi id="S2.E1.m3.2.2.2.1.1.2.3" xref="S2.E1.m3.2.2.2.1.1.2.3.cmml">erg</mi></mrow><mo id="S2.E1.m3.2.2.2.1.1.1" xref="S2.E1.m3.2.2.2.1.1.1.cmml">/</mo><mi mathvariant="normal" id="S2.E1.m3.2.2.2.1.1.3" xref="S2.E1.m3.2.2.2.1.1.3.cmml">s</mi></mrow><mo stretchy="false" id="S2.E1.m3.2.2.2.1.3" xref="S2.E1.m3.2.2.2.2.1.cmml">]</mo></mrow></mrow></math>, <math><mrow id="S2.E2.m3.2.2.1" xref="S2.E2.m3.2.2.1.1.cmml"><mrow id="S2.E2.m3.2.2.1.1" xref="S2.E2.m3.2.2.1.1.cmml"><msup id="S2.E2.m3.2.2.1.1.3" xref="S2.E2.m3.2.2.1.1.3.cmml"><mrow id="S2.E2.m3.2.2.1.1.3.2.2" xref="S2.E2.m3.1.1.cmml"><mo id="S2.E2.m3.2.2.1.1.3.2.2.1" xref="S2.E2.m3.1.1.cmml">(</mo><mstyle displaystyle="true" id="S2.E2.m3.1.1" xref="S2.E2.m3.1.1.cmml"><mfrac id="S2.E2.m3.1.1a" xref="S2.E2.m3.1.1.cmml"><mi id="S2.E2.m3.1.1.2" xref="S2.E2.m3.1.1.2.cmml">ψ</mi><mrow id="S2.E2.m3.1.1.3" xref="S2.E2.m3.1.1.3.cmml"><mpadded width="+2.8pt" id="S2.E2.m3.1.1.3.2" xref="S2.E2.m3.1.1.3.2.cmml"><msub id="S2.E2.m3.1.1.3.2a" xref="S2.E2.m3.1.1.3.2.cmml"><mi id="S2.E2.m3.1.1.3.2.2" xref="S2.E2.m3.1.1.3.2.2.cmml">M</mi><mo id="S2.E2.m3.1.1.3.2.3" xref="S2.E2.m3.1.1.3.2.3.cmml">⊙</mo></msub></mpadded><mo id="S2.E2.m3.1.1.3.1" xref="S2.E2.m3.1.1.3.1.cmml">⁢</mo><msup id="S2.E2.m3.1.1.3.3" xref="S2.E2.m3.1.1.3.3.cmml"><mi id="S2.E2.m3.1.1.3.3.2" xref="S2.E2.m3.1.1.3.3.2.cmml">yr</mi><mrow id="S2.E2.m3.1.1.3.3.3" xref="S2.E2.m3.1.1.3.3.3.cmml"><mo id="S2.E2.m3.1.1.3.3.3.1" xref="S2.E2.m3.1.1.3.3.3.1.cmml">-</mo><mn id="S2.E2.m3.1.1.3.3.3.2" xref="S2.E2.m3.1.1.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mfrac></mstyle><mo id="S2.E2.m3.2.2.1.1.3.2.2.2" xref="S2.E2.m3.1.1.cmml">)</mo></mrow><mrow id="S2.E2.m3.2.2.1.1.3.3" xref="S2.E2.m3.2.2.1.1.3.3.cmml"><mn id="S2.E2.m3.2.2.1.1.3.3.2" xref="S2.E2.m3.2.2.1.1.3.3.2.cmml">1.2</mn><mo id="S2.E2.m3.2.2.1.1.3.3.1" xref="S2.E2.m3.2.2.1.1.3.3.1.cmml">±</mo><mn id="S2.E2.m3.2.2.1.1.3.3.3" xref="S2.E2.m3.2.2.1.1.3.3.3.cmml">0.09</mn></mrow></msup><mo id="S2.E2.m3.2.2.1.1.2" xref="S2.E2.m3.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m3.2.2.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.2.cmml"><mo stretchy="false" id="S2.E2.m3.2.2.1.1.1.1.2" xref="S2.E2.m3.2.2.1.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m3.2.2.1.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.1.1.cmml"><mrow id="S2.E2.m3.2.2.1.1.1.1.1.2" xref="S2.E2.m3.2.2.1.1.1.1.1.2.cmml"><msup id="S2.E2.m3.2.2.1.1.1.1.1.2.2" xref="S2.E2.m3.2.2.1.1.1.1.1.2.2.cmml"><mn id="S2.E2.m3.2.2.1.1.1.1.1.2.2.2" xref="S2.E2.m3.2.2.1.1.1.1.1.2.2.2.cmml">10</mn><mn id="S2.E2.m3.2.2.1.1.1.1.1.2.2.3" xref="S2.E2.m3.2.2.1.1.1.1.1.2.2.3.cmml">39</mn></msup><mo id="S2.E2.m3.2.2.1.1.1.1.1.2.1" xref="S2.E2.m3.2.2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S2.E2.m3.2.2.1.1.1.1.1.2.3" xref="S2.E2.m3.2.2.1.1.1.1.1.2.3.cmml">erg</mi></mrow><mo id="S2.E2.m3.2.2.1.1.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.1.1.1.cmml">/</mo><mi mathvariant="normal" id="S2.E2.m3.2.2.1.1.1.1.1.3" xref="S2.E2.m3.2.2.1.1.1.1.1.3.cmml">s</mi></mrow><mo rspace="7.5pt" stretchy="false" id="S2.E2.m3.2.2.1.1.1.1.3" xref="S2.E2.m3.2.2.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.E2.m3.2.2.1.2" xref="S2.E2.m3.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS2.p5.10.m6.1.1" xref="S2.SS2.p5.10.m6.1.1.cmml"><msub id="S2.SS2.p5.10.m6.1.1.3" xref="S2.SS2.p5.10.m6.1.1.3.cmml"><mi id="S2.SS2.p5.10.m6.1.1.3.2" xref="S2.SS2.p5.10.m6.1.1.3.2.cmml">L</mi><mi id="S2.SS2.p5.10.m6.1.1.3.3" xref="S2.SS2.p5.10.m6.1.1.3.3.cmml">γ</mi></msub><mo id="S2.SS2.p5.10.m6.1.1.2" xref="S2.SS2.p5.10.m6.1.1.2.cmml">∝</mo><msup id="S2.SS2.p5.10.m6.1.1.1" xref="S2.SS2.p5.10.m6.1.1.1.cmml"><mrow id="S2.SS2.p5.10.m6.1.1.1.1.1" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS2.p5.10.m6.1.1.1.1.1.2" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p5.10.m6.1.1.1.1.1.1" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p5.10.m6.1.1.1.1.1.1.2" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.2.cmml">ψ</mi><mo id="S2.SS2.p5.10.m6.1.1.1.1.1.1.1" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2.SS2.p5.10.m6.1.1.1.1.1.1.3" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p5.10.m6.1.1.1.1.1.1.3.2" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.3.2.cmml">M</mi><mi id="S2.SS2.p5.10.m6.1.1.1.1.1.1.3.3" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.3.3.cmml">gas</mi></msub></mrow><mo stretchy="false" id="S2.SS2.p5.10.m6.1.1.1.1.1.3" xref="S2.SS2.p5.10.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.SS2.p5.10.m6.1.1.1.3" xref="S2.SS2.p5.10.m6.1.1.1.3.cmml">1</mn></msup></mrow></math>, <math><mrow id="S2.SS2.p5.11.m7.1.2" xref="S2.SS2.p5.11.m7.1.2.cmml"><msub id="S2.SS2.p5.11.m7.1.2.2" xref="S2.SS2.p5.11.m7.1.2.2.cmml"><mi id="S2.SS2.p5.11.m7.1.2.2.2" xref="S2.SS2.p5.11.m7.1.2.2.2.cmml">L</mi><mi id="S2.SS2.p5.11.m7.1.2.2.3" xref="S2.SS2.p5.11.m7.1.2.2.3.cmml">γ</mi></msub><mo id="S2.SS2.p5.11.m7.1.2.1" xref="S2.SS2.p5.11.m7.1.2.1.cmml">∝</mo><msup id="S2.SS2.p5.11.m7.1.2.3" xref="S2.SS2.p5.11.m7.1.2.3.cmml"><mrow id="S2.SS2.p5.11.m7.1.2.3.2.2" xref="S2.SS2.p5.11.m7.1.2.3.cmml"><mo stretchy="false" id="S2.SS2.p5.11.m7.1.2.3.2.2.1" xref="S2.SS2.p5.11.m7.1.2.3.cmml">(</mo><mi id="S2.SS2.p5.11.m7.1.1" xref="S2.SS2.p5.11.m7.1.1.cmml">ψ</mi><mo stretchy="false" id="S2.SS2.p5.11.m7.1.2.3.2.2.2" xref="S2.SS2.p5.11.m7.1.2.3.cmml">)</mo></mrow><mn id="S2.SS2.p5.11.m7.1.2.3.3" xref="S2.SS2.p5.11.m7.1.2.3.3.cmml">1</mn></msup></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1202.6569

Formulas:

1-

R e = ρ D U / μ

2-

W e = ρ D U^{2} / σ

3-

K = W e \sqrt{R e}

4-

R e \approx 2000 - 6000

5-

U = 4.04 m / s, R e = 3.55 \times 10^{3}, K = 7.44 \times 10^{4}

6-

K = 7.44 \times 10^{4}

7-

t^{*} = t / τ

8-

r_{J} = \sqrt{t^{*} (1 - t^{*})}

9-

R e = 1, 000

10-

R e = 4, 000

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1210.7802

Formulas:

1-

H_{0} = 71 km / s / Mpc, Ω_{m} = 0.27, Ω_{Λ} = 0.73

2-

\sim 50 \frac{Γ}{1 + z} MeV

3-

p^{+} + γ \to Δ^{+}

4-

Δ^{+} \to π^{+} + n

5-

Δ^{+} \to π^{0} + p^{+}

6-

π^{0} \to γ + γ

7-

ν_{p} (1 + z) / Γ

8-

γ_{p} ≳ 2 \times 10^{4} Γ_{2} ν_{p, 6}^{- 1} {(1 + z)}^{- 1},

9-

X_{n} \equiv \frac{X}{10^{n}}

10-

\sim 100 Γ_{2}^{2} ν_{p, 6}^{- 1} {(1 + z)}^{- 2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0708.2055

Formulas:

1-

L_{i j} = - w_{j i}

2-

L_{i i} = \sum_{j \neq i} L_{i j}

3-

\sum_{j} L_{i j} = 0

4-

0 = λ^{1} < λ^{2} \leq \dots \leq λ^{N}

5-

{\dot{𝐱}}_{i} = F (𝐱_{i}) + σ \sum_{j = 1}^{N} L_{i j} H (𝐱_{j}),

6-

𝐱_{i} (t) = 𝐬 (t)

7-

{\dot{ξ}}_{i} = D F (𝐬) ξ_{i} + σ \sum_{j = 1}^{N} L_{i j} D H (𝐬) ξ_{i}

8-

{\dot{ζ}}^{α} = D F (𝐬) ζ^{α} + σ λ^{α} D H (𝐬) ζ^{α}, α = 1, \dots, N .

9-

α = 2, \dots, N

10-

l_{1} \leq σ λ^{α} \leq l_{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0303035

Formulas:

1-

U^{'} = \exp (- i H^{'} t)

2-

\exp (- i H^{'} t)

3-

H = c_{11} x_{A} x_{B} + c_{12} x_{A} p_{B} + c_{21} p_{A} x_{B} + c_{22} p_{A} p_{B},

4-

H_{A} = x_{A}^{2} + p_{A}^{2}

5-

H_{B} = x_{B}^{2} + p_{B}^{2}

6-

U = \exp (- i t H_{A L}),

7-

H_{A L} = κ x_{A} p_{B}

8-

G = V_{N}^{†} e^{- i H t_{N}} V_{N} \dots V_{2}^{†} e^{- i H t_{2}} V_{2} V_{1}^{†} e^{- i H t_{1}} V_{1} .

9-

V_{j} = e^{- i ϕ_{j A} a^{†} a} \otimes e^{- i ϕ_{j B} b^{†} b},

10-

V_{j + 1}^{†} V_{j}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1511.04451

Formulas:

1-

θ_{LD} = 42.3 \pm 0.4

2-

λ / D \approx 14 - 19

3-

θ_{LD} = 2.20 \pm 0.02

4-

0.45 ″ \times 0.45 ″

5-

m_{V} = 0.17 \pm 0.05

6-

m_{R} = - 1.07 \pm 0.05

7-

θ_{50 %} (V) = 40.1

8-

θ_{50 %} (CntH α) = 37.1

9-

θ_{50 %} (NH α) = 39.3

10-

θ_{50 %} (TiO717) = 42.6

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1207.0537

Formulas:

1-

E_{pk} - E_{γ, iso} - E_{X, iso}

2-

E_{X, iso} \sim 6 \times 10^{51} erg

3-

E_{X, iso} \propto E_{γ, iso}^{0.8}

4-

E_{X, iso} \propto E_{γ, iso}^{0.7}

5-

E_{X, iso}^{s h o r t} < 10^{51} erg

6-

N H_{HG}^{s h o r t} = 10^{21.4} {cm}^{- 2}

7-

N H_{HG} = 10^{21.8} {cm}^{- 2}

8-

p (K S) = 34 %

9-

N H_{HG} = 10^{21.8} {cm}^{- 2}

10-

E_{pk} - E_{γ, iso} - E_{X, iso}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1506.08631

Formulas:

1-

G = (V, E)

2-

p_{u, v} (t)

3-

r_{u, v} (t) = \frac{p_{u, v} (t)}{p_{u, u} (t)} t \geq 0

4-

G = (V, E)

5-

p_{u, v} (t)

6-

r_{u, v} (t) = \frac{p_{u, v} (t)}{p_{u, u} (t)} t \geq 0 .

7-

r_{u, v} (0) = 0

8-

{lim}_{t \to \infty} r_{u, v} (t) = 1

9-

r_{u, v} (t) > 1

10-

r_{u, v} (t)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0805.0433

Formulas:

1-

f (\frac{a + b}{2}) ≦ \frac{1}{b - a} \int_{a}^{b} f (x) d x ≦ \frac{f (a) + f (b)}{2} .

2-

\frac{f (a) + f (b)}{2} - \frac{1}{8} S ≦ \frac{1}{b - a} \int_{a}^{b} f (x) d x ≦ f (\frac{a + b}{2}) + \frac{1}{8} S,

3-

S = (f^{'} (b) - f^{'} (a)) (b - a)

4-

a, b \in I, a < b .

5-

m = inf_{x \in [a, b]} f^{''} (x)

6-

M = sup_{x \in [a, b]} f^{''} (x)

7-

f (\frac{a + b}{2}) + \frac{m}{24} {(b - a)}^{2} ≦ \frac{1}{b - a} \int_{a}^{b} f (x) d x ≦ \frac{f (a) + f (b)}{2} - \frac{m}{12} {(b - a)}^{2}

8-

\frac{f (a) + f (b)}{2} - \frac{M}{12} {(b - a)}^{2} ≦ \frac{1}{b - a} \int_{a}^{b} f (x) d x ≦ f (\frac{a + b}{2}) + \frac{M}{24} {(b - a)}^{2} .

9-

f^{''} ≧ 0, t \in [a, b]

10-

f^{''} ≦ 0, t \in [a, b]

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1708.07543

Formulas:

1-

Q_{p}^{'} \approx 10^{6}

2-

a_{per, 0} = (1 - e_{0}) a_{0}

3-

a_{cir} = (1 + e_{0}) a_{per, 0}

4-

≃ 2 a_{per, 0}

5-

a_{per, 0} = a_{R}

6-

a_{cir} \approx 2 a_{per, 0}

7-

a ≲ 2 a_{R}

8-

\partial f / \partial \log P_{orb, 0} \propto P_{orb, 0}^{0.47}

9-

\partial f / \partial e_{0} = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

10-

\partial f / \partial \log R_{p} \propto R_{p}^{- 0.66}

Formulas (html):

<math><mrow id="id4.3.m3.1.1" xref="id4.3.m3.1.1.cmml"><msubsup id="id4.3.m3.1.1.2" xref="id4.3.m3.1.1.2.cmml"><mi id="id4.3.m3.1.1.2.2.2" xref="id4.3.m3.1.1.2.2.2.cmml">Q</mi><mi mathvariant="normal" id="id4.3.m3.1.1.2.3" xref="id4.3.m3.1.1.2.3.cmml">p</mi><mo id="id4.3.m3.1.1.2.2.3" xref="id4.3.m3.1.1.2.2.3.cmml">′</mo></msubsup><mo id="id4.3.m3.1.1.1" xref="id4.3.m3.1.1.1.cmml">≈</mo><msup id="id4.3.m3.1.1.3" xref="id4.3.m3.1.1.3.cmml"><mn id="id4.3.m3.1.1.3.2" xref="id4.3.m3.1.1.3.2.cmml">10</mn><mn id="id4.3.m3.1.1.3.3" xref="id4.3.m3.1.1.3.3.cmml">6</mn></msup></mrow></math>, <math><mrow id="S1.p2.8.m8.3.3" xref="S1.p2.8.m8.3.3.cmml"><msub id="S1.p2.8.m8.3.3.3" xref="S1.p2.8.m8.3.3.3.cmml"><mi id="S1.p2.8.m8.3.3.3.2" xref="S1.p2.8.m8.3.3.3.2.cmml">a</mi><mrow id="S1.p2.8.m8.2.2.2.4" xref="S1.p2.8.m8.2.2.2.3.cmml"><mi id="S1.p2.8.m8.1.1.1.1" xref="S1.p2.8.m8.1.1.1.1.cmml">per</mi><mo id="S1.p2.8.m8.2.2.2.4.1" xref="S1.p2.8.m8.2.2.2.3.cmml">,</mo><mn id="S1.p2.8.m8.2.2.2.2" 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id="S1.p2.8.m8.3.3.1.3" xref="S1.p2.8.m8.3.3.1.3.cmml"><mi id="S1.p2.8.m8.3.3.1.3.2" xref="S1.p2.8.m8.3.3.1.3.2.cmml">a</mi><mn id="S1.p2.8.m8.3.3.1.3.3" xref="S1.p2.8.m8.3.3.1.3.3.cmml">0</mn></msub></mrow></mrow></math>, <math><mrow id="S1.p2.9.m9.3.3" xref="S1.p2.9.m9.3.3.cmml"><msub id="S1.p2.9.m9.3.3.3" xref="S1.p2.9.m9.3.3.3.cmml"><mi id="S1.p2.9.m9.3.3.3.2" xref="S1.p2.9.m9.3.3.3.2.cmml">a</mi><mi id="S1.p2.9.m9.3.3.3.3" xref="S1.p2.9.m9.3.3.3.3.cmml">cir</mi></msub><mo id="S1.p2.9.m9.3.3.2" xref="S1.p2.9.m9.3.3.2.cmml">=</mo><mrow id="S1.p2.9.m9.3.3.1" xref="S1.p2.9.m9.3.3.1.cmml"><mrow id="S1.p2.9.m9.3.3.1.1.1" xref="S1.p2.9.m9.3.3.1.1.1.1.cmml"><mo stretchy="false" id="S1.p2.9.m9.3.3.1.1.1.2" xref="S1.p2.9.m9.3.3.1.1.1.1.cmml">(</mo><mrow id="S1.p2.9.m9.3.3.1.1.1.1" xref="S1.p2.9.m9.3.3.1.1.1.1.cmml"><mn id="S1.p2.9.m9.3.3.1.1.1.1.2" xref="S1.p2.9.m9.3.3.1.1.1.1.2.cmml">1</mn><mo id="S1.p2.9.m9.3.3.1.1.1.1.1" xref="S1.p2.9.m9.3.3.1.1.1.1.1.cmml">+</mo><msub id="S1.p2.9.m9.3.3.1.1.1.1.3" xref="S1.p2.9.m9.3.3.1.1.1.1.3.cmml"><mi id="S1.p2.9.m9.3.3.1.1.1.1.3.2" xref="S1.p2.9.m9.3.3.1.1.1.1.3.2.cmml">e</mi><mn id="S1.p2.9.m9.3.3.1.1.1.1.3.3" xref="S1.p2.9.m9.3.3.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo rspace="4.2pt" stretchy="false" id="S1.p2.9.m9.3.3.1.1.1.3" xref="S1.p2.9.m9.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p2.9.m9.3.3.1.2" xref="S1.p2.9.m9.3.3.1.2.cmml">⁢</mo><msub id="S1.p2.9.m9.3.3.1.3" xref="S1.p2.9.m9.3.3.1.3.cmml"><mi id="S1.p2.9.m9.3.3.1.3.2" xref="S1.p2.9.m9.3.3.1.3.2.cmml">a</mi><mrow id="S1.p2.9.m9.2.2.2.4" xref="S1.p2.9.m9.2.2.2.3.cmml"><mi id="S1.p2.9.m9.1.1.1.1" xref="S1.p2.9.m9.1.1.1.1.cmml">per</mi><mo id="S1.p2.9.m9.2.2.2.4.1" xref="S1.p2.9.m9.2.2.2.3.cmml">,</mo><mn id="S1.p2.9.m9.2.2.2.2" xref="S1.p2.9.m9.2.2.2.2.cmml">0</mn></mrow></msub></mrow></mrow></math>, <math><mrow id="S1.p2.10.m10.2.3" xref="S1.p2.10.m10.2.3.cmml"><mi id="S1.p2.10.m10.2.3.2" xref="S1.p2.10.m10.2.3.2.cmml"/><mo id="S1.p2.10.m10.2.3.1" xref="S1.p2.10.m10.2.3.1.cmml">≃</mo><mrow id="S1.p2.10.m10.2.3.3" xref="S1.p2.10.m10.2.3.3.cmml"><mpadded width="+1.7pt" id="S1.p2.10.m10.2.3.3.2" xref="S1.p2.10.m10.2.3.3.2.cmml"><mn id="S1.p2.10.m10.2.3.3.2a" xref="S1.p2.10.m10.2.3.3.2.cmml">2</mn></mpadded><mo id="S1.p2.10.m10.2.3.3.1" xref="S1.p2.10.m10.2.3.3.1.cmml">⁢</mo><msub id="S1.p2.10.m10.2.3.3.3" xref="S1.p2.10.m10.2.3.3.3.cmml"><mi id="S1.p2.10.m10.2.3.3.3.2" xref="S1.p2.10.m10.2.3.3.3.2.cmml">a</mi><mrow id="S1.p2.10.m10.2.2.2.4" xref="S1.p2.10.m10.2.2.2.3.cmml"><mi id="S1.p2.10.m10.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.cmml">per</mi><mo id="S1.p2.10.m10.2.2.2.4.1" xref="S1.p2.10.m10.2.2.2.3.cmml">,</mo><mn id="S1.p2.10.m10.2.2.2.2" xref="S1.p2.10.m10.2.2.2.2.cmml">0</mn></mrow></msub></mrow></mrow></math>, <math><mrow id="S1.p2.14.m14.2.3" xref="S1.p2.14.m14.2.3.cmml"><msub id="S1.p2.14.m14.2.3.2" xref="S1.p2.14.m14.2.3.2.cmml"><mi id="S1.p2.14.m14.2.3.2.2" xref="S1.p2.14.m14.2.3.2.2.cmml">a</mi><mrow 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width="+1.7pt" id="S1.p2.15.m15.2.3.3.2" xref="S1.p2.15.m15.2.3.3.2.cmml"><mn id="S1.p2.15.m15.2.3.3.2a" xref="S1.p2.15.m15.2.3.3.2.cmml">2</mn></mpadded><mo id="S1.p2.15.m15.2.3.3.1" xref="S1.p2.15.m15.2.3.3.1.cmml">⁢</mo><msub id="S1.p2.15.m15.2.3.3.3" xref="S1.p2.15.m15.2.3.3.3.cmml"><mi id="S1.p2.15.m15.2.3.3.3.2" xref="S1.p2.15.m15.2.3.3.3.2.cmml">a</mi><mrow id="S1.p2.15.m15.2.2.2.4" xref="S1.p2.15.m15.2.2.2.3.cmml"><mi id="S1.p2.15.m15.1.1.1.1" xref="S1.p2.15.m15.1.1.1.1.cmml">per</mi><mo id="S1.p2.15.m15.2.2.2.4.1" xref="S1.p2.15.m15.2.2.2.3.cmml">,</mo><mn id="S1.p2.15.m15.2.2.2.2" xref="S1.p2.15.m15.2.2.2.2.cmml">0</mn></mrow></msub></mrow></mrow></math>, <math><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">a</mi><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">≲</mo><mrow id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mpadded width="+1.7pt" id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml"><mn id="S1.p3.1.m1.1.1.3.2a" xref="S1.p3.1.m1.1.1.3.2.cmml">2</mn></mpadded><mo id="S1.p3.1.m1.1.1.3.1" xref="S1.p3.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml"><mi id="S1.p3.1.m1.1.1.3.3.2" xref="S1.p3.1.m1.1.1.3.3.2.cmml">a</mi><mi mathvariant="normal" id="S1.p3.1.m1.1.1.3.3.3" xref="S1.p3.1.m1.1.1.3.3.3.cmml">R</mi></msub></mrow></mrow></math>, <math><mrow id="S2.p2.10.m10.4.5" xref="S2.p2.10.m10.4.5.cmml"><mrow id="S2.p2.10.m10.4.5.2" xref="S2.p2.10.m10.4.5.2.cmml"><mrow id="S2.p2.10.m10.4.5.2.2" xref="S2.p2.10.m10.4.5.2.2.cmml"><mo id="S2.p2.10.m10.4.5.2.2.1" xref="S2.p2.10.m10.4.5.2.2.1.cmml">∂</mo><mo id="S2.p2.10.m10.4.5.2.2a" xref="S2.p2.10.m10.4.5.2.2.cmml">⁡</mo><mi id="S2.p2.10.m10.4.5.2.2.2" xref="S2.p2.10.m10.4.5.2.2.2.cmml">f</mi></mrow><mo id="S2.p2.10.m10.4.5.2.1" xref="S2.p2.10.m10.4.5.2.1.cmml">/</mo><mrow id="S2.p2.10.m10.4.5.2.3" xref="S2.p2.10.m10.4.5.2.3.cmml"><mrow id="S2.p2.10.m10.4.5.2.3.1" xref="S2.p2.10.m10.4.5.2.3.1.cmml"><mo id="S2.p2.10.m10.4.5.2.3.1.1" xref="S2.p2.10.m10.4.5.2.3.1.1.cmml">∂</mo><mo id="S2.p2.10.m10.4.5.2.3.1a" xref="S2.p2.10.m10.4.5.2.3.1.cmml">⁡</mo><mi id="S2.p2.10.m10.4.5.2.3.1.2" xref="S2.p2.10.m10.4.5.2.3.1.2.cmml">log</mi></mrow><mo id="S2.p2.10.m10.4.5.2.3a" xref="S2.p2.10.m10.4.5.2.3.cmml">⁡</mo><msub id="S2.p2.10.m10.4.5.2.3.2" xref="S2.p2.10.m10.4.5.2.3.2.cmml"><mi id="S2.p2.10.m10.4.5.2.3.2.2" xref="S2.p2.10.m10.4.5.2.3.2.2.cmml">P</mi><mrow id="S2.p2.10.m10.2.2.2.4" xref="S2.p2.10.m10.2.2.2.3.cmml"><mi id="S2.p2.10.m10.1.1.1.1" xref="S2.p2.10.m10.1.1.1.1.cmml">orb</mi><mo id="S2.p2.10.m10.2.2.2.4.1" xref="S2.p2.10.m10.2.2.2.3.cmml">,</mo><mn id="S2.p2.10.m10.2.2.2.2" xref="S2.p2.10.m10.2.2.2.2.cmml">0</mn></mrow></msub></mrow></mrow><mo id="S2.p2.10.m10.4.5.1" xref="S2.p2.10.m10.4.5.1.cmml">∝</mo><msubsup id="S2.p2.10.m10.4.5.3" xref="S2.p2.10.m10.4.5.3.cmml"><mi id="S2.p2.10.m10.4.5.3.2.2" xref="S2.p2.10.m10.4.5.3.2.2.cmml">P</mi><mrow id="S2.p2.10.m10.4.4.2.4" xref="S2.p2.10.m10.4.4.2.3.cmml"><mi id="S2.p2.10.m10.3.3.1.1" xref="S2.p2.10.m10.3.3.1.1.cmml">orb</mi><mo id="S2.p2.10.m10.4.4.2.4.1" xref="S2.p2.10.m10.4.4.2.3.cmml">,</mo><mn id="S2.p2.10.m10.4.4.2.2" xref="S2.p2.10.m10.4.4.2.2.cmml">0</mn></mrow><mn id="S2.p2.10.m10.4.5.3.3" xref="S2.p2.10.m10.4.5.3.3.cmml">0.47</mn></msubsup></mrow></math>, <math><mrow id="S2.p2.12.m12.1.1" xref="S2.p2.12.m12.1.1.cmml"><mrow id="S2.p2.12.m12.1.1.2" xref="S2.p2.12.m12.1.1.2.cmml"><mrow id="S2.p2.12.m12.1.1.2.2" xref="S2.p2.12.m12.1.1.2.2.cmml"><mo id="S2.p2.12.m12.1.1.2.2.1" xref="S2.p2.12.m12.1.1.2.2.1.cmml">∂</mo><mo id="S2.p2.12.m12.1.1.2.2a" xref="S2.p2.12.m12.1.1.2.2.cmml">⁡</mo><mi id="S2.p2.12.m12.1.1.2.2.2" xref="S2.p2.12.m12.1.1.2.2.2.cmml">f</mi></mrow><mo id="S2.p2.12.m12.1.1.2.1" xref="S2.p2.12.m12.1.1.2.1.cmml">/</mo><mrow id="S2.p2.12.m12.1.1.2.3" xref="S2.p2.12.m12.1.1.2.3.cmml"><mo id="S2.p2.12.m12.1.1.2.3.1" xref="S2.p2.12.m12.1.1.2.3.1.cmml">∂</mo><mo id="S2.p2.12.m12.1.1.2.3a" xref="S2.p2.12.m12.1.1.2.3.cmml">⁡</mo><msub id="S2.p2.12.m12.1.1.2.3.2" xref="S2.p2.12.m12.1.1.2.3.2.cmml"><mi id="S2.p2.12.m12.1.1.2.3.2.2" xref="S2.p2.12.m12.1.1.2.3.2.2.cmml">e</mi><mn id="S2.p2.12.m12.1.1.2.3.2.3" xref="S2.p2.12.m12.1.1.2.3.2.3.cmml">0</mn></msub></mrow></mrow><mo id="S2.p2.12.m12.1.1.1" xref="S2.p2.12.m12.1.1.1.cmml">=</mo><mtext id="S2.p2.12.m12.1.1.3" xref="S2.p2.12.m12.1.1.3a.cmml">𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡</mtext></mrow></math>, <math><mrow id="S2.p2.14.m14.1.1" xref="S2.p2.14.m14.1.1.cmml"><mrow id="S2.p2.14.m14.1.1.2" xref="S2.p2.14.m14.1.1.2.cmml"><mrow id="S2.p2.14.m14.1.1.2.2" xref="S2.p2.14.m14.1.1.2.2.cmml"><mo id="S2.p2.14.m14.1.1.2.2.1" xref="S2.p2.14.m14.1.1.2.2.1.cmml">∂</mo><mo id="S2.p2.14.m14.1.1.2.2a" xref="S2.p2.14.m14.1.1.2.2.cmml">⁡</mo><mi id="S2.p2.14.m14.1.1.2.2.2" xref="S2.p2.14.m14.1.1.2.2.2.cmml">f</mi></mrow><mo id="S2.p2.14.m14.1.1.2.1" xref="S2.p2.14.m14.1.1.2.1.cmml">/</mo><mrow id="S2.p2.14.m14.1.1.2.3" xref="S2.p2.14.m14.1.1.2.3.cmml"><mrow id="S2.p2.14.m14.1.1.2.3.1" xref="S2.p2.14.m14.1.1.2.3.1.cmml"><mo id="S2.p2.14.m14.1.1.2.3.1.1" xref="S2.p2.14.m14.1.1.2.3.1.1.cmml">∂</mo><mo id="S2.p2.14.m14.1.1.2.3.1a" xref="S2.p2.14.m14.1.1.2.3.1.cmml">⁡</mo><mi id="S2.p2.14.m14.1.1.2.3.1.2" xref="S2.p2.14.m14.1.1.2.3.1.2.cmml">log</mi></mrow><mo id="S2.p2.14.m14.1.1.2.3a" xref="S2.p2.14.m14.1.1.2.3.cmml">⁡</mo><msub id="S2.p2.14.m14.1.1.2.3.2" xref="S2.p2.14.m14.1.1.2.3.2.cmml"><mi id="S2.p2.14.m14.1.1.2.3.2.2" xref="S2.p2.14.m14.1.1.2.3.2.2.cmml">R</mi><mi mathvariant="normal" id="S2.p2.14.m14.1.1.2.3.2.3" xref="S2.p2.14.m14.1.1.2.3.2.3.cmml">p</mi></msub></mrow></mrow><mo id="S2.p2.14.m14.1.1.1" xref="S2.p2.14.m14.1.1.1.cmml">∝</mo><msubsup id="S2.p2.14.m14.1.1.3" xref="S2.p2.14.m14.1.1.3.cmml"><mi id="S2.p2.14.m14.1.1.3.2.2" xref="S2.p2.14.m14.1.1.3.2.2.cmml">R</mi><mi mathvariant="normal" id="S2.p2.14.m14.1.1.3.2.3" xref="S2.p2.14.m14.1.1.3.2.3.cmml">p</mi><mrow id="S2.p2.14.m14.1.1.3.3" xref="S2.p2.14.m14.1.1.3.3.cmml"><mo id="S2.p2.14.m14.1.1.3.3.1" xref="S2.p2.14.m14.1.1.3.3.1.cmml">-</mo><mn id="S2.p2.14.m14.1.1.3.3.2" xref="S2.p2.14.m14.1.1.3.3.2.cmml">0.66</mn></mrow></msubsup></mrow></math>

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: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1109.6721

Formulas:

1-

\sqrt{s_{N N}} = 2.76

2-

\frac{d N}{d ϕ} \propto 1 + 2 \sum_{n = 1}^{\infty} v_{n}^{a} \cos (n (ϕ - Ψ_{n}))

3-

Δ ϕ = ϕ_{t r i g g e r} - ϕ_{p a r t n e r}

4-

Δ η = η_{t r i g g e r} - η_{p a r t n e r}

5-

C (Δ ϕ) \propto 1 + 2 \sum_{n = 1}^{\infty} v_{n, n} (p_{T}^{a}, p_{T}^{b}) \cos (n Δ ϕ)

6-

v_{n, n} (p_{T}^{a}, p_{T}^{b}) = v_{n} (p_{T}^{a}) \times v_{n} (p_{T}^{b})

7-

| η | \in (3.3, 4.8)

8-

v_{n}^{o b s} = ⟨ \cos n (ϕ - Ψ_{n}) ⟩

9-

v_{n}^{o b s}

10-

v_{n} = v_{n}^{o b s} / R e s o l u t i o n (Ψ_{n})

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/nlin/0303036

Formulas:

1-

a_{\pm}, b_{\pm}, c

2-

T = Γ r_{2} [ℒ_{L} \dots ℒ_{1}]

3-

ℒ_{j} = (\begin{matrix} a_{+} σ_{j}^{+} σ_{j}^{-} + b_{+} σ_{j}^{-} σ_{j}^{+} & d σ_{j}^{+} + c σ_{j}^{-} \\ c σ_{j}^{+} + d σ_{j}^{-} & b_{-} σ_{j}^{+} σ_{j}^{-} + a_{-} σ_{j}^{-} σ_{j}^{+} \end{matrix})

4-

F (a_{\pm}, b_{\pm}, c, d) = 0, \frac{c d}{a_{+} b_{-} + a_{-} b_{+}} = I_{1}^{F}, \frac{a_{+}^{2} + b_{-}^{2} - a_{-}^{2} - b_{+}^{2}}{a_{+} b_{-} + a_{-} b_{+}} = I_{2}^{F}

5-

a_{\pm} = b_{\mp}, c = d o r a_{\pm} = b_{\pm}, c = d

6-

a_{+} = a_{-}, b_{+} = b_{-}, \frac{c d}{a_{+} b_{+}} = I_{1}^{B}, \frac{F (a_{\pm}, b_{\pm}, c, d)}{a_{+} b_{+}} = I_{2}^{B}

7-

F (a_{\pm}, b_{\pm}, c, d) = a_{+} a_{-} + b_{+} b_{-} - c^{2} - d^{2}

8-

I_{1, 2}^{F, B}

9-

a_{\pm} a_{\pm}^{^{'}} d^{"} + d c^{^{'}} a_{\mp}^{"} = c d^{^{'}} a_{\pm}^{"} + b_{\mp} b_{\mp}^{^{'}} d^{"}

10-

d b_{\pm}^{^{'}} c^{"} + a_{\pm} d^{^{'}} b_{\mp}^{"} = b_{\pm} d^{^{'}} a_{\pm}^{"} + c b_{\mp}^{^{'}} d^{"}

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/0708.2374

Formulas:

1-

λ^{- 4} \propto ⟨ Δ B^{2} ⟩ \propto σ_{s c}^{2}

2-

σ_{s c} \propto λ^{- 2}

3-

σ_{s c}^{2} = σ^{2} - σ_{n m}^{2}

4-

σ_{a b} \propto λ_{a b}^{- 2}

5-

σ_{s c} \propto λ^{- 2}

6-

i, j, k = a, b, c

7-

λ_{j k}^{- 2} = {(λ_{j} λ_{k})}^{- 1} \propto σ_{j k} = \sqrt{σ_{j} σ_{k}}

8-

σ_{a b} \propto λ_{a b}^{- 2} = {(λ_{a} λ_{b})}^{- 1}

9-

σ_{a b} (T)

10-

σ (T) = σ^{s} (T) + σ^{d} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/nlin/0601040

Formulas:

1-

ω = [0, 1, 11, 1, 1, \dots]

2-

x_{n} + a (1 - y_{n + 1}^{2})

3-

y_{n} - b \sin (2 π x_{n}),

4-

(x, y) \in 𝕋 \times ℝ

5-

\partial x_{n + 1} (x_{n}, y_{n}) / \partial y_{n} \neq 0

6-

S (x, y) = (x + 1 / 2, - y)

7-

ω = [0, 1, 11, 1, 1, \dots]

8-

ω_{i} = x_{i} / i

9-

| ω_{i} - ω_{i - 1} | < 10^{- 7}

10-

ω_{i} < \max_{n < i} ω_{n}

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0510413

Formulas:

1-

m_{F U V} \sim 25

2-

(F U V - V)

3-

F U V - V

4-

E (B - V) ≲ 0.02

5-

r < 1 \overset{'}{.} 5

6-

(m - M) = 31.12

7-

m_{S T M A G} = - 2.5 \log_{10} (\frac{counts \times PHOTFLAM}{EXPTIME}) - 21.10,

8-

E (B - V) = 0.022

9-

A_{F U V} = 0.183

10-

A_{N U V} = 0.178

Formulas (html):

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xref="id5.4.m3.1.1.1.1.2.2.cmml">F</mi><mo id="id5.4.m3.1.1.1.1.2.1" xref="id5.4.m3.1.1.1.1.2.1.cmml">⁢</mo><mi id="id5.4.m3.1.1.1.1.2.3" xref="id5.4.m3.1.1.1.1.2.3.cmml">U</mi><mo id="id5.4.m3.1.1.1.1.2.1a" xref="id5.4.m3.1.1.1.1.2.1.cmml">⁢</mo><mi id="id5.4.m3.1.1.1.1.2.4" xref="id5.4.m3.1.1.1.1.2.4.cmml">V</mi></mrow><mo id="id5.4.m3.1.1.1.1.1" xref="id5.4.m3.1.1.1.1.1.cmml">-</mo><mi id="id5.4.m3.1.1.1.1.3" xref="id5.4.m3.1.1.1.1.3.cmml">V</mi></mrow><mo stretchy="false" id="id5.4.m3.1.1.1.3" xref="id5.4.m3.1.1.1.1.cmml">)</mo></mrow></math>, <math><mrow id="S1.SS1.p6.2.m2.1.1" xref="S1.SS1.p6.2.m2.1.1.cmml"><mrow id="S1.SS1.p6.2.m2.1.1.2" xref="S1.SS1.p6.2.m2.1.1.2.cmml"><mi id="S1.SS1.p6.2.m2.1.1.2.2" xref="S1.SS1.p6.2.m2.1.1.2.2.cmml">F</mi><mo id="S1.SS1.p6.2.m2.1.1.2.1" xref="S1.SS1.p6.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S1.SS1.p6.2.m2.1.1.2.3" xref="S1.SS1.p6.2.m2.1.1.2.3.cmml">U</mi><mo id="S1.SS1.p6.2.m2.1.1.2.1a" xref="S1.SS1.p6.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S1.SS1.p6.2.m2.1.1.2.4" xref="S1.SS1.p6.2.m2.1.1.2.4.cmml">V</mi></mrow><mo id="S1.SS1.p6.2.m2.1.1.1" xref="S1.SS1.p6.2.m2.1.1.1.cmml">-</mo><mi id="S1.SS1.p6.2.m2.1.1.3" xref="S1.SS1.p6.2.m2.1.1.3.cmml">V</mi></mrow></math>, <math><mrow id="S1.SS2.p1.2.m2.1.1" xref="S1.SS2.p1.2.m2.1.1.cmml"><mrow id="S1.SS2.p1.2.m2.1.1.1" xref="S1.SS2.p1.2.m2.1.1.1.cmml"><mi id="S1.SS2.p1.2.m2.1.1.1.3" xref="S1.SS2.p1.2.m2.1.1.1.3.cmml">E</mi><mo id="S1.SS2.p1.2.m2.1.1.1.2" xref="S1.SS2.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.p1.2.m2.1.1.1.1.1" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.SS2.p1.2.m2.1.1.1.1.1.2" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.p1.2.m2.1.1.1.1.1.1" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S1.SS2.p1.2.m2.1.1.1.1.1.1.2" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.2.cmml">B</mi><mo id="S1.SS2.p1.2.m2.1.1.1.1.1.1.1" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.1.cmml">-</mo><mi id="S1.SS2.p1.2.m2.1.1.1.1.1.1.3" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.3.cmml">V</mi></mrow><mo stretchy="false" id="S1.SS2.p1.2.m2.1.1.1.1.1.3" xref="S1.SS2.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS2.p1.2.m2.1.1.2" xref="S1.SS2.p1.2.m2.1.1.2.cmml">≲</mo><mn id="S1.SS2.p1.2.m2.1.1.3" xref="S1.SS2.p1.2.m2.1.1.3.cmml">0.02</mn></mrow></math>, <math><mrow id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">r</mi><mo id="S2.SS1.p1.1.m1.1.1.1" xref="S2.SS1.p1.1.m1.1.1.1.cmml"><</mo><mrow id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml"><mn id="S2.SS1.p1.1.m1.1.1.3.2" xref="S2.SS1.p1.1.m1.1.1.3.2.cmml">1</mn><mo id="S2.SS1.p1.1.m1.1.1.3.1" xref="S2.SS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mover id="S2.SS1.p1.1.m1.1.1.3.3.2" xref="S2.SS1.p1.1.m1.1.1.3.cmml"><mi mathvariant="normal" id="S2.SS1.p1.1.m1.1.1.3.3.2.2" xref="S2.SS1.p1.1.m1.1.1.3.3.1.cmml">.</mi><mo mathsize="142%" stretchy="false" id="S2.SS1.p1.1.m1.1.1.3.3.2.3" xref="S2.SS1.p1.1.m1.1.1.3.3.1.cmml">′</mo></mover><mo id="S2.SS1.p1.1.m1.1.1.3.1a" xref="S2.SS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mn id="S2.SS1.p1.1.m1.1.1.3.4" xref="S2.SS1.p1.1.m1.1.1.3.4.cmml">5</mn></mrow></mrow></math>, <math><mrow id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mrow id="S2.SS1.p1.2.m2.1.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.2.m2.1.1.1.1.2" xref="S2.SS1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p1.2.m2.1.1.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.1.1.1.1.1.2.cmml">m</mi><mo id="S2.SS1.p1.2.m2.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.1.1.1.cmml">-</mo><mi id="S2.SS1.p1.2.m2.1.1.1.1.1.3" xref="S2.SS1.p1.2.m2.1.1.1.1.1.3.cmml">M</mi></mrow><mo stretchy="false" id="S2.SS1.p1.2.m2.1.1.1.1.3" xref="S2.SS1.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">=</mo><mn id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">31.12</mn></mrow></math>, <math><mrow id="S2.E1.m1.2.2.1" xref="S2.E1.m1.2.2.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1" xref="S2.E1.m1.2.2.1.1.cmml"><msub id="S2.E1.m1.2.2.1.1.3" xref="S2.E1.m1.2.2.1.1.3.cmml"><mi id="S2.E1.m1.2.2.1.1.3.2" xref="S2.E1.m1.2.2.1.1.3.2.cmml">m</mi><mrow id="S2.E1.m1.2.2.1.1.3.3" xref="S2.E1.m1.2.2.1.1.3.3.cmml"><mi mathsize="71%" id="S2.E1.m1.2.2.1.1.3.3.2" xref="S2.E1.m1.2.2.1.1.3.3.2.cmml">S</mi><mo id="S2.E1.m1.2.2.1.1.3.3.1" xref="S2.E1.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.E1.m1.2.2.1.1.3.3.3" xref="S2.E1.m1.2.2.1.1.3.3.3.cmml">T</mi><mo id="S2.E1.m1.2.2.1.1.3.3.1a" xref="S2.E1.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.E1.m1.2.2.1.1.3.3.4" xref="S2.E1.m1.2.2.1.1.3.3.4.cmml">M</mi><mo id="S2.E1.m1.2.2.1.1.3.3.1b" xref="S2.E1.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.E1.m1.2.2.1.1.3.3.5" xref="S2.E1.m1.2.2.1.1.3.3.5.cmml">A</mi><mo id="S2.E1.m1.2.2.1.1.3.3.1c" xref="S2.E1.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.E1.m1.2.2.1.1.3.3.6" xref="S2.E1.m1.2.2.1.1.3.3.6.cmml">G</mi></mrow></msub><mo id="S2.E1.m1.2.2.1.1.2" xref="S2.E1.m1.2.2.1.1.2.cmml">=</mo><mrow id="S2.E1.m1.2.2.1.1.1" xref="S2.E1.m1.2.2.1.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.cmml"><mo id="S2.E1.m1.2.2.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.2.cmml">-</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml"><mn id="S2.E1.m1.2.2.1.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.1.3.cmml">2.5</mn><mo id="S2.E1.m1.2.2.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.2.cmml"><msub id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.E1.m1.2.2.1.1.1.1.1.1.1a" xref="S2.E1.m1.2.2.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.2.cmml"><mo maxsize="210%" minsize="210%" id="S2.E1.m1.2.2.1.1.1.1.1.1.1.2.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml"><mfrac id="S2.E1.m1.1.1a" xref="S2.E1.m1.1.1.cmml"><mrow id="S2.E1.m1.1.1.2" xref="S2.E1.m1.1.1.2.cmml"><mi id="S2.E1.m1.1.1.2.2" xref="S2.E1.m1.1.1.2.2.cmml">counts</mi><mo id="S2.E1.m1.1.1.2.1" xref="S2.E1.m1.1.1.2.1.cmml">×</mo><mi id="S2.E1.m1.1.1.2.3" xref="S2.E1.m1.1.1.2.3.cmml">PHOTFLAM</mi></mrow><mi id="S2.E1.m1.1.1.3" xref="S2.E1.m1.1.1.3.cmml">EXPTIME</mi></mfrac></mstyle><mo maxsize="210%" minsize="210%" id="S2.E1.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E1.m1.2.2.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.2.cmml">-</mo><mn id="S2.E1.m1.2.2.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.3.cmml">21.10</mn></mrow></mrow><mo id="S2.E1.m1.2.2.1.2" xref="S2.E1.m1.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS3.p6.2.m2.1.1" xref="S2.SS3.p6.2.m2.1.1.cmml"><mrow id="S2.SS3.p6.2.m2.1.1.1" xref="S2.SS3.p6.2.m2.1.1.1.cmml"><mi id="S2.SS3.p6.2.m2.1.1.1.3" xref="S2.SS3.p6.2.m2.1.1.1.3.cmml">E</mi><mo id="S2.SS3.p6.2.m2.1.1.1.2" xref="S2.SS3.p6.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.p6.2.m2.1.1.1.1.1" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS3.p6.2.m2.1.1.1.1.1.2" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p6.2.m2.1.1.1.1.1.1" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS3.p6.2.m2.1.1.1.1.1.1.2" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.2.cmml">B</mi><mo id="S2.SS3.p6.2.m2.1.1.1.1.1.1.1" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.1.cmml">-</mo><mi id="S2.SS3.p6.2.m2.1.1.1.1.1.1.3" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.3.cmml">V</mi></mrow><mo stretchy="false" id="S2.SS3.p6.2.m2.1.1.1.1.1.3" xref="S2.SS3.p6.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p6.2.m2.1.1.2" xref="S2.SS3.p6.2.m2.1.1.2.cmml">=</mo><mn id="S2.SS3.p6.2.m2.1.1.3" xref="S2.SS3.p6.2.m2.1.1.3.cmml">0.022</mn></mrow></math>, <math><mrow id="S2.SS3.p6.3.m3.1.1" xref="S2.SS3.p6.3.m3.1.1.cmml"><msub id="S2.SS3.p6.3.m3.1.1.2" xref="S2.SS3.p6.3.m3.1.1.2.cmml"><mi id="S2.SS3.p6.3.m3.1.1.2.2" xref="S2.SS3.p6.3.m3.1.1.2.2.cmml">A</mi><mrow id="S2.SS3.p6.3.m3.1.1.2.3" xref="S2.SS3.p6.3.m3.1.1.2.3.cmml"><mi mathsize="71%" id="S2.SS3.p6.3.m3.1.1.2.3.2" xref="S2.SS3.p6.3.m3.1.1.2.3.2.cmml">F</mi><mo id="S2.SS3.p6.3.m3.1.1.2.3.1" xref="S2.SS3.p6.3.m3.1.1.2.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.SS3.p6.3.m3.1.1.2.3.3" xref="S2.SS3.p6.3.m3.1.1.2.3.3.cmml">U</mi><mo id="S2.SS3.p6.3.m3.1.1.2.3.1a" xref="S2.SS3.p6.3.m3.1.1.2.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.SS3.p6.3.m3.1.1.2.3.4" xref="S2.SS3.p6.3.m3.1.1.2.3.4.cmml">V</mi></mrow></msub><mo id="S2.SS3.p6.3.m3.1.1.1" xref="S2.SS3.p6.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS3.p6.3.m3.1.1.3" xref="S2.SS3.p6.3.m3.1.1.3.cmml">0.183</mn></mrow></math>, <math><mrow id="S2.SS3.p6.4.m4.1.1" xref="S2.SS3.p6.4.m4.1.1.cmml"><msub id="S2.SS3.p6.4.m4.1.1.2" xref="S2.SS3.p6.4.m4.1.1.2.cmml"><mi id="S2.SS3.p6.4.m4.1.1.2.2" xref="S2.SS3.p6.4.m4.1.1.2.2.cmml">A</mi><mrow id="S2.SS3.p6.4.m4.1.1.2.3" xref="S2.SS3.p6.4.m4.1.1.2.3.cmml"><mi mathsize="71%" id="S2.SS3.p6.4.m4.1.1.2.3.2" xref="S2.SS3.p6.4.m4.1.1.2.3.2.cmml">N</mi><mo id="S2.SS3.p6.4.m4.1.1.2.3.1" xref="S2.SS3.p6.4.m4.1.1.2.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.SS3.p6.4.m4.1.1.2.3.3" xref="S2.SS3.p6.4.m4.1.1.2.3.3.cmml">U</mi><mo id="S2.SS3.p6.4.m4.1.1.2.3.1a" xref="S2.SS3.p6.4.m4.1.1.2.3.1.cmml">⁢</mo><mi mathsize="71%" id="S2.SS3.p6.4.m4.1.1.2.3.4" xref="S2.SS3.p6.4.m4.1.1.2.3.4.cmml">V</mi></mrow></msub><mo id="S2.SS3.p6.4.m4.1.1.1" xref="S2.SS3.p6.4.m4.1.1.1.cmml">=</mo><mn id="S2.SS3.p6.4.m4.1.1.3" xref="S2.SS3.p6.4.m4.1.1.3.cmml">0.178</mn></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0109048

Formulas:

1-

0.5 μ m \leq λ \leq 3

2-

L_{⋆} \approx 0.2 L_{⊙}

3-

M_{disk} \approx 1.5 \times 10^{- 3} M_{⊙}

4-

L_{acc} ≲ 0.2 L_{⋆}

5-

\dot{M} ≲ 4 \times 10^{- 9} M_{⊙} {yr}^{- 1}

6-

M_{disk} \sim 0.5 M_{⊙}

7-

M_{disk} ≳ 10^{- 4} M_{⊙}

8-

τ_{e q} \propto κ_{λ} M_{disk}

9-

M_{disk} \approx 10^{- 4} M_{⊙}

10-

M_{disk} = 0.065 M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2005.11208

Formulas:

1-

\log (ℒ / ℒ_{0})

2-

p o i n t l i k e

3-

g t l i k e

4-

| b | > 5 °

5-

| b | < 5 °

6-

2^{(n - 1)}

7-

p o i n t l i k e

8-

p o i n t l i k e

9-

p o i n t l i k e

10-

p o i n t l i k e

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0709.2684

Formulas:

1-

N + 1 \dots 2 N

2-

V_{i} (t)

3-

τ_{i}^{(m)} \frac{d V_{i} (t)}{d t} = - V_{i} (t) + \sum_{j} w_{i j} r_{j} (t - τ_{i j}) + I_{i} (t) .

4-

r_{j} = S_{j} (V_{j}),

5-

I_{i} (t)

6-

τ_{i}^{(m)} = τ^{(m)}

7-

τ^{(m)} = 1

8-

τ_{i j} = τ

9-

S_{j} (V_{j}) = S (V_{j})

10-

i = 1, 2, \dots, N

Formulas (html):

Correct Categorie: physics

Guessed Categorie: q-bio

Result: incorrect

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/astro-ph/9707250

Formulas:

1-

λ λ 1031.9, 1037.6

2-

λ λ 1238.8, 1242.8

3-

λ λ 1548.2, 1550.8

4-

λ λ 1031.9, 1037.6

5-

λ λ 1238.8, 1242.8

6-

λ λ 1548.2, 1550.8

7-

λ λ 1031.9, 1037.6

8-

λ λ 1031.9, 1037.6

9-

λ λ 1031.9, 1037.6

10-

λ λ 1031.9, 1037.6

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/0906.4528

Formulas:

1-

| Ψ ⟩ = {[a | 0_{s} 0_{i} ⟩ + b | 1_{s} 1_{i} ⟩]}_{𝟏} \otimes {| 0_{s} 0_{i} ⟩}_{𝟐},

2-

{| a |}^{2} + {| b |}^{2} = 1

3-

{| 0 ⟩, | 1 ⟩}

4-

| Ψ^{'} ⟩ = a {| 0_{s} 0_{i} ⟩}_{𝟏} {| 0_{s} ⟩}_{𝟐} + b {| 1_{s} 1_{i} ⟩}_{𝟏} {| 1_{s} ⟩}_{𝟐} .

5-

{(| 0_{s} ⟩ + | 1_{s} ⟩)}_{𝟏} / \sqrt{2}

6-

| Ψ_{HES} ⟩ = a {| 0_{s} ⟩}_{𝟐} {| 0_{i} ⟩}_{𝟏} + b {| 1_{s} ⟩}_{𝟐} {| 1_{i} ⟩}_{𝟏} .

7-

| Ψ ⟩ = [a {| H ⟩}_{s} {| H ⟩}_{i} + b {| V ⟩}_{s} {| V ⟩}_{i}] \otimes [c {| F ⟩}_{s} {| F ⟩}_{i} + d {| A ⟩}_{s} {| A ⟩}_{i}],

8-

{| a |}^{2} + {| b |}^{2} = 1

9-

{| c |}^{2} + {| d |}^{2} = 1

10-

{| H ⟩}_{s} {| F ⟩}_{s}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1305.0893

Formulas:

1-

{1, 2, \dots, n}

2-

{1^{σ (1)}, 2^{σ (2)}, \dots, n^{σ (n)}}

3-

{a_{1}, a_{2}, \dots, a_{n}}

4-

i = 1, 2, \dots, n

5-

j_{i} \in {1, 2, \dots, n}

6-

{a_{1}, a_{2}, \dots, a_{n}}

7-

{1, 2, \dots, n}

8-

ℙ := {2, 3, 5, \dots}

9-

2 s + 1 \in ℙ

10-

υ_{p} (m)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

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: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1803.00161

Formulas:

1-

n = {(a_{k} a_{k - 1} \dots a_{0})}_{b}

2-

a_{k - i} = a_{i}

3-

i = 0, 1, 2, \dots, k

4-

s_{b} < s_{b^{'}}

5-

2 \leq b < b^{'}

6-

a, b, m, n, k, ℓ

7-

n = {(a_{k} a_{k - 1} \dots a_{0})}_{10}

8-

a_{i} = a_{k - i}

9-

0 \leq i \leq ⌊ \frac{k}{2} ⌋

10-

1, 2, 3, \dots, 9

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0701264

Formulas:

1-

η (λ) = - \lg (\frac{P_{sig} (λ)}{P_{ref} (λ)}) \approx \frac{P_{abs} (λ)}{\ln (10) P_{ref} (λ)},

2-

P_{sig} (λ)

3-

P_{ref} (λ)

4-

[λ, λ + Δ λ]

5-

P_{abs} (λ) = P_{ref} (λ) - P_{sig} (λ) ≪ P_{ref} (λ)

6-

η_{free} (λ) = θ σ (λ) / \ln (10),

7-

1 / \cos (ϕ)

8-

θ_{\min} \approx \ln (10) η_{\min} (λ) / σ (λ) .

9-

σ (λ) / A_{eff}

10-

A_{eff} (λ) = P_{ref} (λ) / I_{surf} (λ) .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1209.2223

Formulas:

1-

Q_{N_{0}, Z_{0}}

2-

\sum \prod \frac{ω_{N, Z}^{n_{N, Z}}}{n_{N, Z}!}

3-

N_{0} = \sum N \times n_{N, Z}

4-

Z_{0} = \sum Z \times n_{N, Z}

5-

Q_{N_{0}, Z_{0}}

6-

{⟨ n_{N, Z} ⟩}_{c}

7-

ω_{N, Z} \frac{Q_{N_{0} - N, Z_{0} - Z}}{Q_{N_{0}, Z_{0}}}

8-

Q_{N_{0}, Z_{0}}

9-

μ_{n} N + μ_{p} Z

10-

{⟨ n_{N, Z} ⟩}_{g c}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/2005.12318

Formulas:

1-

A = {A_{0}, A_{1} \dots A_{t}}

2-

l^{p} = {l_{0}^{p}, l_{1}^{p} \dots l_{t}^{p}}

3-

l_{t}^{p} \in ℝ^{68 \times 2}

4-

a = {a_{0}, a_{1}, \dots a_{t}}

5-

a_{t} \in ℝ^{16 \times 29}

6-

A = {A_{0}, A_{1} \dots A_{t}}

7-

l^{m} = {l_{0}^{m}, l_{1}^{m} \dots l_{t}^{m}};

8-

l_{t}^{m} \in ℝ^{68 \times 2}

9-

δ l^{m} = {δ l_{0}^{m}, δ l_{1}^{m} \dots δ l_{t}^{m}}

10-

δ l_{t}^{m} \in ℝ^{68 \times 2}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1311.5490

Formulas:

1-

N_{f} = 2 + 1 + 1

2-

N_{f} = 2 + 1 + 1

3-

η, η^{'} \to γ γ

4-

U_{A} (1)

5-

N_{f} = 2 + 1 + 1

6-

D 45.32 s c

7-

r_{0} = 0.45 (2) fm

8-

𝒞 {(t)}_{q q^{'}} = ⟨ 𝒪_{q} (t^{'} + t) 𝒪_{q^{'}} (t^{'}) ⟩, q, q^{'} \in l, s, c,

9-

𝒪_{l} = (\bar{u} i γ_{5} u + \bar{d} i γ_{5} d) / \sqrt{2}

10-

𝒪_{s} = \bar{s} i γ_{5} s

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/0708.2918

Formulas:

1-

19^{h} 38^{m} 41^{s}

2-

54^{\circ} 58^{'} 26^{''}

3-

0 \overset{''}{.} 02512 (56)

4-

7 \overset{d}{.} 6407568 (6)

5-

87.0 {(1)}^{\circ}

6-

1.346 (23) R_{⊙}

7-

1.323 (23) R_{⊙}

8-

v s i n i

9-

7 \overset{d}{.} 6407568 \pm 6 \times 10^{- 7}

10-

1.346 \pm 0.023 R_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Run 11274858 (Agent925)